Find each exact function value. See Example 2. ...

Question

Find each exact function value. See Example 2. cos (―4π/3)

Accepted Answer

Welcome back. I am so glad you're here. We are asked to determine the exact value of the given function. Our given function is the cosine of negative seven pi divided by three. Our answer choices are answer choice, a one half answer choice, B negative one half answer choice C square root three divided by two and answer choice D negative square root three divided by two. All right. So we're given this angle of negative seven pi divided by three. And if you recall from previous lessons, and if you take a look at the figure of the unit circle in the textbook negative seven pi divided by three isn't on there and neither is seven pi divided by three. But we can figure out if there's a co terminal angle that's on the unit circle. And the way to do that is we can add to pi. So if we take negative seven pi divided by three and add one full rotation two pi to get a co terminal angle that will give us negative pi divided by three. And hey, Pi divided by three is on the unit circle. So let's find, let's add another two pi to find out our co terminal angle between zero and two pi. So negative pi divided by three plus two pi equals five pi divided by three. Yep, that one is on the unit circle. So now I'm going to draw a rough sketch of the unit circle. I am not including all of the details. Definitely refer to the figure of the unit circle in your textbook. For all the details, we've got a vertical Y axis and a horizontal X axis. They come together at the origin in the middle. The unit circle overlies that every point on the unit circle is one unit away from the origin and certainly not what I drew here. But you can imagine that every point on this circle is one unit away from the origin. Now, for our five pi divided by three, if we start at the initial side along the positive part of the X axis, we're going to head counterclockwise and this is going to take us all the way into the fourth quadrant just a little bit past the negative part of the Y axis. And there's our five pi divided by three looking at those figures in the textbook, that's the same as a 300 degree angle. And our X and Y values there, our X value is at one half and our Y value is at negative square root three divided by two. Now, why is all that important? Well, we recall from previous lessons that the cosine of an angle on the unit circle, we'll call that s is equal to X. So it's equal to the X value at that point. And here the cosine of negative seven pi divided by three, which we know is the same as five pi divided by three. They're co terminal is the X value is equal to one half. So the cosine of negative seven pi divided by three is a positive one half. We look at our answer choices and that matches with answer choice. A well done. We'll catch you on the next one.

Find each exact function value. See Example 2. cos (―4π/3)

Key Concepts

Unit Circle

Reference Angles

Trigonometric Function Values

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