Find the exact values of s in the given interval that satisfy ...

Question

Find the exact values of s in the given interval that satisfy the given condition.

[0 , 2π) ; cos² s = 1/2

Accepted Answer

Welcome back. I am so glad you're here. We are told considering the interval from 0 to 2 pi that's one full rotation. Find the exact values of P. Then we are given that the cosine squared of P equals 3/4. Our answer choices are answer choice. A pi divided by 65 pi divided by 67 pi divided by six and 11 pi divided by six answer choice B pi divided by 32 pi divided by 34 pi divided by three and five pi divided by three answer choice. C pi divided by 43 pi divided by 45 pi divided by four and seven pi divided by four. And answer choice D pi divided by three and two pi divided by three. OK. So we've got that interval of a full rotation. That means these values could be anywhere within one full rotation. And cosine squared P equals 3/4 looks a lot like we recall from previous lessons that the cosine of an angle on the unit circle P equals the X value of where that angle intercepts the unit circle. But we've got cosine squared. So what do we do? Well, we can take the square root of both sides. And so on the left, now we have the cosine of P like we want equals. And then on the right, when we take the square root that's going to be plus or minus, that part's important. And then in the numerator, the square root of three is the square root of three divided by, in the denominator, the square root of four is two. So we've got the cosine of P equals plus or minus the square root of three divided by two, which means that wherever our X value equals plus or minus the square root of three divided by two, that's going to be a solution for our angle P. So let's take a look at the unit circle. You should definitely check out the figure of the unit circle in your textbook and look for where X is equal to plus or minus square root of three divided by two. I'm going to draw some of the unit circle but not all of it. So we've got a vertical Y axis and a horizontal X axis. They come together at the origin in the middle. The unit circle lies over that every point on the unit circle is exactly one unit away from the origin. That's not true of what I drew. But you can imagine that that is now where does X equal plus or minus square root of three divided by two? If we start in quadrant, one closer to the positive part of the X axis than the negative than the positive part of the Y axis. We have that first angle on the unit circle pi divided by six. And its xy coordinates are square root three divided by two. There it is and one half. So there's our square root three divided by two X value one of the positive ones that we're looking for. So one of our solutions for P is the angle pi divided by six. Now the other three angles are going to be very similar to this where we have the positive or negative values for X of square root three divided by two. The next one in the second quadrant that's going to be the same distance along the X axis. It's going to be closer to the negative part of the X axis than the positive part of the Y axis. And that angle is our five pi divided by six. It has xy coordinates of negative square root three divided by two positive one half. There's our negative square root three divided by two X value. So another angle P is five pi divided by six. The next one in the third quadrant closer to the negative part of the X axis than the negative part of the Y axis is going to be the angle seven pi divided by six. Its xy coordinates are negative square root three divided by two and negative one half, there's our negative square root three divided by two. So another value for P is seven pi divided by six. Now, finally, we're looking for an angle in the fourth quadrant and it's going to be that same distance along the X axis as the one in the first quadrant. So that's going to be closer to the positive part of the X axis than the negative part of the Y axis. That's going to be 11 pi divided by six. It's xy coordinates are square root three divided by two and negative one half. There's our last square root three divided by two X value. And so our final P value is 11 pie divided by six. There are all of the solutions within one full rotation for where the cosine squared of pi or excuse me of P equals 3/4. We look at our answer choices and this matches with answer choice. A well done. We'll catch you on the next one.

Find the exact values of s in the given interval that satisfy the given condition.

[0 , 2π) ; cos² s = 1/2

Key Concepts

Trigonometric Functions and Their Squares

Unit Circle and Angle Intervals

Solving Trigonometric Equations

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