In Exercises 61–86, use reference angles to find the exact value ...

Question

In Exercises 61–86, use reference angles to find the exact value of each expression. Do not use a calculator.

sin (-17𝜋/3)

Accepted Answer

Welcome back. I am so glad you're here. We're asked to evaluate the following expression manually by using reference angles. Our expression is the sign of negative 35 pi divided by six. Our answer choices are answer choice. A negative one half, answer choice B two, answer choice, C one half and answer choice. D negative two. All right. First, let's figure out where negative 35 pi divided by six falls on the unit circle. That's a negative angle. So let's add a full rotation and see where we are. What's a full rotation when we've got a denominator of six, that's going to be a 12 pi divided by six. So we'll take our negative 35 pie divided by six plus 12 pi divided by six. And that will get us up to a negative 23 pi divided by six. All right. Not on the unit circle yet. Let's add another 12 pi divided by six that will get us up to a negative 11 pi divided by six. All right, you might know where that one is, but just in case we can go one more time around, we'll add 12 pi divided by six. And that's going to give us pi divided by six. All right, we know where that one is. So we've got the sign of pi divided by six, we can draw a rough sketch of a coordinate plane with a horizontal X axis, a vertical y axis that come together at the origin in the middle. Now, overlaying that with a unit circle, we know that pi divided by six is our first hashmark as we go counterclockwise from our initial side, which is the positive part of the X axis. So our terminal side is going to be right here on the first quadrant heading up toward positive infinity for X and Y closer to the X axis than the Y axis. So there's our angle. Now what we need to know, we recall from previous lessons on the unit circle that that is going to be our 30 degrees pi divided by six is 30 degrees and we can use a 30 60 90 triangle. And of course, this is approximate, we've got a 30 60 90 triangle. So 30 degrees is the smallest angle, 90 degrees is our 90 degree angle and then 60 degrees is the midsized angle. And we recall from previous lessons that across. So the opposite of our 30 degrees that is going to be one adjacent to our 30 degrees. So between 30 and 90 that is going to be the square root of three. And our high pous here between 30 degrees and 60 degrees that is going to be two and sine is going to be opposite over or divided by hypotenuse. So the opposite is one and the hypotenuse is two. So we get one half. Now, the last question is, is this positive or negative? Well, we're in the first quadrant and sine is always positive in the first quadrant. So this is a positive one half and we look at our answer choices and this matches with answer choice. See, well done. We'll catch you on the next one.

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In Exercises 61–86, use reference angles to find the exact value of each expression. Do not use a calculator. sin (-17𝜋/3)