Find each exact function value. See Example 2.
sin (-4π/...

Question

Find each exact function value. See Example 2.

sin (-4π/3)

Accepted Answer

Welcome back. I am so glad you're here. We're asked to determine the exact value of the given function. Our given function is the sign of negative five pi divided by three. Our answer choices are answer choice, a negative one half, answer choice, B one half answer choice, C negative square root two divided by two and answer choice D square root three divided by two. All right. So we can solve for this exact value using the unit circle. We recall from previous lessons that the unit circle overlays the coordinate plane. We have a vertical Y axis and a horizontal X axis. They come together at the origin in the middle, the unit circle has a radius of one. So everything is one unit from the origin. That is certainly not what I drew. But you can imagine that everything is a one unit and that, that's an actual circle. Now, if we were talking about just five pi divided by three, that is on the unit circle. So if we started at the initial side along the positive part of the X axis, and if we were to head counter clockwise from the initial side, five pi divided by three is in the fourth quadrant. So down into the right of the origin. And so looking at the unit circle from the textbook or from previous lessons, we can see that five pi divided by three is the same as 300 degrees. So for negative five pi divided by three, we would start at that same initial side. But this time, we would head clockwise because it's negative, but we would go the same distance. So heading clockwise around the unit circle, that same distance 300 degrees puts us in the first quadrant. And that's a negative five pi divided by three or a negative 300 degrees which corresponds to on the unit circle pi divided by three or positive 60 degrees. And we know from that unit circle that the XY coordinates at pi divided by three radians are one half square root three divided by two. So now we've gone through all of this. But why is this important? Well, we recall from previous lessons that the sign of an angle, the sine of S on the unit circle is equal to Y. So here the sign of our negative five pi divided by three is going to be equal to the Y value there. And the Y value here at negative five pi divided by three is square root three divided by two. So that is the exact value of our sine of negative five pi divided by three. We look at our answer choices and this matches with answer choice d well done. We'll catch you on the next one.

Find each exact function value. See Example 2.
sin (-4π/3)

Key Concepts

Unit Circle and Radian Measure

Sine Function on the Unit Circle

Reference Angles

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