Use the given information to find each of the following.

Question

cos x/2 , given cot x = -3, with π/2 < x < π

Accepted Answer

Welcome back. I am so glad you're here. We are asked to determine the exact value of the given trigonometric expression using the half angle identity. Our given trigonometric expression is the cosine of X divided by two. And we have a couple conditions. We're told that the cotangent of X equals negative five and that X is greater than pi divided by two but less than pi. Our answer choices are answer choice. A negative square root of and inside of the radical, there is a fraction in the numerator. We have 26 minus five square root 26. And that's all divided by 52. Answer choice B we have the square root of and then a fraction inside of the radical. In the numerator, we have 26 minus five square root 26. And that's all divided by 52 answer choice C we have the square root of again, a fraction inside of the radical in the numerator 26 plus five square root 26 all divided by 52. And answer trace D negative square root of inside of the radical is a fraction in the numerator 26 plus five square root 26 all divided by 52. All right. So we're using a half angle identity. The half angle identity for cosine is that the cosine of a divided by two is equal to plus or minus the square root of. And then we have a fraction inside of the radical in the numerator. It's one plus the cosine of a all divided by two. Here, we can see comparing our given trigonometric expression with the half angle identity that are A equals X. So we can rewrite our cosine of X divided by two is equal to what the cosine of A divided by two is equal to plus or minus the square root of inside of the radical in the numerator, one plus the cosine of not A but X and that's all divided by two except what is the cosine of X. Well, we know the cotangent of X. Can we go from cotangent to cosine? Yes, we can. There are several ways to do that. I'm just choosing one of them. It's all right. If you use a different way, we recall from previous lessons that the co tangent of an angle, we'll say theta is equal to the X value divided by the Y value. And so here we're told that our cotangent is equal to negative five. And so that means we're going to have five divided by one and either our X value or our Y value is going to be negative but which one is it? Well, let's see our X angle is in between pi divided by two. And pi let's draw a quick sketch of a coordinate plane and see where that is. We've got a vertical Y axis, a horizontal X axis. They come together at the origin in the middle. The positive part of the Y axis is pi divided by two. And the negative part of the X axis is pi if we're in between that, we are in quadrant two and cotangent is negative in quadrant two. And we can see that our X values are negative in quadrant two. So, and that means it's a negative five divided by a positive one. So here we've identified our X is equal to negative five and our Y is equal to a positive one. Our cosine of theta is going to be equal to X divided by R. We know our X that's negative five. But we don't know our R yet, but we can solve for R because we recall from previous lessons that R equals the square root of X squared plus Y squared. So I'm going to move this to a different part of the screen we'll solve for, we'll solve for R real quick and then we'll get back to it. So R equals the square root of X is negative five squared plus Y is one squared, negative five squared is 25. So this is the square root of 25 plus one squared is 125 plus one is 26. So this is the square root of 26. Our R is the square root of 26. That one's always positive because it's a distance. So now we can plug that in for our cosine X divided by R is negative five divided by the square root of 26. And that would be lovely except we typically want to rationalize that denominator. So we're going to multiply both the numerator and the denominator by the square root of 26. So that our cosine is going to be negative five square root 26 divided by 26. Lovely. All right. So, but now we can go back to our half angle identity and plug this in. So our cosine of X divided by two is equal to plus or minus the square rut of and inside the radical is the fraction in the numerator. We have one plus. Now we have the exact value for the cosine of X which is negative five square root 26 divided by 26. And then that's all divided by two. So this, I mean now for, for the most part, we're just simplifying and it's kind of yucky simplification but it is what it is. So in the numerator, we have one plus negative five square root 26 divided by 26 we want to add those together one's a fraction. So we need a common denominator that one turns into 2626. And then when we add 2626 to negative five square root 26 divided by 26 that's just going to be 26 minus five square root 26 all divided by 26. So intuitive, right, that's still in the numerator of our fraction that's inside of the radical. So that's all divided by two, taking the square root of it and it's plus or minus. So now if we take our numerator 26 minus five square root 26 all divided by 26 and divide that by two, we are going to have plus or minus the square root of, we still have 26 minus five square root 26 in our numerator. And now it's divided by 26 multiply by two, which is 52 and by the grace of whatever, we don't have to rationalize this one. So thank you people who wrote this problem, we don't have to rationalize this, but we do have to figure out if it's a plus or a minus. So we're talking about the angle which is X divided by two, we know that X is between pi divided by two and pi. But to find out what X divided by two is between, we have to take our pi divided by two and divide that by two and our pi and divide it by two. And so that's going to simplify to be where our X divided by two is greater than pi divided by two. Divided by two is pi divided by four and pi divided by two is greater than our X divided by two. So where are we? Well, pi divided by two, we know is the positive part of the Y axis. Pi divided by four is in between the positive parts of the X and Y axis. So we are in quadrant one, everything, all of our trigonometric functions are positive and quadrant one. So that means we can disregard that negative results that we got. And it's just going to be a positive square, it of 26 minus five square root 26 all divided by 52. So we look at our answer choices and that one matches with answer choice. B well done. We'll catch you on the next one.

Use the given information to find each of the following.
cos x/2 , given cot x = -3, with π/2 < x < π

Key Concepts

Cotangent and Its Relationship to Sine and Cosine

Quadrant and Sign Determination

Half-Angle Formulas for Cosine

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