GRADE 12 MATHS P2 NSC JUNE 2026 | TRIGONOMETRY EXPLAINED

追加: 2026-05-18

[00:00:01]All right, welcome back again learners in this video.

[00:00:05]I'll be focusing on the DBE May-June 2026 trigonometry question number five.

[00:00:13]Okay?

[00:00:14]Now, question five, let's look at 5.1.

[00:00:19]Now, on this 5.1, I was telling the real learners on this one that there are times where you will be asked to derive formula for cos from cos or from formula for sign from sign or sign from cos, but in this case we are being told to derive this formula.

[00:00:41]Uh it's sign into A plus B.

[00:00:46]You need to change this to be the same with what is given.

[00:00:50]So, you need to say sign into A minus B.

[00:00:56]All right?

[00:00:58]Uh no, no, no, no, no. You need to say sign into A minus into minus B. Then you expand this. That's sign A cos negative B minus cos A sign negative B. That is what we are having. Now, with this information, we are going to say sign A cos B. Why? Because it's the fourth quadrant. So, this one will be minus cos A uh minus cos A negative sign B. All right?

[00:01:42]For which it will be sign A cos B. This will be positive.

[00:01:49]Plus cos A sign B.

[00:01:55]That is the answer for question number five. That is how you answer question number five, right?

[00:02:04]Okay, 5.1.1.

[00:02:08]Okay, now let's look at this hands.

[00:02:11]Uh without using a calculator, solve for X in this interval if, all right?

[00:02:17]If root three cos X is equal to You see this thing? They say replace this in your formula. So, that is A and that is B. So, I'm going to copy the formula that I've proven.

[00:02:35]Uh for A, what is A?

[00:02:38]50° + X. What is B?

[00:02:43]+ 10° - X.

[00:02:47]Which is root three cos X is equal to sin 60° Therefore, three cos X is equal to root three over two, right?

[00:03:04]Divide everything by root three.

[00:03:07]Divide everything by root three. So, it's going to be cos X is equal to 1 over two. You see?

[00:03:16]Now, X is equal to arc cos 1 over two, which is 60°.

[00:03:24]That's the first condition.

[00:03:26]We also go to the fourth quadrant and say 360° - 60. So, the answer is 300.

[00:03:34]So, these are the two answers in this interval from zero to 360°.

[00:03:41]I believe that that's making sense.

[00:03:45]All right? Let's go to the next one.

[00:03:48]Uh question 5.2.

[00:03:53]Now, sin 2x you change it into sin x cos x. You are choosing that's left-hand side, learners.

[00:04:03]over 2 cos squared x, that's what we are having.

[00:04:08]plus sin x right?

[00:04:16]plus sin x all over What about tan?

[00:04:23]It's sin x over cos x.

[00:04:27]into 1 plus sin x by reduction formula Okay?

[00:04:36]All right?

[00:04:42]You can see two and two will cancel, cos and cos will cancel.

[00:04:50]You are left with sin x over cos x plus Let's interchange.

[00:05:12]sin x all over sin x plus 1 sin x over cos x Let's interchange and see.

[00:05:28]sin x over cos x plus sin x multiplied by cos x over sin x into 1 plus sin x. This and this will cancel.

[00:05:48]So, we are left with sin x over cos x plus cos x over 1 plus sin x.

[00:06:01]All right, cross multiply.

[00:06:06]Cross multiply.

[00:06:10]sin x plus sin squared x then we got there cos squared x over cos x. The reason why I don't multiply, I'm looking for cos.

[00:06:24]This is the same as sin x plus sin squared x plus 1 minus sin squared x over cos x into 1 plus sin x.

[00:06:43]This one and this one will cancel.

[00:06:45]You'll be left with sin x plus 1 sin x plus 1 cos x into 1 plus sin x. You see they cancel.

[00:07:02]So, the answer is 1 over cos x.

[00:07:07]Left hand side is equal to right hand side.

[00:07:14]Right, that is how you answer question 5.2.

[00:07:21]Okay?

[00:07:22]Let's go to the next one.

[00:07:24]5.

[00:07:26]3.

[00:07:28]I love this question.

[00:07:31]So, sin 260° is equal to negative the root of cos 20° plus 1 over 2 Now, I start to check the relationship if this one is 20 it means I must have 10 on the left.

[00:07:55]So, I'll start by saying sign on the left.

[00:08:02]I'll start by saying sign 180° + 80 cuz I see 10 on 80.

[00:08:10]It's equal to negative sign 80° which is the same as negative sign into 90° minus 10 which is negative cos 10°. That is left-hand side.

[00:08:35]Right?

[00:08:37]Now, I check right-hand side.

[00:08:42]I want to change this. Negative root I need 10 there.

[00:08:47]So, I'll say cos 2 into 10 + 1 over 2 That's left-hand side. Negative 2 cos squared 10 minus 1 + 1 over 2 Negative root 2 cos squared 10 over 2 Negative root cos squared 10 which is negative cos 10°.

[00:09:30]That's brilliant. That's how we do it, learners.

[00:09:34]Easy stuff and beautiful stuff.

[00:09:37]Okay?

[00:09:40]That's all about question 5.3.

[00:09:45]Question 5.3.

[00:09:47]That's what we're supposed to do.

[00:09:53]Right, let's continue. Let's go to question um The reason why I just wanted to work left-hand side and right-hand side.

[00:10:03]Okay, let's go to question 5. I mean question 6.

[00:10:10]Question 6.

[00:10:13]6.1.1.

[00:10:16]cos 330 is the same as 360 minus 30 degrees.

[00:10:26]cos 360 degrees plus 15. Do you know why I'm using 15? I saw it on 175. That's That's the reason.

[00:10:36]That's the main reason.

[00:10:38]sin 180 minus 15 degrees.

[00:10:46]And then we say sin 180 plus 30 degrees. This is a reduction. We are reducing.

[00:10:57]Okay?

[00:10:59]This is cos 30 degrees.

[00:11:03]This is cos 15 degrees.

[00:11:09]This is sin 15 degrees.

[00:11:14]negative sin 30 degrees which is cos 30 degrees cos 15 degrees minus sin 15 degrees sin 30 degrees. I see the compound angle for cos.

[00:11:34]So that is cos 30 degrees plus 15 degrees, which is cos 45 degrees.

[00:11:43]The answer is root two over two.

[00:11:47]That is the answer for five for other 6.11.

[00:11:52]That was five marks for free.

[00:11:55]Right?

[00:11:59]Okay, let's go to the next one. Hence 6.1.2, they say hence without using a calculator simplify the above expression to a single trigonometric or rather I did. Hence determine sine 70 in terms of P.

[00:12:16]In terms of P, right?

[00:12:33]Okay.

[00:12:34]Now, the question was what? Let's see sine 15 75. Yes.

[00:12:48]Oh, I made a mistake. I made a mistake.

[00:12:53]Let's go back. Thank you.

[00:12:56]Let's go back to the next one. Let's I mean to the previous one.

[00:13:03]It'll be cos 30 degrees.

[00:13:10]It's a five. We use five.

[00:13:24]Even here it's you know?

[00:13:34]80 minus five plus sign five degrees.

[00:13:43]Negative sign 30 like that.

[00:13:51]Okay? So, it'll be cos into 30 degrees.

[00:14:00]Because this one will be cos 30 cos five minus sign five sign 30.

[00:14:11]So, it'll be plus five. See?

[00:14:14]Which is cos 35 degrees. That is the answer for the first one.

[00:14:21]Now, the reason why I I saw that something is wrong is when now we know that hence, it means cos 35 is equal to P because this one they say hence is equal to P.

[00:14:37]Right?

[00:14:38]Now, which means you're going to draw a right angle triangle on the first quadrant.

[00:14:46]Then you say 35.

[00:14:49]This one you say 55.

[00:14:53]Cos it's adjacent over hypotenuse.

[00:14:57]This one will be one minus P squared.

[00:15:01]Now, the question we are looking for it's sign 70 degrees.

[00:15:07]Remember, this is sign two into 35, right?

[00:15:13]Which is two sign 35 degrees cos 35 degrees which is going to be two into sin 35 1 minus p squared.

[00:15:28]cos 35 that is p. So, therefore, the answer is 2p the root of 1 minus p squared.

[00:15:37]That is how you answer the question 3 6.1.2, right?

[00:15:45]Now, let's go to the next one. We say 6.

[00:15:50]2.

[00:15:52]If it is given that 2 to the power of 3 sin squared x is equal to we are cross multiplying.

[00:16:07]I will change that four into two squared times two to the sin x.

[00:16:23]All right? Now, two three sin squared x is equal to two sin x plus two. When the bases are the same, you equate the exponents. So, it will be three sin squared x is equal to sin x plus two.

[00:16:48]Three sin squared x minus sin x minus two equal to zero, right?

[00:16:56]Then, what do you do, learners? We are We are looking for the general what?

[00:17:00]Solutions. So, three sin x and this is sin x.

[00:17:06]It should be a two and a one there.

[00:17:09]Should be a minus and a plus.

[00:17:12]Therefore, sin x is equal to minus two over three or sine x is equal to one.

[00:17:22]I hope you are getting this.

[00:17:28]So, for this, x is equal to 90° plus 360 k.

[00:17:40]And then for this one, we look for a ref angle. You say arc sine x is equal to arc sine s- um sine two over three. You don't press the negative, right?

[00:17:59]Therefore, it will be x is equal to 108°.

[00:18:10]Plus this ref angle that you got plus 360 k, where k is an element of integers.

[00:18:20]Or x is equal to on the third quad- on the fourth quadrant.

[00:18:25]Minus ref angle plus 360 k.

[00:18:32]So, that is how we're supposed to work on this question. Till we meet again on the next video. Don't forget to subscribe. Bye-bye.