Grade 11 Trigonometry "In terms of" questions - Exam Questions

Added: 2026-05-15

[00:00:00]to do an exam question for the in terms of trigonometric section as well as another practice question. So, let's start with this one first and before we get into this, I just want to remind you of the steps that we learned in the previous video. There's the steps over here. If you need help knowing how to use the steps or how it works and you want a basic example first, go watch that because we're going to be applying these steps to the example questions that I showed you and to the past paper questions.

[00:00:27]If you like seeing steps broken down and like this in math, please please hit the subscribe button because I do this in all the sections of math. I can help you more. Let you pause the screen and try this yourself first.

[00:00:39]So, firstly, let's draw our diagram. sin 40 is equal to m.

[00:00:44]So, this angle over here would be a 40° angle. And remember, I'm giving you sin 40. So, we're going to put m over 1 and sin the angle sin I mean sin the ratio sin or sine is opposite over hypotenuse.

[00:00:58]So, opposite the 40°, it's is connect to the X, connect to the X, make it a 90° triangle. While we are here, if that's 90 and that's 40, this angle up here would be a 50° angle. Let's just fill that in.

[00:01:12]Now, sin 40 is equal to opposite over hypotenuse. Opposite corresponds to m.

[00:01:17]So, opposite the 40° will be length m.

[00:01:21]And then hypotenuse would be 1.

[00:01:25]What that means is I can find this missing length over here using Pythagoras. So, we're looking for a short side here. Remember in the previous question when we use Pythagoras, we were looking for a long side. We were looking for the 90°, the side opposite the 90°, which is why we added the other two sides squared. Here, I'm looking for a short side. So, basically, I'm going to call it X because it's a X value. So, X squared would be equal to you take your hypotenuse squared minus your other short side squared. And of course we're doing Pythagoras. So if you're looking for a short side, you minus and you take the hypotenuse squared minus the short side squared, square root both sides. So I'm going to get 1 squared minus m squared. And again, 1 squared is just 1. So I'm going to write it like that. Let's fill that in on our diagram just because it's something that we have found and we may use. So this is going to be 1 minus m squared.

[00:02:23]So remember, you might get a mark for that diagram. So it's very important that we see it even though we don't ask for it in the question. Now, sin of 320, I can write that as sin or sine of 360 minus 40. The reason I know that I can do that is because remember this is all stations to Cape Town. This is theta, this is 180 minus theta, 180 plus theta, 360 minus theta. And we know that 320° would fall in this quadrant over here. And so we would rewrite it as 360 minus and in this case it's minus 40, which is that acute angle. When we reduce, we are in the cos quadrant, but we've got a sin or sine. So it's going to be negative sin 40. And we know that sin 40 is equal to m. That was given in the question over here. Do you remember?

[00:03:14]Sin 40, oopsie, sin 40 is m. So this part turns into an m. So it's going to be negative m. Okay, next one.

[00:03:22]Sin 140. So 140 would be in this quadrant over here. It's less than 180 but bigger than 90. So we're going to rewrite that as sin of 180 minus 40.

[00:03:35]Now, because it's 180 minus, now when we reduce it, we're in the sin quadrant and this is sin. So it's going to be positive sin 40.

[00:03:44]And in the beginning, we said sin 40 is m. So my answer here is m. So over here we've We've cos 230.

[00:03:51]And like I said, 230 is in this quadrant over here. Bigger than 180 but less than 270, so it's going to be cos of 180. In this case, it's plus 50. Okay, this is interesting. Now, when we reduce it, I hope you can see that it's going to be negative cos 50. The reason why it's going to be negative cos 50 is because tan is positive there, but I've got a cos. So, negative cos 50.

[00:04:19]And now you might be thinking, "But my original information doesn't give me 50.

[00:04:24]My original information speaks about 40.

[00:04:27]So, now what?" Remember, we filled in on our diagram that if this angle was 40, this one's 90, this one up here is 50.

[00:04:37]This one right here. So, we don't have to stress. We can just use our diagram.

[00:04:41]So, it's going to be negative, then cos is going to be adjacent over hypotenuse.

[00:04:46]What is adjacent or next to the 50°? The M.

[00:04:50]And what is the hypotenuse? One. So, it's negative M over one, which, you know, you can just write as negative M.

[00:04:58]There are different ways to do this, and I know some people will be tempted to use 270 if they see something like this.

[00:05:03]So, for example, you could say, "Well, ma'am, it could be 270 minus 40."

[00:05:09]And that is true. Remember, when we did cofunctions or co-ratios, we said that this quadrant over here can also be called 270 minus theta, and this one over here can be called 270 plus theta.

[00:05:22]That is true. So, 270 minus theta is here. In this tan cot quadrant. Okay, interesting, but we've got a cos. So, now 270 minus theta, remember when we reduce it, if it if we use a 270 or a 90° here for reduction, we have to use the co-ratio or the cofunction. So, we change it from a cos to a sin of 40, but it's going to be a negative sin 40 because we are in the tan quadrant where tan is negative.

[00:05:54]And then we get back to the same place.

[00:05:55]Negative sin 40, sin 40 is equal to m, so negative sin 40 is going to be negative m. Exactly the same answer.

[00:06:03]Cos 40, this one is nice and easy. We just look at our diagram. So here's 40, cos 40 is equal to adjacent over hypotenuse. So adjacent to 40 is going to be this over here, 1 - m squared.

[00:06:19]Hypotenuse is 1.

[00:06:21]And lastly, tan 220. So 220 would be in this quadrant over here. It's bigger than 180, but smaller than 270. So I'm going to write that as 180 + 40. And then we reduce. So we are in the tan quadrant, tan is positive, and this is a tan, so it's going to be tan of 40. Then we look at my good old diagram over here and we say, "Okay, tan is equal to opposite." So m over adjacent. Adjacent to 40 is that over there, root 1 - m squared. And we leave it like that, we can't simplify further.

[00:07:01]For the exam question, they say, "Given sin 24 is equal to p, write the following in terms of p without the use of a calculator and with the aid of a diagram." So here they're telling me to use a diagram, but you know that that is step one anyway, just like we discussed in the previous video.

[00:07:18]So 24° would be somewhere over here. It doesn't have to be beautiful and according to scale. Then you connect to the x like that. This would be a 90° triangle. What I would do immediately at this point is also fill in this angle over here, which you would get by saying 180 - 90 - 24 because of angle sum triangle.

[00:07:39]And we get 66° for that little corner angle over there.

[00:07:43]Then the next thing that you would do is use the given information over here and we would write or fill in the rest of the diagram according to what's given.

[00:07:51]So, this is P over 1. As you know, sin or sine is opposite over hypotenuse. So, opposite the 24, in other words, this side over here, here's your 24, opposite that would be here, that I can label as P and my hypotenuse is opposite the 90°, so that would be 1. Then what I can do is I can find this X value over here using Pythagoras. So, it would be X squared is equal to you take your long side squared minus your short side squared.

[00:08:22]Then you square root both sides. So, we get the square root of 1 squared minus P squared. And as I've said in previous questions like this, 1 squared is equal to 1, so I'm just going to leave it like that and I'm going to fill it in on my diagram because I just found it. So, this is 1 minus P squared. Then what I can do next is I filled in all the angles and all the sides on my diagram so I can answer the questions.

[00:08:46]So, A is cos 24 and we use our diagram.

[00:08:50]Cos is adjacent over hypotenuse.

[00:08:53]Adjacent to the 24 is this side over here. So, we can say root 1 minus P squared. That is my adjacent side and my hypotenuse is 1. So, that's my answer for cos 24. B is a little bit more difficult. So, first things first, we've got a negative, so I'm going to carry that negative down. This negative here is that negative here. Then I have sin of a negative acute angle. Now, first what I would do is let's do reduction over there for this piece of the equation.

[00:09:26]So, we know that this is all stations to Cape Town. When I reduce a negative acute angle, remember this is my positive acute angle, this is 180 minus theta, 180 plus theta. This is 360 minus theta or I can also write this quadrant as negative theta. And that's what I have here. I have a negative theta, a negative acute angle. So, I'm in the cos quadrant, but I've got a sin or sine.

[00:09:52]So, if I'm in the cos quadrant, it means that cos would be positive here.

[00:09:55]Remember, I'm in this quadrant because I'm looking at a negative acute angle.

[00:09:58]So, cos would be positive, but I've got a sine, which means it would be negative sin 66.

[00:10:05]Multiply that by tan 204. Now, where would 204 be on my car cast diagram? It would be in this quadrant over here, the 180 plus quadrant. So, I'm going to reduce that, use my reduction formulae.

[00:10:19]I'm going to write it as 180 plus 24.

[00:10:24]Because 180 plus 24, that is equal to that. And then I'm reducing it or I'm using reduction formulae to write it in terms of my acute angle. In the next step, I will do the reduction over here.

[00:10:38]But also, what I can also do is a negative next to a negative, that turns into a positive. So, positive sin 66 multiplied by, let's do the reduction.

[00:10:48]180 plus, that again, like I said, is in this quadrant where tan is positive and I've got a tan. So, good. It's going to be positive tan 24. Then, my one of my last steps is I can use my diagram. My diagram being this one over here. So, sin of 66, ah, isn't that nice? That is why I filled the 66 in on my diagram over there earlier. Sin or sine is opposite. So, what's opposite the 66?

[00:11:14]Yeah, this side over here. Roots 1 minus P squared. So, opposite over hypotenuse.

[00:11:21]The hypotenuse is 1.

[00:11:23]We're going to multiply that by tan of 24. And as you know, tan is opposite over adjacent. So, what's opposite the 24?

[00:11:31]P.

[00:11:33]Opposite over adjacent adjacent or next to the 24 is this.

[00:11:40]1 - P squared. Remember it said without a calculator but we can do this as is easy. We cross cancel those and we're left with P divided by 1 which is simply P. There we go. My answer is in terms of P.

[00:11:53]I hope you found that useful. I can't wait to see you in more trig videos in this playlist. Remember to subscribe if you haven't yet. Bye everyone.