Find the exact value of s in the given interval that has the g...

Question

Find the exact value of s in the given interval that has the given circular function value.

[ 0, π/2] ; cos s = √2/2

Accepted Answer

Welcome back. I am so glad you're here. We're told, considering the interval from zero to pi divided by two, find the exact value of X. And we're given that the cosine of X equals square three divided by two. Our answer choices are answer choice. A pi divided by six answer choice B pi divided by three answer choice C five pi divided by six and answer choice D seven pi divided by six. All right. So we're trying to figure out the angle here and we know the exact value that our cosine is equal to and this square root three divided by two that's familiar and that is on the unit circle. So we can use the unit circle to solve this one. Going to draw a rough sketch of the unit circle. I am not going to include all of the details. So definitely check out the image of the unit circle in your textbook. For one with all the details, we have a vertical y axis, a horizontal X axis. They come together at the origin in the middle lying on top of that is a unit circle. Every point on the unit circle is exactly one unit away from the origin. That is not true of what I drew. But you can imagine that it is the interval from zero to pi divided by two. Well zero is the radiant angle measure of the positive part of the X axis and pi divided by two is the radiant angle measure of the positive part of the y axis. So from that interval, we're talking about quadrant one up and to the right of the origin. So for our angles, they're only going to be in quadrant one. And I'm going to start drawing those angles that are in quadrant one. So closer to the positive part of the X axis than the positive part of the Y axis, we have the angle pi divided by six and its xy coordinates are square at three divided by 21 half. And then right in the middle of the first quadrant, we have the angle pi divided by four. It's xy coordinates are square root two divided by two square at two divided by two. And then closer to the positive part of the Y axis than the positive part of the X axis is the angle pi divided by three. Its xy coordinates are one half square root three divided by two. Now we recall from previous lessons that the cosine of an angle we'll call it s on the unit circle is equal to the X value. And here the X value is equal to square root three divided by two. So let's look at our X values for the three angles that we identified for pi divided by six. That X value is square root three divided by two. That's the angle we're looking for. So our angle is pi divided by six and that's the answer to our X here. So X equals the angle pi divided by six. And we'll look at our answer choices and that matches with answer choice. A well done. We'll catch you on the next one.

Find the exact value of s in the given interval that has the given circular function value.
[ 0, π/2] ; cos s = √2/2

Key Concepts

Cosine Function

Unit Circle

Principal Values of Trigonometric Functions

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