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Problem 24

Evaluating Trigonometric Functions In Exerrises $21-24$ , evaluate the sine, cosine, and tangent of each angle. Do not use a calculator.

$$\begin{array}{ll}{\text { (a) } 750^{\circ}} & {\text { (b) } 510^{\circ}} & {\text { (c) } \frac{10 \pi}{3}} & {\text { (d) } \frac{17 \pi}{3}}\end{array}$$

Answer

( SEE SOLUTION )

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## Discussion

## Video Transcript

all right. Problem Number 24 asks us to evaluate the sign co sign and tangent of each of the following angles, and it asks us to not use a calculator. So for number A wee look at that angle and we see that it's 750 degrees now. 750 degrees is a very large angle, and one tactic we can use to help solve this problem would be to make it smaller by finding a smaller co terminal angle. So we can do that by subtracting 360 degrees. Now 7 50 minus 3 60 is 3 90 and that's still larger than 360. And we want to keep our our final angler Most simple angle within about 360 degrees. So I'm going to subtract 360 degrees again, and we're gonna be left with 30 degrees. Now, where does 30 degrees exist on our unit circle? We're going to use that unit circle toe, find our exact values of sign and coastline, and from there we can calculate tangent. So 30 degrees exists right here, and that's it. The point where are co sign value? You know this distance right? here is the same as square root three over two and our signed value, which is just that. Why? Height right here is equal to 1/2. Okay. And so for party, Sign of 750 degrees equals 1/2 co sign a 100 or 700 and 50 degrees. Equal square, three over two, and then tangent of 7 50 degrees equals 1/2 over square three over two equals one over square root three. So those are the answers for party. Now for part B, we're gonna do the same exact thing where we start with our large angle and begin by subtracting 360 to make it stress smaller. You know, the 5 10 minus 3 60 is 150 degrees, which is less than 360. So we know that it exists within our first full revolution of the unit circle. And that exists right about here. Okay. And so at that point, I'm just gonna draw narrow up here where I put these values Our color sign value is now negative. Square root three over two and sign is 1/2. So sign of 1 50 equals 1/2 co sign a 1 50 equals the negatives for three over two in tangent of 1 50 Equal sign divided by co sign equals one are negative. One over. Spirit three. All right, so that's the answer for part B. Now, let's do part. See, well, you notice about the angle, given in part sees its and radiance. So when we want to find that smaller co terminal angle instead of subtracting 360 degrees, we're going to be subtracting two pi multiples of two pipe just until we get it in less than two pi. So we'll start with 10. Hi thirds minus two pi. Now, if we want this to be expressed, you know, as a fraction of thirds we could multiply by free over three, which gives us 10 pie thirds minus six. Pie thirds equals four pi thirds. Now, where does four pi thirds exist on our unit circle? Well, right here would be 12123 hi thirds, which is one pie. So over here, right down here is our four pi thirds. And I'm just gonna list those angles that we just found previously. 30 degrees. This right here is 1 50 There you go. All the way over here. It's four pi thirds. All right. Now, our values of sign and co sign at this point I'm co sign is negative. 1/2 and sign is negative. Swear or three over two. Okay, so that helps us find our answer. Signed a temp. I thirds was negative. Square three over two. Co sign of 10 pie thirds. It's negative 1/2. The intention is just gonna be signed about. My co sign equals Square E three. Yeah, of it. There's an answer for that and finally on the party. So we start with 17 pie thirds, and now we're pros at this at this point, so we can go little bit quicker. We're going to subtract to pie. We're gonna attracted by which the same a six hour thirds it's gonna give us. It's gonna give us 11 pi thirds pretty big. So let's subtract another 6 5/3 That's gonna give us five high thirds. So that's less than two pine greater than zero. You know, that's the first revolution of our unit circle. So let's find that Well, this is for by thirds. Then ride over here is gonna be our five by thirds relatively new, probably more over here. But it's all good, just a sketch. And we know that our sign or will our coast? I advise him 1/2 and our sign is negative. Square three over two. So let's write that down here. Sign of 17 pie thirds. It was negative. Square three over two. Co sign of 17 pie thirds equals 1/2 and tangent of 17 pie thirds. It's just gonna be the sign value divided by the coastline. Value equals negative square root three. And there you have it. We were able to find the sign co sign in tangents of each of these angles without our calculator by utilizing our knowledge of the unit circle and its exact values of sign and coastline at varying degree measures.

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