S.5 TRIGONOMETRY LESSON 2 : SPECIAL ANGLES | REFERENCE ANGLES ||60 ,30 & 45 degrees

Ajouté : 2026-05-09

[00:00:00]that you have given us to share in your presence. Father, may we understand you whatever that we are going to learn today. Loving father, thank you Jesus for enabled us to wake up loving father to attend this lesson in the same way.

[00:00:14]Loving father, may you give us the brains to understand in the might and believe.

[00:00:21]>> Amen. Uh thank you so much uh for the word of prayer.

[00:00:26]uh we are going to look at trigonometry and as we start the session uh yesterday we started on this topic and I want us to have a recap so I'm going to ask you in the chat to quickly type for me uh the trigon what the the meaning of these trigonometric ratios when I write cosec of x tell me in the chat quickly Just type what does cosec mean cosec what is it relation type in the chat what you think cosec x is equal to I want it equivalency in terms of the non sin cos tan when we put cosec x what is it equivalent to I don't know the members have got a question so I want you to tell me that cosec is the same as one out of sin x. Thank you Dorothy and Danella. Good members. What about sec of x? What is se x equaling to?

[00:01:38]What about sec? What? Aha. Thank you Dorothy. Aha. So sec means one out of co of x. Good. Uhhuh. What about cot of eggs?

[00:01:51]What about cot of eggs?

[00:01:56]Uhhuh. Aha. Cot is one out of the 10 of X. Very good. Yes. Now as we go on with the topic of trigonometry, make sure that these ones at your fingertips and also when we write code X, what about C in terms of S and co? How can you write it in terms of S and co when you have C?

[00:02:19]Uhhuh. Thank you D. Uhhuh. I said is the same as cos out of sin of x. Perfect.

[00:02:28]Now make sure that as we go on with this topic some of those things at your fingertips. The moment someone talks about cosec you know that is saying sec that is co that is about what turn and that's where now we are going to build up going forward. So allow me now to tell you that today we are going to look at what we call trigonometric ratios of those special angles. We have what call special angles and some of those special angles we are going to look at. We are going to first look at the first type and here we are going to have what call a right angled isoclesles triangle. That is what we are going to look at. And after looking at that right angled isocles triangle, we are going to go on and look at another one. We know about an isocles triangle. And we know that an isocles triangle will always have two sides that are equal. So I'm going to to draw my triangle. I'm going to let me see whether I will be able to come up with a good Uhhuh.

[00:03:43]Uhhuh.

[00:03:45]Yeah. So this is my triangle. And yesterday in our discussion we agreed that whenever we are coming up with with this we need always to know where the hypotenuse is. So if this is our 90° what are we going to do? A right angled isoc is whose two equal sides are of length unit one. So here members and this side are going to be equal and their length is one and one.

[00:04:18]Now what am I going to find? I'm going to be able to obtain what is my hypotenuse. Members, can we find the hypotenuse using your knowledge of Python?

[00:04:44]Uhhuh. Everyone is trying out to see what we are coming up with our hypotenuse.

[00:04:52]Uhhuh. So don't forget we shall use our knowledge of a² + b²= c². So you get for me don'thu da is getting square <unk>2.

[00:05:03]Good. Uhhuh. Are we getting there? So we shall say 1 2 + 1 2 = c 2 1 2 is 1 + 1 which is = c². So we shall have 2 = c 2.

[00:05:19]So in order for us to get C it means we shall get the root on both the sides. So we shall have our C equals to roo<unk> of 2. So here we shall have root of two.

[00:05:30]Now the next question I'm going to ask members what do you think our angle is?

[00:05:37]This angle is inside. Which angle do you think is inside there?

[00:05:42]Which angle do you think is inside there? Uhhuh. Aon you are correct. Amara you are correct. Now my question is which angle this is an isosles triangle.

[00:05:53]Which angle do you think is inside there?

[00:06:05]Uhhuh. Uhhuh. Ben is saying 30. Uhhuh.

[00:06:08]And here remember I want you always to remember some of you know you are used to search when they talk when two sides are equal it means that the base angles are also equal. So this means it is a it is a. So the same thing here it means this is a and this is a. So you add up a + a + 90° should give us 180. So members can you find out the angle inside now?

[00:06:37]Aha, thank you Danielle. 45 45 D 45 and Emma. Very good.

[00:06:45][clears throat] So when you work out the mathematics here, you have 2 A + 90 = 180. It means that 2 A is 180 - 90. 2 A will = 90 / 2 / 2 and that angle is 45°.

[00:07:04]So that's how members are getting the 45. So inside there we are going to have 45°.

[00:07:11]So you are going to allow me that inside here no matter the position we shall have 45.

[00:07:19]So here you have 45°.

[00:07:21]Now when you look at our triangle like we say that the side that faces the 90 is hypotenuse. So this is going to be our hypotenuse. The side that sits this is going to be adjacent and this is going to be our opposite basing on the angle 45. Now whenever they talk about special angles one of the special angles is 45° this triang this one we have now like we said we have what trigonometric ratios and the first one is going to be our sign and which sign are we looking for for 45. So I'll say sine of 45 but do not forget from your so to that sign is given by opposite out of the hypotenuse. So members our s of 45 will equal to opposite which is one out of hypotenuse which is roo<unk> of 2.

[00:08:20]Now if you want you can now rationalize such that you don't have a root. So in case you want to rationalize you say roo<unk> of 2 *<unk> 2 * roo<unk> of 2 and therefore you remain with roo<unk> of 2 out of roo<unk> of 4 which will give us roo<unk> of 2 out of two and members the moment you place s of 45 you'll end up with this yeah on your now what is your task members can we go on and find what is of 45.

[00:08:57]Can you go on and find what is tan of 45? Members, can you go on and find what is of 45?

[00:09:06]When you get the answer, feel free to raise up your hand and you let us know what you are getting.

[00:09:14]Uh Kh, you can Okay. Uhhuh. I'm going to start from Ian.

[00:09:27]>> Uh cos 45 is uh roo<unk> of 2 / 2.

[00:09:33]>> Root of 2 out of two. Okay. Thank you, Ian. Uhhuh. Any other who's getting another one? 10 45. What is quot of 45?

[00:09:48]Uh turn of 45 M is telling us is getting one. That is from M. Uhhuh. Uhhuh. Yes, Daniel.

[00:10:02]>> I'm getting one as well.

[00:10:05]>> Okay. Uhhuh. What about court of 45 also one >> also you are coming up with one good members what about co of 45 get also for coc Sec.

[00:10:47]Uh-huh. You are getting root two. Okay.

[00:10:50]What about Thank you, Daniel. other members. What are you getting for? Oh, Aaron, never write decimals.

[00:11:00]Never write decimals. The reason why we call them special angles, it is because you can get an exact value.

[00:11:09]One of the angles where you can get an exact value is 45 for whatever trigonometric ratio you are dealing with. So here we don't write decimal points but we write that exact value you obtain.

[00:11:24]Okay AC >> uh teacher I have a concern on co of five cos of 45.

[00:11:33]>> Uhhuh.

[00:11:35]uh cos of 45 according to it says co of 45= to adjacent over hypotenuse >> of this >> you have said adjacent uh >> over hypotenuse >> which is what is your adjacent >> adjacent is one >> out is root of two >> okayh but I see you had written as roo<unk> of 2 / 2.

[00:12:05]>> Oh, now a why it is like this. Your answer is also correct. But because someone [clears throat] has rationalized >> so when you this answer you get when you multiply by roo<unk> of two and roo<unk> of two down that is the rationalizing.

[00:12:20]So 1 * <unk>2 is roo<unk> of 2 <unk>2 * roo<unk>2 is roo<unk> of 4 and roo<unk> of 4 is two.

[00:12:28]>> Okay. Okay please. Okay please.

[00:12:31]>> Okay. Uh me.

[00:12:34]>> Okay.

[00:12:35]45. What are you getting?

[00:12:38]Someone is getting one.

[00:12:45]>> Mhm.

[00:12:47]Uh do we use the quot X to calculate C of 45? Yes. Uhhuh. Yes, Melissa. The moment you see quot like we said quot means cos of 45 out of sine of 45. Alternatively you know that quot of 45 like we said is the same as 1 out of 10 of 45. So you can use the knowledge of any of the two.

[00:13:18]Yes, hope it is clear. But since we had already obtained 45 as one so that's why we we can just conclude or use that oneh hope m that is okay people are saying cosec is one okay we know that what is cosec because that one we need to be very careful we know that cosec of 45 is the same as 1 out of sin of 45 but what did we get for sin 45 when you come here s of 45 was something like this. So members it cannot be one. It cannot be one. Can we get it again? I don't know how members are getting one for co sake.

[00:14:02]Use the relation very well. Something is not okay. Root of two. Perfect. Yes it is root of two because uh when you check here check check are you prec? Yes. Yes, I agree. Now members have started. So remember cosec 45 is the same as 1 out of sine of 45.

[00:14:31]Now when you substitute s of 45 here you have 1 out of s of 45. So s of 45 you know you get one out of root of two and members this will give you root of two.

[00:14:43]Alternatively when you are dealing with a cosec you can know that cosec can be round I hope to not confuse you cosec now for it it will be hypotenuse out of the opposite in other words so it will be root of two out of one which is root of two okay members in case you have a question ask now yes mi Excuse me sir.

[00:15:16]>> Yes.

[00:15:18]>> On this part of co 45 why are we using one out of <unk>2 yet we had gotten roo<unk>2 out of two as our last answer of sin 45.

[00:15:33]>> Anything we we shall come up with the same. M you stay on must you have roo<unk> of 2 out of two not so >> which is the same as 1 / roo<unk> of 2 out of two true >> yes >> which is the same as 1 * 2 out of roo<unk> of 2 >> yes >> so 2 out of root of two remember you don't want to write a root in your denominator so you rationalize roo<unk> of 2 out of root of two so 2 * roo<unk> of 2 will give you 2 roo<unk> of 2 roo<unk> of 2 * roo<unk> of 2 gives you roo<unk> of 4 which is 2. So the two and the two cancel so you remain with root of two.

[00:16:14]>> Okay. Thank you sir.

[00:16:16]>> Okay. Uh Genesis >> good morning sir.

[00:16:23]>> Yes good morning.

[00:16:26]So I'm just inquiring like I'm seeing here of 45 we are having co of 45 out of sign of 45.

[00:16:35]>> I'm requesting for a pardon on that. I don't get that very well.

[00:16:40]>> Oh now this one comes from the definition. Did you attend yesterday's lesson?

[00:16:46]>> A half of it.

[00:16:48]>> Uhhuh. So we will go back and review because we looked at we said that by definition when I have any angle is the same as co out of s.

[00:17:01]So this is from definition of what is because you know that of theta is the same as 1 out of tan of any angle but you know that tan is the same as sin theta of cos of theta. So when you do the mathematics 1 / sin theta of cos of theta you end up with 1 * cos of theta out of s of theta. So this is where it comes from.

[00:17:30]>> Okay. Thank you sir.

[00:17:32]>> Okay members let me know through the chat whether we are good to continue.

[00:17:37]Should we continue?

[00:17:42]Canita says yes. Are we on the same Okay. Okay. Good. Now, whatever thing we have obtained, we are going to use.

[00:17:54]>> Now, the second one. So, one of the special angles we have looked at is 45°.

[00:18:01]Now, we are going to look at another special angle. Now, to get other special angles, we are going to use what we call an equal triangle. And you know that equality triangle all the sides of an equal triangle are equal. So this side this side and this side are all equal.

[00:18:20]Members in the chart if all the sides are equal it means that also all the angles are equal. Which angles do we have inside triangle?

[00:18:32]>> 30.

[00:18:34]>> Not yet. Not yet. Not yet.

[00:18:37]>> 45.

[00:18:38]>> 45. Not yet. Not yet. Not yet.

[00:18:42]>> Not yet. Not yet. Not. Remember, when you add up all the angles, you must be able to get 180.

[00:18:48]>> The angles that are in this triangle.

[00:18:56]>> Ah. So from the chat people saying very good. Remember if this angle is a this one will also be a and this one is going to be a. So that means a + a + a is equal to 18. So we shall have 3 a = 180ide by 3x3 and your a will equal to 60°. So that's how we come up with our 60. So meaning that all angles when we have an equal triangle the moment you equal it means that all sides are equal and all angles are going to be equal.

[00:19:35]Now I'm going to come up with my with my my my my triangle. So I have my triangle here whereby all the sides are equal. My sketch may not be the best but uh for you must come up with a better one.

[00:19:54]Come up with a better one. So I'm trying to see how I can come up with a better sketch.

[00:20:02]Ah that's what I've managed to do. So since we have said all sides are equal it means that also all the angles we have are going to be equal. So this is 60° this is 60 and 60. But what have we said? We have said that if all the sides are equal here we are using two units.

[00:20:23]So that means that here we shall have two we shall have two and also here we are going to have two. Now inside an equal triangle we can divide it into two equal parts. So when I divide here I have divided into equal parts. Meaning that this part here and this part here are going to be equal members. What is going to be the length from this part up to here? What will be our length? Put in the chat quickly. What is the length?

[00:20:56]Ah people are saying it is one. Perfect.

[00:21:00]So that is one members. Now we look at the angle. Thank you members. We look at the angle. Tell me which angle shall we have in this area here. Here up up.

[00:21:11]Uhhuh. Shami says 30 30 30.

[00:21:16]Very good. So the angle there is going to be 30 because initially you had 60°.

[00:21:22]When you divide it into two you get 30.

[00:21:25]60 / 2. Now you're going to allow me.

[00:21:28]Now I'm going to extract one right angled triangle.

[00:21:35]I'm going to extract this right angled triangle of this nature.

[00:21:41]And what am I going to have here? I'm going to have my 60° here. Remember as you are seeing I have two here I have one and here I have 30° and here you have your 90°. Now what is the task? The task members is to help me find out what is the distance here. What is the length of our height? Let's use our knowledge of pythograph theorem and find out what is the height. Calculate calculate. Work it out. Work it out.

[00:22:16]Uhhuh.

[00:22:23]Okay. I'm seeing some answers coming in.

[00:22:26]Someone can quickly take us through.

[00:22:29]Mhm. Make sure you work it out. Don't guess.

[00:22:37]Uhhuh. Yes. Amara.

[00:22:40]>> Yes. So, uh, teacher, how the answer came about?

[00:22:45]>> What I needed to do? We're supposed to use the method of pyth= c².

[00:22:54]>> Okay.

[00:22:58]>> After that you continue and you replace.

[00:23:00]So the a² would be 1 >> + b² which no sorry a² would be 1 2 + b² = 2.

[00:23:14]Okay.

[00:23:23]Continue.

[00:23:30]So teacher after that.

[00:23:32]So it would be 1 2 + b ² is equal to 2.

[00:23:37][snorts] So as we continue you're going to get 1 + b ^ 2 = 4.

[00:23:54]So then it would be 4 - 1 that would be 3. So b² would equal to 3. So b after that you cancel out the square of b by putting a square root.

[00:24:07]Then you put a square root also on three. then b will equal to<unk> 3.

[00:24:12]>> Okay, perfect. Thank you Amara. So members here we are going to place our root of three. Now on our special angles we are going to add special angles are those angles where you can get the exact trigonometric ratio without getting points. You can get the exact value. So on your angles the first one was 45. So we are going to add on a 60 and we are going to add on a 30.

[00:24:45]Now we start cos of 60° what shall we have? Now if I'm dealing with co I want you always to you label it very well where the angle faces is opposite. This side is going to be the hypotenuse. This side is going to be adjacent. So what is going to be our cost of 60 members? Put it in the chat.

[00:25:11]What is your cost of 60?

[00:25:19]Uhhuh.

[00:25:21]Get people saying it's going to be a half.

[00:25:26]Thank you. Go on. Can we also get sign of 60? Mhm.

[00:25:34]Don't for We shouldn't leave you behind.

[00:26:15]Okay, loot three out of two. Very good members. In case we are leaving you Aron, I don't know what question you have. Alon, I'm not getting your question. I'm saying Aaron you may ask uh members we continue as Aaron you ask 10 of 60 A askhuhan saying you get moving continue.

[00:27:15]Okay. So 10 people are saying the answer is roo<unk>3. Very good. So we have been dealing with the cos 60. Now, what about allow me to remove these ones and I'm going to allow you to label for yourself. I'm not going to label for you, but you are going to to tell us members. I'm looking at 30. You tell me what is on our opposite for 30°?

[00:27:46]Which value is on the opposite for the 30°? Oh. Uhhuh. Very good. members are saying our opposite is going to be here.

[00:27:56]Members, what's going to be our hypotenuse? What is the on the hypotenuse of 30?

[00:28:03]Uhhuh. People are saying two. Very good.

[00:28:07]That's perfect. So members, can we now go on and find sin of 30, cos of 30 and tan of 30. So go on and find all those ones there. trigonometric ratios.

[00:29:20]Okay.

[00:29:28]Okay. Okay. I'm seeing the answers.

[00:29:31]Okay. Sign of 30. People getting a half co of 30 people getting root three out of two t of 30 uh people getting one out of root of three or people can rationalize and have root three out of three. Okay.

[00:30:01]How? Yes. Now remember tan of 30 is given by tan is given by toa opposite.

[00:30:10]So opposite is 1 and adjacent is roo<unk>3. So when you rationalize you'll have roo<unk> of 3 out of roo<unk> 3 1 * roo<unk>3 is roo<unk>3 roo<unk>3 * roo<unk>3 is roo<unk> of 9 which is 3. So that's how it comes about.

[00:30:30]10 is three out of two. How am check out?

[00:30:37]Okay.

[00:30:40]No, I don't which one I all of one out of root three and root3 out of three. It is the same thing. You get the same answer. So, both of them are correct and both of them are correct.

[00:30:55]After rationalizing yes, all of them are correct. But the best is the one where you don't have a root in the denominator. It becomes easy to work out with.

[00:31:06]Okay members let me know sign when you look at sign s of 30 s is given by opposite out of hypotenuse. So your opposite for 30 is one hypotenuse is two. So that's how we are getting it.

[00:31:30]Oh okay. Okay. But still am I don't know where the two is coming from.

[00:31:39]Okay. All answers are correct. All answers. I don't know whether fact something is confused. Kindly ask.

[00:32:00]Okay. I failed to get her team >> sir.

[00:32:10]>> Yes.

[00:32:11]>> But the opposite I'm saying it's roo<unk>3 and here for the sign 30 we have >> which angle which angle are you looking at? for the sign 30.

[00:32:22]>> Uhhuh. Which angle is facing 30? Look very well.

[00:32:26]>> It's 60 >> here. No, you look at the length.

[00:32:33]>> Okay.

[00:32:34]>> It's one.

[00:32:35]>> Yeah.

[00:32:38]>> Okay. Thank you, sir.

[00:32:41]Pasco.

[00:32:47]Okay. Members, are we on the same page?

[00:32:49]Let me know through the chat such that we can continue.

[00:33:09]Okay. Okay. I failed to get her but from the chat members are saying we are on the same page. We are on the same page.

[00:33:20]So we are going to continue. Now all what we have looked at members we are going to apply. We are going to apply them. Now where are you going to apply?

[00:33:31]You are going to be working out some numbers. You'll meet scenarios and they will tell you that give your answer in an exact form.

[00:33:39]When they tell you to give it an exact form, you'll notice that sometimes there are some angles that are special that have been given to you. Meaning that your answer you leave it in the sad form.

[00:33:52]So they don't want you to write in points.

[00:33:55]So we are going to apply all what we have looked at. So allow me that we are going to start from here. So what am I going to do? I'm going to draw like I've said that always I want you to understand the concepts. Don't cram because when you cramp you may mess up.

[00:34:13]So here we have two angles. We have 30 and 45. So for 45 we use our isocles triangle. So I have 45. I know I have one. I have one. I have root of two.

[00:34:27]Then since I have 30, you use the isocles one. So you still come up with your simple sketch that this is my isosles whereby I have here two I have one I have root of three and here I have my 30. Now when they tell us to find exact value they have told us to find exact value of sin of 30 and cos of 45.

[00:34:52]What does it mean? Substitute in the way you have them. So sine of 30 I'll come here and I say sine of 30 is given by opposite out of hypotenuse cos of 45 here you say cos of 45 is given by adjacent out of hypotenuse so you substitute in these values so sin 30 I'm going to put a half time cos of 45 I'll put 1 out of roo<unk> of 2 so mathematically 1 * 1 is 1 out 2 * roo<unk> of 2 which is 2<unk> 2. So you can continue from there. If you want to continue from here, you can now go on and rationalize. So when you rationalize, you'll have 1 out of 2<unk> 2 *<unk> 2 * roo<unk> of 2. So what shall we have? 1 * roo<unk> of 2 is roo<unk> of 2. Out of here we shall have roo<unk> of 2 * roo<unk> of 2 which is roo<unk> of 4<unk> of 4 which is 2 * 2 which is 4. So you get an answer of that nature.

[00:35:58]So you can end here or here. So that is how you express it in an exacting form.

[00:36:08]Where are you getting the root of two?

[00:36:12]No members. Now remember whatever now we are doing we are applying from the first from the start. The first thing we looked at, you go back and look at your at what we started with. When this is one and this is one to get hypotenuse, use your Pythograph theorem.

[00:36:32]We are using those two triangles we looked at at first. Go back and look at the first diagram we had.

[00:36:40]Let me hope that is okay.

[00:36:45]The root of two sunk. Look at the first right angle triangle we looked at.

[00:36:50]That's why we are getting how did you get triangle for co of 45? It came from the start.

[00:37:01]Okay.

[00:37:03]And what about two? Still we are applying anything we are doing now we are applying from the previous knowledge. Remember it is a continuation.

[00:37:13]Uhhuh. Yes. Get >> good morning sir.

[00:37:22]>> Yes, good morning >> sir. This part of sin 30 >> it's of course sign is opposite that of hypotenuse but here I see the hypotenuse you have root on the triangle hypotenuse is two.

[00:37:39]I come here.

[00:37:43]What is the opposite? This is the opposite and this is the hypotenuse.

[00:37:52]>> Oh, okay. Now get it. Here we were expressing we were now converting the co.

[00:37:58]>> Okay. Uhhuh. Joan.

[00:38:02]>> Okay.

[00:38:03]Joan.

[00:38:05]>> Yeah. Good morning, sir. Morning.

[00:38:10]>> My question was is it that we are going to use the this these angles we are using here. I mean these triangles we are using here in every question we are going to use such angles.

[00:38:25]>> Yes. Yes. That's why we we call them special.

[00:38:30]Wherever you are going to be finding 30 45 60 these triangles apply.

[00:38:39]>> So for you the moment you find 30 45 60 those three special angles always these triangles will apply.

[00:38:49]>> Okay.

[00:38:49]>> Okay.

[00:38:52]Now teacher I was asking that during that part where you have rationalized >> do we only rational do we only take the that number under the root sign and then we leave the constant the two or >> I thought to take both >> even if you take both there's no problem even if you do like that >> all Right?

[00:39:21]>> Instead you get 1 * 2<unk> 2 you get 2<unk> 2. 2 * 2 you get what? 4 <unk>2 * roo<unk> of two you get two. So you'll end up with this two and the two cancelling. So you get root of two out of four.

[00:39:36]>> Okay.

[00:39:39]>> Okay. Uh Joshua >> master for that part I'm not understanding why you're having a half * 1 out of <unk>2.

[00:39:50]>> We are substituting what sin 30 is and what cos 45 is sin 30 is the same as 1 out of 2. 45 is the same as 1 out of root of two. Oh, >> it was not. Okay.

[00:40:04]>> Yes.

[00:40:06]>> Thank you.

[00:40:07]>> Okay. Uhhuh. techno.

[00:40:15]Okay. Okay. I think uh we are going to continue. Members, whenever you find these special angles, the moment you see 45, you see 30, you see 60, run very fast to these triangles.

[00:40:29]We go to another one. So, allow me to clean my screen.

[00:40:35]And now we go to the next one. Now when you look here uh there are quite many but we are going to only do three or two. In fact we are going to do two of them.

[00:40:48]Uh can you quickly do for me part A everyone place your angles very well. Always come up with those triangles and with time you'll be able to get them in the head.

[00:41:02]So we are doing part A. I'll ask someone at a later time to raise up the hand and you take us through party A.

[00:41:11]Yes, Cyprien.

[00:41:18]Okay, Crian, I'm not getting you techno.

[00:41:28]Okay, I'm also not getting techno.

[00:41:31]Okay, someone who wants to take us through part A, feel free to raise up your hand.

[00:41:38]Okay, Joshua.

[00:41:42]>> Yeah. To take you through part A.

[00:41:45]>> Mhm.

[00:41:46]>> So, part A is sin 30= 60.

[00:41:49]>> Okay.

[00:41:50]>> So, sin 30 is is equal to opposite over hypotenuse. We using the second triangle. Okay.

[00:42:02]So that sin theta is equal to the opposite to the angle is 1 and the hypotenus is two. Okay.

[00:42:12]>> So therefore s is a half then cose 60 is adj.

[00:42:26]That means cos 60 is equal to the adjacent to 60 is 1.

[00:42:35]>> Uhhuh.

[00:42:37]>> And the hypotenus is 2.

[00:42:41]>> Okay.

[00:42:44]>> So now we multiply sin 30 * cos 60. So that is a half a* a half is 1.

[00:42:57]Come again. Come again.

[00:43:02]>> Okay. Okay. Thank you, Joshua.

[00:43:06]>> Okay. So, members, that's how you deal with that. Nancy, unmute.

[00:43:19]Nancy, unmute.

[00:43:24]Okay.

[00:43:25]Uh Daniela >> teacher.

[00:43:31]>> Yes.

[00:43:32]>> Good morning teacher.

[00:43:33]>> Morning >> teacher. I thought for 60 >> since the 60 was in the equilateral triangle.

[00:43:44]>> So we we consider the twos where we put on the sides. So I thought when you're getting the adjacent and the hypotenuse the answer will be 2 over two. Okay, you stay on. If this is your triangle, Yeah. And this is where >> when you look at at this triangle, what could be your adjacent on a >> B and C?

[00:44:16]>> Two.

[00:44:18]>> It will be it will be which side? I have A, B and C.

[00:44:23]A >> yeah it is true but don't forget we say that to get adjacent hypotenuse and opposite always we must deal with a right angled triangle this triangle is not a right angled triangle so there is nothing we can do about it I don't know that you're getting it >> so that's Why we have to cut out? So what we cut out is what forms our right angled triangle. So we can now confidently use it.

[00:45:01]>> Thank you teacher.

[00:45:02]>> Okay.

[00:45:06]Okay. Danella Dell, are you still on?

[00:45:17]Okay.

[00:45:19]I'm failing to get her. Can we have someone to take us through now? Part B.

[00:45:26][clears throat] >> Part B.

[00:45:31]Uh-huh. And it unmute.

[00:45:45]Oh, I'm failing to get Annette.

[00:45:49]Okay.

[00:45:50]Arajab uh sir good morning.

[00:45:58]>> Good morning.

[00:46:01]>> Uh I'm taking I'm going to take through party B.

[00:46:06]>> Mhm.

[00:46:09]>> Uh here we shall consider our triangle first triangle.

[00:46:16]>> Okay.

[00:46:19]by sin 45 mm will be equals to 1 over <unk>2.

[00:46:31]>> Okay.

[00:46:34]So now shall have 1. So sin² 45° will be 1 <unk>2 >> everything squared.

[00:46:49]Yeah. Everything squared.

[00:46:51]>> Mhm.

[00:46:54]>> So now there we shall have 1 1 / <unk>2 is the same as 1 / <unk>2.

[00:47:08]>> Okay.

[00:47:10]We shall have a half.

[00:47:13]>> Okay. Yeah. Thank you. Thank you. Yes.

[00:47:19]Thank you. Alison, any questions?

[00:47:25]>> No, I just wanted to take you through but the other one has taken.

[00:47:29]>> Okay, Alison, no worries. Alison, we are going to run together through party. Which part do people want in the in the chat members? Choose one part we are going to do, then the rest will be yours.

[00:47:45]Which part? Uhhuh.

[00:47:48]I'm going to take the first. Okay.

[00:47:51]People saying F. Good. So Alison, you we are going to go run through together part F.

[00:48:01]So Alison, yes. Now members they have told us that sin² of 30 + cos² of 30 uh divide by 2 sin 45 cos 45. So members we first rewrite that thing the first solve for the for the we but we are going to first use those things that are on the numerator and of which sin square of 30 and we know that sin 30 from our from our second triangle sin 30 from our second triangle sin is what?

[00:48:54]Opposite over hypotenuse and our opposite here is a one and our opposite is a two. So we say a half but everything squared plus now cos 30 we know that from our course >> cos 30 we also going to use triangle 2 and you know from our course is adjacent over hypotenuse and here our adjacent is a root theory over hypotenuse which is a two. So we say root the over hypotenuse which is 2 and everything here is squared.

[00:49:36]Okay. So we divide this by we divide this by 2 into so 2 into we have sin 40 sin 45. So sin 45 members are going to use triangle one. And we know that our sign is also opposite over hypotenuse. And our opposite here using triangle one. Our opposite is a one and our hypotenuse is a <unk>2. So we say 1 / <unk>2. [snorts] All right. So this one this bracket is multiplied by another bracket which is cos 45. And you know that cos 45 we use the first triangle cos 45 from our we adjacent over hypotenuse. So our adjacent here is a one and our hypotenuse here is a <unk>2.

[00:50:38]So our adjacent is a one and hypotenuse is al <unk>2.

[00:50:45]So members this two has to multiply these two brackets. So you can put a very big bracket on them showing that two multiplies everything after getting a product of those things that are of after getting a product of the two brackets. So [snorts] members from this we know that when we bring this square when we get the square a half squared when it is in a bracket But everything we getting 1.

[00:51:27]So solving for the numerator we get 1 / 4 + 1 / 4 out of So when you solve for this bracket multiplies this bracket you find out that we are going to get 2 into 1 over. So when you get<unk>2 * <unk>2 we know that we are going to get a roo<unk>4 and roo<unk>4 is a 2. So inside we get a half. So everything when is multiplied by two we know that we are going to get a quarter what do you get members?

[00:52:08]So a quarter plus a quarter we know that we shall get let me first add here a quarter plus a quarter.

[00:52:22]All right members I get a half. So if I get a half a quarter plus a quarter you get a half. This half is divided by so when you get when you multiply this two inside this fraction that is inside we get 2 over two and that two cancels with a two down. So our our denominator becomes a one and of which when it is a one half by a one we get our answer as a half.

[00:52:52]>> Okay. Uh there is something Alison you forgot here when it came to squaring. I don't know that you noticed something when you were squaring.

[00:53:04]>> All right.

[00:53:06]Yeah, I forgot I made a mistake there.

[00:53:09]So here I thank you sir.

[00:53:14]>> Okay.

[00:53:17]from our addition. Uh, when you add a quart [snorts] 3 out of 4, a quarter + 3 out of 4, we get a one.

[00:53:32]Okay. Thank you, sir.

[00:53:36]Okay.

[00:53:39]Okay. So, members, are we on the same page? Anyone that is lost?

[00:53:53]Okay.

[00:53:55]Danila, anyone that is lost members, let me know through the chat whether we are good to continue.

[00:54:02]teacher on my 2 sin 45° when I got the opposite over hypoten 45° then I multip Yes.

[00:54:33]>> Yes teacher.

[00:54:34]>> H. So roo<unk>2 * roo<unk>2 da what do you get?

[00:54:39]>> Shall you get two?

[00:54:41]>> Uhhuh. So by 21 by 21.

[00:54:47]>> Oh okay. Thank you teacher.

[00:54:49]>> Okay. Lastly Shadia.

[00:54:57]Okay. So members uh for the rest of the numbers take a screenshot. You'll respond to them uh uh you'll respond to them at a later time. Okay. So Shadia I failed to get to you but we are going to continue. So members that's how you deal with that.

[00:55:23]So uh now uh we are going to deal with what we call the general definition of angles. Now when it comes to trigonometry I want you not to do there are two things at all level you looked at trigonometry you looked at bearing many times you children usually confuse bearings and trigonometry but those are totally different things. Now when it comes to trigonometry, when it comes to angles, we measure angles from the positive x axis. This is your x axis and this is your y ais. Now when we are measuring angles, we shall always place our protractors in this format. That is how you place your protractor. What does it mean? It means that your zero your zero will start from here. Then you go to your 30, 40, uh, 80 and 90 up to 180.

[00:56:27]This is how we place our protractor. Now that means that your 0° is starts from here. And which direction do we take? We always move in this direction here. So all angles that we measure in the ant clockwise direction are positive angles.

[00:56:48]This is how we move the axis. So when you measuring the angle, the moment you get a positive angle, it means that that person is moving in the positive x ais.

[00:57:01]The moment someone writes 30°, it means the person is moving is starting to measure his angles from zero going in the antlockwise direction. But if someone writes 30° now negative shows that someone is measuring angles but in the clockwise direction that is when someone will come here you place your protractor in this format. So 30° is an angle here. It means that the direction you are taking is the clockwise direction. But always when we are measuring angles we start from the positive xy axis.

[00:57:43]Members have you understood that concept for angle measurement.

[00:58:00]Uhhuh. Okay. Now when you look at what I have, when you look at what I have, we have quadrants. Quadrants means quarter that something has been divided into four equal parts. Now this is what we call the first quadrant, second, third and fourth. Now I'm going to write my angles and you're going to use your estimation. Remember this is 0, this is 90, this is 180, 270 and then lastly we come back to 360. Now I'm going to write an angle and you are going to tell me the quadrant where that angle lies. Let me say when I write 45, you just tell me that is in the one. When I write - 45, you tell me that - 455 will lie in the fourth quadrant. That's what I'm saying.

[00:58:52]Uhhuh. So I am starting you type in the chart me I'm going to be looking there and I will say that now you have understood because that one is going to show me that now you know where angles you can estimate where angles are found in the different quadrants and now you can distinguish whether it is clockwise or whether it is anticlockwise.

[00:59:12]Now the first one the first one is is 95. Just put for me the quadrant 1 2 3 4. Mhm.

[00:59:21]Perfect. Perfect. Aha. You are very good. You are very good. Uhhuh. The next one is here.

[00:59:29]Uhhuh.

[00:59:33]Uhhuh. I've put for you another one.

[00:59:36]Uhhuh. Now here I'm testing. Have you understood? In case you find 130° here, we have not yet understood. I'm seeing answers are varying. Some are saying three, others are saying two. Uhhuh. Now the threes are many. Very good. Very good. Yes, members, we have said negative means you're moving in the clockwise direction. So you place your protractor like this. So you not that 132 is going to be somewhere here. So that angle is is here. So that angle will be in the third quadrant.

[01:00:13]Let me hope you have understood that the moment you find a negative on an angle it means that that angle was measured in the clockwise direction but from the positive x axis. I put another one. I put another one.

[01:00:30]Ah I have another one.

[01:00:53]Uhhuh. Uhhuh. Uhhuh. Very good. Very good. Very good. Very good. Now, someone is still putting another answer.

[01:01:04]Uh-huh. You need to know that this is where 270 is. So, you estimate.

[01:01:10]Okay. Okay. Now, if I don't get any person who fails this, I'll know that we have understood. Yes. Yes. Yes.

[01:01:23]If I don't get anyone who fails this, I know we have understood now.

[01:01:30]Uhhuh.

[01:01:34]I'm I'm starting to count from someone has put another answer.

[01:01:42]We are still having mixed reactions here. Someone has put a okay can someone I want someone now to just raise up the hand and you explain in your own words the way you have understood such that also others can pick uh eda >> okay good morning sir >> yes good morning ed >> so inveh >> if we limit ourselves to only 180 everyone is going to get confused.

[01:02:17]>> Mhm.

[01:02:17]>> So me I've understood it as all my angles if I want to make them if they are negative all my angles will start from zero where there's the x-axis.

[01:02:28]>> Okay.

[01:02:30]>> Then I'll be moving anticlockwise up to when I come back to 360. Of course where there is a zero there's a 360.

[01:02:37]>> Mhm.

[01:02:39]>> Yeah.

[01:02:39]>> Repeat that statement.

[01:02:41]where there is a zero.

[01:02:43]>> Mhm.

[01:02:44]>> On position there's 360.

[01:02:48]>> Of course, when you rotate around the whole quadrant.

[01:02:53]>> So me as though have understood if it is a negative, you're going to be moving in the clockwise direction up to when you reach 360.

[01:03:03]Uhhuh.

[01:03:05]>> So you find your 240 exceeds 180 and will be in the second quadrant.

[01:03:11]>> Okay. Perfect.

[01:03:13]>> Uhhuh.

[01:03:15]Thank you editor. Yes.

[01:03:17]>> Welcome.

[01:03:18]>> Okay. Hope everyone. Now let me put the last one. The last one. Members make sure you don't fail this one. If everyone gets it, I'll know that we are there.

[01:03:32]Ah just place in which quadrant is your answer.

[01:03:38]I've started someone has spoiled the thing. The the quadrants members the quadrants they are constant the way you see them.

[01:03:54]This is the first second third fourth.

[01:03:56]They don't change.

[01:03:59]Someone has sp has sp it. Someone has spo it. I've seen an answer different.

[01:04:04]But I'm happy that many of you now have got the concept that we measure. So can you summarize in your own words? Let me give you one or two minutes. Just summarize in your book the way you have understood angle measurements, how we measure angles and which direction. It's going to be important. Yeah. But uh Shia, your hand is still up.

[01:04:58]Okay. Okay. Members, have you finished to summarize in your own words?

[01:05:04]You just continue and then we proceed.

[01:05:14]Uhhuh. A minute. Okay.

[01:05:42]Okay. Uhhuh. I'm waiting for more to three people. Always learn to summarize.

[01:05:48]Even if the teacher tells you something, always put it in in your own words. The way you have picked it, the way you have understood, you write in your own write in your own words that maybe this and this. Yes. Okay.

[01:06:13]Okay.

[01:06:15]Your hand is up.

[01:06:19]>> Good morning, sir. Good morning.

[01:06:23]>> Uh I this for these quadrants I'm now okay but the other day we are talking about and on the previous assignment that you were that that man was doing I think the two was affecting only sign 45 but not even 45 because there were because there were no brackets.

[01:06:47]>> Okay.

[01:06:49]Yes.

[01:06:50]>> What is your answer here?

[01:06:56]>> Where?

[01:06:57]>> Are you seeing what I'm writing?

[01:07:00]>> Yes. 2 * >> d 60 / 2.

[01:07:06]>> Uh-huh. Which is >> CD which is Uhhuh. Now what we had was 2 * 1 out of roo<unk> of 2 * 1 out of roo<unk> of 2 not so >> it was >> 2 cos of 45 sin of 45 something like this >> hope you know that this is multiplication of two things. Yes.

[01:07:34]>> Which is the same thing like here.

[01:07:41]>> So it is when you relate it back relate it to this that you have done that you have multiplied 2 * 3 * 1 to get six you get the same.

[01:07:57]>> Yeah.

[01:08:00]>> I have seen it now.

[01:08:02]>> Okay. So members let's now continue.

[01:08:07]Okay. Now this is where we are going to to start. They have asked us that draw a diagram showing the quadrant in which the rotating line OP lies. For each of the following angles in each diagram indicate clearly the direction of rotation and state the acute angle that that line makes with the x axis. I'm going to start with a.

[01:08:34]Now we have said that when they talk about 100° I'm going to come I draw I draw.

[01:08:44]Remember this is 0° this is 90 this is 180. Now they want us to show the quadrant in which the rotating line OP rotating line is this line that I'm going to show you the rotating line. Now we are going to indicate 100. So you do not that 100 is going to be somewhere here. Now the angle we are locate remember we are rotating 100. This is 100° here and this is line O to P. Now take note that this is your let me make this long enough such that people can see. So they want us to indicate line OP which is this line here I have indicated showing the direction of rotation. Now the direction of rotation is the clockwise anticlockwise positive or negative. So the direction we are taking here is the ant clockwise direction and then two state theute angle. Yesterday we looked at the angles members in the chat.

[01:09:53]Give just one example of an angle. Put it in the chat to to check. Uhhuh. HPS. Perfect.

[01:10:02]Perfect. Uhhuh. Yes. So they want us to get the attute angle. The line OP makes with the X-axis. Members I want you to take note that this is what we call the Xaxis. This line here is what we call the what? x axis. Now whenever you are looking for the angle it makes always look at the angle that is near it. Now members the angle near the x-axis is this angle here and that is the angle we are going to find. So which angle is here? When you look at the angle you have here you ask yourself how many degrees are we remaining? If we have stopped at 100, how many degrees are we remaining with here? Put in the chat what is the answer. Very good. So the angle we have here is 80°.

[01:10:56]So that's how we going to do it. So the angle it makes with the x-axis is 80°.

[01:11:02]And this is the sketch for that diagram.

[01:11:05]I do one more. I'm going to do part part. Let me do part D. When you look at part D, I'll still draw my axis. The angle I have here is 150. But we say that negative means that I move in the clockwise zy direction from the positive x-axis. So I move remember this is like 90° and this is 180. So I will move up to somewhere here. So this is where we have our angle of 150°.

[01:11:41]Now when you reach here which angle is close to the x-axis this angle here members tell me in the chat which angle will that behu [clears throat] members are not seeing that that angle is going to be 30°.

[01:11:57]So here the angle that is cross with the x-axis is 30°.

[01:12:04]members, have we understood what to do?

[01:12:10]Is that okay?

[01:12:14]Uhhuh. As I pick on a hand, then be doing part B and C as I respond to questions. Yes, N.

[01:12:33]Okay, I'm failing to get Nazi a Trinity.

[01:12:44]Oh, I'm failing to get Trinity.

[01:12:48]>> Okay.

[01:12:49]>> Okay. Yes. Yes. Ask >> for part A8.

[01:12:57]Not only >> Oh, you thought it's going to be >> 80.

[01:13:03]>> Uh-huh. Why 80?

[01:13:05]>> Cuz remember you said what when you're measuring, you begin from the x-axis, right?

[01:13:10]>> And when you put our protractor on that line to measure the remaining angle, it will be in the clockwise direction >> cuz remember I said we have to show the direction of the arrow and the angle beginning xaxis to that line. Oh, now now here we are only looking at the angle that is cross to the x-axis without looking at the direction because they told state the angle that the line OP makes with the x-axis. This is the line OP and this is the x-axis. So the angle that it is making here is just 80°.

[01:13:48]>> So we don't consider the signs.

[01:13:51]>> Uh no no need for the signs.

[01:13:53]>> Okay. Thank you.

[01:13:55]Okay.

[01:14:04]The same question cuz when I tried out part >> to get 80° I think you're going to type. You're going to type.

[01:14:34]I'm not getting you.

[01:14:40]Okay. Uh, you can type I'm not clearly getting you. Okay. So from the chat our members are telling me that part B they have obtained 80 and for pass C they got uh 50. That is what I'm learning from uh the members.

[01:15:01]Okay which is okay. Thank you Tracy Abrai.

[01:15:07]Okay, let's finish and oh no, we are just looking at the angle.

[01:15:16]We are not looking at the direction because clearly indicated the direction of rotation. You have indicated the direction already. So that's why we don't put the 30 into constellation. You have indicated that you are moving in this direction.

[01:15:32]Yeah.

[01:15:34]Okay. So we are going to continue lay this animute.

[01:15:52]Okay. So we are going to continue.

[01:15:57]Uh we are going to continue. Okay. So uh I have another activity here. The okay for each of the following diagrams find the basic angle theta. Do you know the basic angle? Now the any angle that is always close to the xaxis we call it the reference angle.

[01:16:26]reference angles are going to be very very important.

[01:16:31]Now take an example when for the diagram we have had the previous diagram we have had the diagram which had 100 this angle that is very close to the x-axis we call it the best angle. So it was 80. If for the other second one we had around here this angle that is close to the x-axis is called the besi angle. In other words if I have let me say diagram like this the angle close to this x-axis is always called a reference angle. This angle here is what a reference what angle. In case is in this format the angle here is called a reference angle. The angle here is called a reference what angle. So whenever you are talking about a reference angle you are talking at for what angle is very close 110 is close but it's not the closest to the x ais so always you must look out for that angle a case in point when you look at this angle here this is the angle closest to the x-axis then the 110 so I want you for each of these diagrams can you find What is the best angle?

[01:17:49]What is that angle that is very close to the x-axis? So find for me theta for each of those diagrams and put your answers in the chat.

[01:18:23]Mhm. Please. You must be able to get that bang that people saying 70. Good. Uhhuh. Continue.

[01:18:34]Genesis.

[01:18:37]>> Mr. Kazwa, could you please pardon on that part? I've not gotten it well. like the reference angle is always measured from the xaxis but it is that angle that is the closest to the x-axis take an example in case I have here this is my diagram and I have an angle moving like this the angle very close to the x-axis here is what we call a b angle have you got it Yes.

[01:19:13]>> Okay. You're taking an assumption people are getting 70 on a how is it coming?

[01:19:19]>> Uh-huh. Now you ask yourself from here up to here you notice 180 not so.

[01:19:26]>> Yes teacher.

[01:19:27]>> Now you ask yourself what angle is here?

[01:19:30]Because when you look at this angle and the angle you have moved which one is very close is it here or it is B? This angle here.

[01:19:40]It is a.

[01:19:42]>> It is a. Now you ask how many degrees is a making for you to reach a 180 along the x-axis. How many degrees should you do you need to move?

[01:19:57]Are you getting it?

[01:19:59]>> Yes, teacher.

[01:20:00]>> Ah, so that means you say 180us 110 which will give you the 70°.

[01:20:07]>> Oh, I've understood. Thank you, sir.

[01:20:09]>> Okay. Okay. Don't run away. Come back.

[01:20:10]Come back. Uh-huh. We on party. Are you seeing part C?

[01:20:18]>> Genesis.

[01:20:21]>> Yes, teacher.

[01:20:22]>> We on part C. Not so.

[01:20:25]>> Yes.

[01:20:26]>> Uh-huh. You tell me which angle is going to be close here to the x-axis here.

[01:20:37]Let me see.

[01:20:40]20 20 >> 20 How did you get 20?

[01:20:49]>> Okay, I get to know that here it's 180 and they moved. Okay, they moved 200°.

[01:20:56]>> Mhm.

[01:20:57]>> So the closest I subtracted 200° minus the 120 sorry 180.

[01:21:05]>> Okay. Okay, K. Thank you. Thank you. Thank you.

[01:21:09]>> Now you are there. Okay. Hannah.

[01:21:14]>> Yes sir. On part D.

[01:21:17]>> Uhhuh. Part D.

[01:21:19]>> Yes. The theta first moved. Yes. The 360°.

[01:21:25]>> Uh-huh.

[01:21:27]>> Then after it moving 360° >> it again. What I'm asking, >> did it add the 140° from 360 going anticlockwise?

[01:21:43]>> Uh-huh. Yeah. Because you are seeing Yes, it is anticlockwise.

[01:21:49]>> So, >> this was the first turn. Are you seeing the first turn?

[01:21:53]>> Yes.

[01:21:54]>> And then he continued the second turn.

[01:21:59]So, you tell us the reference angle. So, the reference angle should be this angle. very close to the x-axis. Which angle is that? Don't go away, Hannah.

[01:22:11]>> The refresher angle.

[01:22:13]>> It's 140.

[01:22:16]>> No. Well, remember Hannah, are you saying he has so far moved again? 140.

[01:22:24]Now you ask yourself if this is the difference. You have moved so far 140.

[01:22:31]You ask yourself this angle here and the angle you have moved which one is going to be very close to the x-axis the smallest one.

[01:22:44]>> Oh so the reference angle is 40 >> is 40. Very good. How did you get 40?

[01:22:53]>> Getting Yes. getting 40.

[01:22:58]>> I I subtracted 180.

[01:23:02]>> Uh-huh. 180 minus 140.

[01:23:06]>> 140.

[01:23:07]>> Okay. Perfect. Perfect. Yeah.

[01:23:10]>> Okay. Thank you very much, sir.

[01:23:12]>> Okay. Thank you. Yeah. So, members, that's how you attempt. And I want to let you know that reference angles are going to be very key when you are solving trigonometric functions because your reference angle is going to be supporting you to determine other angles within that particular quadrant in order for you to know other angles because at this level we shall not only be giving one answers. We shall be giving very many what answers depending on the quadrant where our thing lies. So make sure you know how to find the reference angle. And in summary, in summary, let me say let me see whether I have a summary. Oh, unfortunately I don't have a summary. But we are going to come up with a summary. But one thing you have noted that when you reach in this quadrant you were in order for you to get the angle you always said 180 - 110 that's how you got the angle in this quadrant here you said otherwise what did you say here you said 360us the angle you had 360 when you came to this quadrant you noticed that now you had to say 180 minus the 140 here that's what you said when it came here you not that you got 200 minus 180 now whatever thing you are going to be working on you are going to notice that there is something we are going to come up with at the end of the day now now we have what we call trigonometric ratios for the general angles now when it comes to these general angles I know you have already summarized them at all level and we are going to summarize them like this but I wouldn't want you to okay okay okay okay yes any angle the sign the coine and the tan have a way they do lie I'm going to show you one take an example like I've said I wouldn't want you always to cram I want you to understand the basics.

[01:25:39]When I am in this quadrant here and I have the always I want you always to refer to the reference angles. You know that this is the positive x-axis.

[01:25:51]You know that this is the negative x-axis. You know this is the positive yaxis and this is the negative yaxis.

[01:25:58]Now when you check here in case this is our triangle this is your y this is the x this is positive x this is positive y and this one can be our r. Now in order for us to get any let me say theta to get the sign of theta you know that s is going to be given by always opposite out of the hypotenuse. So you notice that sign is going to be opposite which is y out of r that one you know it. So you going to notice that your sign is going to be positive your co is going to be positive and then the other one is also going to be positive. But when it comes to the second quadrant what do you notice that this is positive y and this is x. Now when I draw my my triangle like this with this is the reference angle this is going to be x this is going to be positive y then this one is going to be your r what are you not seeing you not that in case I come in this quadrant and let me say I want to get s of theta what is s given by I will say that s is opposite remember this is opposite hypotenuse and this is adjacent so S is given by opposite out of the hypotenuse. So s is y out of r.

[01:27:27]So meaning that here you notice that you get a positive answer because r is this.

[01:27:33]When it comes to cos of theta cos is given by what cos is adjacent. So you not that adjacent out of the hypotenuse which is r. So you not that cos is negative.

[01:27:47]The answer you get is negative. That's why in the second quadrant we always say that cos is negative. When it comes to tan you know that tan of theta is given by opposite which is y out of adjacent which is x. So you notice that the answer you remain with here is going to be y out of x. That's why here tan we shall always say that tan is negative in this quadrant.

[01:28:16]Members let me know are you understanding what we are doing are you understanding the concept I'm trying to show you why do we say that co is negative here t is negative here are you getting the concept okay perfect let me now show you one more now in case I'm going to use the last quadrant in case I'm using the last quadrant always it must be with reference to the this angle here the reference angle You know that this is y but you know that this is negative y so this is going to be negative but this is positive x and this is your r. So when members I come here and I say I want cos theta cos theta you know that this is the opposite this is the adjacent and this is the hypotenuse. So you know that cos is given by adjacent out of hypotenuse. So cos is positive. But when it comes to sin of theta, sine of theta is going to be given by opposite which is y out of hypotenance which is r. So you notice that s is negative. That's why we say that in the fourth quadrant s is negative.

[01:29:32]So that's how it comes about. Yes, sank.

[01:29:44]Okay. Okay. Members, are we on the same page? I want now to continue. So, even 10 is negative.

[01:29:54]Tell me, are we good to continue?

[01:29:57]Uhhuh. [snorts] Okay. People saying now we can continue.

[01:30:02]So now when we come here to summarize, that is how you can summarize. So summarize your things. Summarize in your table. Show now. Yes, I know you remember that here sign co and tan all of them are what? Positive. Here it's sign cos and tan are negative. Here it's the others are negative. So can you summarize and I know you have your pneumonics you use. So feel free in case you have any you used to remember you can place it in the chat such that HP said all students take cake all science teachers complain all students take care. Okay.

[01:30:50]Uhhuh. Yeah. And then for you put your pneummonic you want to use to remember acts. Uhhuh.

[01:31:00]Acts. Uhhuh. All science teachers can.

[01:31:03]Okay. All science teachers Kane. All All student teachers can. All science teachers can. All science take chemistry. All secondary.

[01:31:14]That one I don't know. All science teachers care. Okay. Okay. Yeah.

[01:31:22][laughter] I think everyone has it own, but that's okay. That's okay.

[01:31:31]Okay.

[01:31:36]Okay. Okay.

[01:31:40]All right.

[01:31:45]Okay. Yeah. So use anyonic you you want.

[01:31:49]So we are going now to continue. Now I I've brought in two concepts reference angle and now I'm going to bring in the quadrants. Now we are going to start.

[01:32:01]Now they want us to express in terms of ratios the acute angle. Acute angle. We said that acute angle are those angles between 0 and 90°.

[01:32:14]What are we going to do here? I want you to understand those sketches.

[01:32:21]So I'm going to come I draw my sketch. I know this is 0 90 180 270.

[01:32:32]Now I estimate where is 140 lying. Now check out the angle you have that the angle you have is a positive angle.

[01:32:43]Is that fine? That it is a positive angle. Now since it is a positive angle you know that all s t and c it is a positive angle of 140.

[01:32:57]So you remove 140 is going to lie here. So this is your 140°.

[01:33:05]Why have we taken this direction?

[01:33:07]Because we know very well that our angle positive we measure from the positive xaxis and then we go in that direction.

[01:33:16]Now they want us to express it in terms of an acute angle. An acute angle means an angle between 0 and 90. Now always get the reference angle. Any angle cross to the xaxis. Members in the chat tell me what is our reference angle. Angle is that angle closest to the x-axis. What is that angle we have there? Uhhuh.

[01:33:45]40. Very good. So the angle we have here is 40°.

[01:33:50]Now 40 is what we call the reference angle obtaining the angle. The next question you are going to ask is I have sign in this particular quadrant. What is the sign of sign?

[01:34:08]Have I have I pronounced it well? Yes.

[01:34:10]What is the sign of sign? That the sign of s is a positive because s is positive. So members I'm going to come here and I say that in terms of the acute angle sin 140 is the same as s of 40°.

[01:34:27]So I don't need to put degrees here is the same as 40.

[01:34:33]Members tell me whether you have understood that I have left it as positive because sign is positive in that quadrant.

[01:34:43]Uh-huh. Understood?

[01:34:46]Okay. Cla >> good morning sir.

[01:34:51]>> Morning >> sir. I'm asking why don't we put degrees on the 40?

[01:34:57]>> Oh 40. Why you don't put degrees? We can put no problem.

[01:35:03]Okay.

[01:35:05]>> In the question they had given us that's why I can put it.

[01:35:10]>> Okay. Uhhuh. God is child.

[01:35:17]But clear let me also make it clear when you find that they have not put degrees then you need to take into consideration what we call radians. But I'll bring it at a later time.

[01:35:30]>> Okay. Yes, Junior.

[01:35:35]>> Um, sir, can you repeat that concept of s of 40 is equals to s of 140?

[01:35:43]>> Hey, now okay. Um, the first thing always, okay, let me use part B. Now, when it comes to part B, the first thing you need to ask yourself that the angle is 130. How are you going to move? I'm going to move in the clockwise direction. So I'll come here. I move my one 130 is somewhere there. That is 130°.

[01:36:11]Now the next thing you ask yourself what is the angle closest to the x-axis here.

[01:36:17]Members in the chat tell me which angle is that that is closest.

[01:36:24]Uhhuh. members are saying it is 50. So the angle you have here is 50°.

[01:36:35]Now the next question you're going to ask that you are dealing with course. So you know that all students take a chemistry. Now the next thing you're going to ask yourself in this particular quadrant what is the sign of co that in this quadrant co is negative I don't know whether that is okay now after knowing that co is negative what does it mean it means that the answer you are going to write the answer you are going to write is going to be now Since cos is negative in this quadrant, I'll write negative but I write cos of 50°.

[01:37:24]The negative here is showing that in this partial quadrant cos is negative but my reference angle is 50. So when you place your calculator of 13 is the same thing asative cos of 50.

[01:37:40]I don't know whether you have understood was it sat Have you understood? I think it was sata.

[01:37:54]Have you understood?

[01:37:58]What was someone? Yes. Members, have you understood the concept?

[01:38:02]>> Yes, sir.

[01:38:04]>> Okay.

[01:38:09]Is it a question?

[01:38:12]I just like sir what if when you do like cos of 130° you say that is equals to cos then into brackets 180us 130 then you get cos 50 and you apply a sign don't you get the same answer for that.

[01:38:44]>> Yes, you get the same answer. But that how does that thing come from? That's what I'm showing you. That's why I always want you to learn the basics.

[01:38:54]You need to learn how does it come about.

[01:38:58]>> Okay sir.

[01:39:00]>> And when you look here that you have even a sketch. So here you are not cramming a lot of things in your head but you are understanding from the basics.

[01:39:10]Hey, that's what I'm looking at.

[01:39:12]>> Okay. Uhhuh. Zit.

[01:39:18]Okay.

[01:39:20]K.

[01:39:21]>> Sir. Yes.

[01:39:25]>> I was asking that is negative cos the same as cosative.

[01:39:33]>> Uhhuh. You're asking is cos oft >> the answer is no.

[01:39:48]>> Is that okay?

[01:39:49]>> Why?

[01:39:50]>> Why?

[01:39:52]>> Yeah. Because like when you place your cos of 50 and you place cos of 50, you'll get a different answer.

[01:40:08]Have you tried it?

[01:40:14]Yes, you try it. You'll see that the answers are different. But that one I'll bring an idea about it when we reach the graphical work.

[01:40:25]>> Yes, Ruth.

[01:40:27]>> Um, excuse me, sir. We have that okay we moving in the clockwise direction >> and our angle close to the x-axis is 50.

[01:40:38]So the fact that it is clockwise doesn't automatically make the 50 a negative or oh no 50 a reference angle doesn't have any it doesn't have like negative or positive it will always be positive your reference angle >> okay >> but now the signs come again >> now if it was Maybe in the anticlockwise direction. Maybe if it was like -260.

[01:41:14]>> Give me give me what? If it was, give me the thing you want. I write.

[01:41:20]>> I have -260.

[01:41:24]>> When it is what? Cos sign. What?

[01:41:28]>> When it is sign >> sign. So you'll come here.

[01:41:34]>> You know that you move here. So it is going to be somewhere here. This is that is the right >> now you ask which angle is here that is very close to the x-axis.

[01:41:50]>> Uhhuh. Which angle is that?

[01:42:01]>> Sure.

[01:42:03]Remember don't forget this is 180.

[01:42:07]>> So the anything you that comes should be able to give you the total thing.

[01:42:15]>> So I'm confused. No like like you have moved from here up to here you have moved not so >> so what do you remain with to make it 240 here because here you know already is 180 >> 60 >> 60 is that fine >> now s of 240° in this quadrant you're going to say that all students that sign is positive so you sign is positive but the angle you have here is 60. So since sign is positive I write a positive sign then I put my reference angle.

[01:42:59]>> Are you getting it?

[01:43:01]>> Yes sir. Thank you.

[01:43:02]>> Yes. So when you place s of -240 and s of 60 you'll get the same thing.

[01:43:09]>> Okay. Thank you.

[01:43:11]>> Okay. So I think members hope you are getting it. Your reference angle will always be positive no matter the direction you are taking. But the quadrant where it falls it will determine the sign of the trigonometric ratio.

[01:43:27]Is it a question members you doing for me? She has given us part C. My part D is here. Get for me the answer weekly master.

[01:43:39]>> Yes.

[01:43:41]Now my question is are you hearing me sir? Yes, I'm hearing >> like you for the cost of 240 >> when you because 240 I've seen that lies in the third quadrant and where it is and where it is it is a negative in the third quadrant and remember I've said co 240 so I'm getting confused Because the >> this was -240.

[01:44:20]>> No, I'm noting that one. I'm bringing it I'm bringing for the cost of 240.

[01:44:27]>> Oh, okay. It lies here.

[01:44:30]>> Mhm.

[01:44:31]>> Then after then when it lies there which means in that quadrant it is a negative.

[01:44:39]>> Okay. You first find what is your all.

[01:44:43]>> Uhhuh. Yes.

[01:44:46]It lies in the third quadrant.

[01:44:48]>> Yes.

[01:44:50]>> Then lying in the third quadrant, what does it mean? It means that the course is negative.

[01:44:56]>> Uhhuh. Negative. Uh-huh.

[01:45:01]>> So, but what is the reference angle here >> in that quadrant?

[01:45:06]>> Yeah.

[01:45:08]>> T 60.

[01:45:11]>> 60. That means that cos of 240 is the same asative cos of 60. Once you place your you get the same answer.

[01:45:21]That's what we are looking for.

[01:45:25]Understood.

[01:45:29]>> Have you picked?

[01:45:32]>> Yes.

[01:45:33]>> Yes. And now where are we heading? In fact, now we are going to reach a point where now we are going to bring in issues concerning the special angles because here someone can even go on and tell you let me say now like our brother has given us cos 240. You may reach there and they tell you that you have remained with cos 240 and maybe they asking you to write it as an exact form.

[01:46:00]So that means you need to know that this is the same asative cos of 60. Then you remember your special angles that now this one is going to be<unk>3 out of out of two. Is it3 out of two or no? It is a half. In fact a half. Yes. So that you need to remember that the idea. So that is where we are heading.

[01:46:23]Okay members let me know through the chat whether we have understood and we are going to continue. Yes.

[01:46:33]you've not.

[01:46:36]>> Yes, teacher.

[01:46:40]>> Yes, ma'am.

[01:46:43]>> Like you say that that cos50 is not equal to cos of 50.

[01:46:54]>> Mhm.

[01:46:57]And now confused why we say that cos of 30= 50.

[01:47:05]>> Come again. Why we say >> you say that cos of 50 is not equal to cos of 50.

[01:47:14]>> Yes. Just place the you you'll get different things.

[01:47:18]>> Now I'm confused why we say that cos of -30 is equal to cos of 50. Why we say cos of >> cos of 30= cos of 50?

[01:47:32]No, it is cos of 130.

[01:47:36]>> Yes.

[01:47:37]>> Is equal to >> cos of 50.

[01:47:41]>> Yes, those two things are equal.

[01:47:46]>> The angles here are different.

[01:47:50]>> Okay teacher, why is why is cos50 not equal toative cos of 50?

[01:47:57]>> Now those two things mean different. cos of50 I told you it means that I'm going to move in this in this direction are you saying that >> yes >> that is the meaning whereas negative co of 50 it is something different you move this side >> okay >> but to make it much easier for you just get your you place co of 50 see which answer you get less negative co of 50 you'll find that one is positive another one is negative >> okay thank you sir >> yes so those are two different directions you are taking >> remember I told that you need to be very careful whenever you find an angle with it this one shows you the direction you are taking when the angle is positive you are moving in another direction >> yes thank you sir >> all right now.

[01:49:04]Okay.

[01:49:07]>> Okay. Now we start. Now we are going to we have started now solving equations.

[01:49:16]Now they have given us that we have now here you are going to be very very careful. They have given us to solve co of this cos of is equal to -3 out of 5. Now always you need to know they have told us that this angle they have given us is between 180 and 270. come you need to know that's why I tell you I'm telling you you need to understand that this is zero this is 90 this is 180 this is 270 what does it mean it means that when you are drawing when you are coming up with your sketch you need to have it in the right way so your sketch will look like this that's how your sketch will look like and now this one is going to be your theta. Now from your knowledge of caua you know that this is adjacent and this is hypotenuse. So you adjacent here as three and then your hypotenuse is going to be five. Now what are we wanting or what are we finding? Now you know that in this quadrant all students take ah the coffee thing. Now may I ask each one of us to first get this side here and first come up with this side. What do you get? Calculate this side. Use your knowledge and calculate.

[01:50:59]>> I'm wondering why we put a triangle in that.

[01:51:02]>> No, the triangle is to help you understand. It is to understand because it is very important when because they want you to find the value of sin theta and tan of theta >> teacher.

[01:51:17]>> Yes.

[01:51:20]>> Why draw the triangle in the in this quadrant where is negative is positive.

[01:51:28]>> Yeah. Because they what they gave us was co not. So >> yes >> and they have and you know that co is clearly negative in the third quadrant and the second quadrant. Is that okay?

[01:51:41]>> Yes. Yes.

[01:51:42]>> But they have told us that the angle is between 18 and 270. So 18 and 270 means that now our triangle is going to be in the third quadrant.

[01:51:56]>> Okay. I understand now.

[01:51:58]>> Okay.

[01:52:00]Mandra people are getting four. Good. So members are getting four. So a 2 + b 2 is = c 2. So this is 3^ 2 + b 2 which is = 5^ 2. So b ^ 2 is equal to 25 - 16 which is a 25 - 9 which is 16 and therefore b will equal 4. So that means here we are going to have four. Now they have asked us to find the value of sin theta. So members this is where understanding is important. So s of theta is going to be remember from your soap that it is opposite out of hypotenuse.

[01:52:51]>> Hypotenuse.

[01:52:52]>> So what is your opposite for this angle?

[01:52:59]>> It's four.

[01:53:00]>> It is four. What is hypotenuse?

[01:53:04]>> Five.

[01:53:05]>> Five. Then next. What is the sign of sign in this quadrant?

[01:53:13]negative >> negative so I'm going to put here negative so that is your sign of theta we go to tan of theta so tan is given by opposite out of adjacent so what is our opposite fouride by adjacent which is three then the next thing you going to Ask yourself what is the sign of tan that in this third quadrant tan is positive. So we shall leave it the way it is. And now those are the values of sin theta and the value of tan of theta.

[01:53:58]Now members are we on the same page?

[01:54:05]Uhhuh. But the square of 16 is pos4 and -4. Uh-huh. Yes. So remember this is the ne remember remember this is the negative y ais. So the best answer you could have taken could be4.

[01:54:29]Okay. Are we on the same page? In case you have a question you may ask. Uh xxi >> teacher.

[01:54:39]>> Yes. Good sir, why don't you put the negative the negative sign on the three when you have drawn the angle?

[01:54:47]>> No, it is okay. You can put it there or you may not. So if you put it there, let me draw the triangle here.

[01:54:55]>> Yes, >> let me put the triangle here. So if I put it here like this. Uh let me draw sorry there are two ways you can do it. You can put it there or you may not. I have no problem. Now in case you have put it here, this will be -3. Now this is going to be -4 and this will be five.

[01:55:19]So if you want you could do it like that. The only thing now you need to remember that this one being the yaxis when you get the this side remember you get the square root. So you don't take in four but you instead take4.

[01:55:34]So here now you come and say what is my sign? My sign is opposite.

[01:55:40]So4 is4 out of3.

[01:55:48]>> Yeah. So that is also okay. You can do it that way.

[01:55:51]>> Okay sir.

[01:55:52]>> Thank you sir.

[01:55:54]>> Okay. Uh let me pick on from Z E something.

[01:56:03]Okay. In case >> I have the same question. I've understood.

[01:56:07]>> You have understood. Okay. What be Adrian and Billion?

[01:56:13]>> Okay. Teacher, my concern is on the on the triangle you've just drawn in the quadrant there.

[01:56:20]>> When they tell us that the angle is in between 180° and 270, >> does it mean that you have to draw that triangle there? Does it have to attach itself on does it always have to attach itself on the x-axis?

[01:56:36]>> No, it does. You don't. You may not do it as long as you have drawn the right in the right as long as it looks like how it's supposed to look like.

[01:56:47]Not really teacher but >> when you identify that the angle is in between 18 and 27 >> and then you draw that op that OP line >> the way you attached that OP line to the x-axis can it attach itself as so to the yaxis or it always has to attach itself on the x-axis >> oh always we start from the from the xf we start from here >> excuse me sir Are you getting it?

[01:57:19]>> Yes. Yes.

[01:57:22]>> Because this is where now we start now separating that here it is positive x-axis this is negative this is positive something like that.

[01:57:33]>> Okay teacher excuse me sir.

[01:57:36]>> Yes sir. Is it a match that the the answer here like as we have got saying over five that we should leave it like that or you can simp go further to simplify?

[01:57:51]>> Oh like simplify to >> like in decimal point.

[01:57:55]>> Oh no leave it in fraction form.

[01:57:59]Whenever you create a decimal point you are making your answer not to be accurate.

[01:58:04]You're making more errors.

[01:58:08]Okay sir.

[01:58:09]>> Okay. Any other?

[01:58:13]>> Excuse me sir.

[01:58:14]>> Yes.

[01:58:16]>> Uh when does this five become negative?

[01:58:22]>> This five.

[01:58:23]>> No your hypotenuse can never be five.

[01:58:27]>> Okay.

[01:58:29]>> Thank you sir.

[01:58:31]>> Okay.

[01:58:32]>> Excuse me sir.

[01:58:33]>> Yes >> sir. I thought that the the turn is opposite over >> opposite out of >> adjacent.

[01:58:44]>> Yes. What is that's what we have here?

[01:58:47]>> The opposite is a4 but I'm seeing a positive4.

[01:58:51]>> No. Yes. 10 is -4ide by adjacent which is -3. So is positive which is three.

[01:59:03]>> Okay. Thank you sir.

[01:59:04]>> Okay.

[01:59:08]How come like the adjacent was negative?

[01:59:17]>> Like I told you like I told you about the axis you know that this is -12.

[01:59:25]So the negative comes from that idea.

[01:59:29]>> Okay. I now understand.

[01:59:33]>> Okay. So members allow me uh to display uh this activity. I I need to be to look at my time.

[01:59:46]Uh-huh. Take a screenshot of this. Take a screenshot now quickly. This is what you're going to do. Take a screenshot of this.

[01:59:56]Hope members have finished to take a screenshot. Take a screenshot of this.

[02:00:01]Uhhuh. Now here this is where I want you to be very careful when they say that without using find exact values of each of the following. So I expect you to first express this angle in terms of an acute angle that this is 120.

[02:00:20]After getting 120 you know that here what will remain is going to be 60 such that you know that cos of 120 is the same as cos of 60. And then you go back to identify that coleal and the other and then you substitute in.

[02:00:40]>> Okay, that's what you do on that part.

[02:00:42]>> Teacher.

[02:00:44]>> Yes, >> I didn't understand how you did that.

[02:00:49]>> Which one?

[02:00:52]>> The cost of 120. Okay, I understood that. But then where you say that we have to substitute? How do we substitute that >> cos 120 like from our quadrants here you know it is going to lie here.

[02:01:06]>> Yes.

[02:01:06]>> Now the reference angle here is 60.

[02:01:11]>> But in this quadrant you know that s is positive but cos is negative. So that means that cos of 120 is equal to cos of 60°.

[02:01:24]Then you go back to the special triangles we had where we had 1 2<unk> 3. Then you ask yourself what is co of six the co is given by adjacent out of iot.

[02:01:39]So it becomes -1 out of two.

[02:01:43]>> Okay.

[02:01:44]>> Yeah that's what I'm saying. So that's what you're going to do on this part. Ah then this is your exercise as well.

[02:01:59]Now you see some of these words obuse.

[02:02:02]Now you need to know that obuse in which quadrant is this thing lying?

[02:02:09]Okay. Have you taken a screenshot? Have you taken a screenshot?

[02:02:14]>> Yes.

[02:02:16]>> Okay. M. Why is the triangle in the T region? Because of the ang the angle.

[02:02:21]They specified the angle that was between 180 and 270°.

[02:02:27]Okay, thank you GR for responding because of the range. Thank you Julie.

[02:02:33]Aha Shark that is okay.

[02:02:36]Yeah, thank you everyone.

[02:02:39]Okay, okay, okay. I think why don't we are going to end here such that we can now go and do housework.

[02:02:52]So thank you so much for attending. Uh let's meet