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University of Southern California

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Problem 22

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

$$\begin{array}{lll}{(a)-30^{\circ}} & {\text { (b) } 150^{\circ}} & {\text { (c) }-\frac{\pi}{6}} & {\text { (d) } \frac{\pi}{2}}\end{array}$$

Answer

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## Discussion

## Video Transcript

all right. Problem Number 22 asks us to evaluate the sign co sign in tangent of each of the following angles. And the one caveat is we're not allowed to use our calculator. So how are we going to do this? Well, we're gonna use the unit circle now. The unit circle has a select or a few key angles that have really nice, um, defined values for coastline and sign. So the first thing we're gonna look at is negative 30 degrees. I'm gonna come into our unit circle. We know that this horizontal line on the right in deposit direction corresponds to zero degrees. So negative 30 degrees is just down like that. And something that's unique about this, um, negative 30 degrees is that the coastline value is equal to square root three over two, and then the sign value is equal to negative 1/2. And that's just a property of the unit circle that over time you will memorize. And so because we know that it becomes fairly easy to evaluate sine, cosine and tangent co sign of negative 30 degrees is going to equal that X value writes. How much is in this direction. And so that equals square it. Three over two. Sign of negative. 30 equals negative 1/2. And then tangent of negative 30 is the same as the sine divided by the coastline. So you can see. Yeah, my device were three over two. These twos cancel out on your left. Negative one over. Square root three. So that bray or for letter A now for part B were given a value of 115. Well, we can think of that. As is This whole isn't. I need this quadrant here is 90 degrees plus nineties. 1 80 minus 30 is gonna equal 1 50 And so right here, we'll find ourselves at 130 degrees because it is a multiple of 30 degrees. We have it as Ah, it's one of those unique values, right? And so the X right here he's going to equal square it. Three over two and the wife was 1/2. And so we go back. Family can do the exact same process again. Co sign of 150 degrees is equal to. That's a negative because of the negative direction is equal to negative square three over two. Sign of 150 degrees sequel, The positive 1/2 and then for a tangent of 150 degrees. We have 1/2 divided by negative square three over two, and that's going to equal one or square three wrote that kind of funky one negative one over square three. All right, so that's the answer. Be See is our first angle and radiance And one thing that we will are you should realize is that negative pie six is equivalent to negative 30 degrees If you're to use that conversion factor for that where you go negative pie six times two pi over 360. You'd soon see to this just equals negative 30 degrees, Which means heart am part See, have the exact same answers So same and then over for being was gonna circle answers right now just cause things look like they're getting kind of messy. And I don't want you guys to get lost now for D We come in here and were given pie halfs soap. I half is a really nice one. It's just right here by half also equals 90 degrees and a pie Halfs X equals zero, right? It was one So our co sign of Pi House equal zero. Our sign of pie halfs equals one. And then our tangent of perhaps which is just the sign over coastline goes one over zero mats undefined. So that doesn't really exist. There you have it because you can't divide, you know, one by zero. And they have it. Those for the sign coast and tangents for each of the four angles listed above.