Use the given information to find each of the following.

Question

sin x/2 , given cos x = - 5/8, 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 expression is the sine of X divided by two. And we're given a couple of conditions. We're told that the cosine of X equals negative 17 divided by 21. And that X is greater than pi divided by two, but less than pi. Our answer choices are answer choice. A square root 399 divided by 21. Answer choice B negative square root 399 divided by 21. Answer choice C negative seven H no negative square root 798 divided by 21 and answer choice D square root 798 divided by 21. All right. So we're told to use a half angle identity. And what we're given looks a lot like a half angle identity. We recall from previous lessons. The half angle identity for S is where the sign of a divided by two is equal to plus or minus the square root of and inside the radical is a fraction in the numerator. It's one minus the cosine of A. And that's divided by two. Here, we can see the parallel that our A is equal to X. So we can substitute in the right hand side of this. Our sine of X divided by two is equal to plus or minus the square root of. We have the radical in the numerator, we have one minus. But we do have an exact value for the cosine of X which is the same as cosine of a, that's negative 17 divided by 21. So we can put that in negative 17 divided by 21 for the cosine of X or the cosine of A and that's all divided by two. So now it's a lot of simplification, we'll take this numerator one minus negative 17 divided by 21 off to the side. Well, that's the same as one plus 17 divided by 21. And if we add those two fractions one plus 17 divided by 21 is going to be 38 divided by 21. So we can put in here, the sine of X divided by two is equal to plus or minus the square root of our numerator simplifies to be 38 divided by 21 and that's still divided by two. So if we come back off to the side and working inside the radical 38 divided by 21 divided by two is going to simplify to be 19 divided by 21. So we can go back up put that in, this is equal to plus or minus the square root of 19 divided by 21. And that looks lovely. But we need to radicalize our denominate, rationalize our denominator, not radicalize it. So we are going to get rid of the radical in the denominator by multiplying the numerator and the denominator by the square root of 21. And when we do that, this will be equal to plus or minus in the numerator, the square root of 19 multiplied by the square root of 21 is the square root of 399 divided by, in the denominator, the square rut of 21 multiplied by the square root of 21 is just 21. And that looks nice, but we're not done yet. We need to figure out this plus or minus. Is it plus or is it minus? So here's where we look back to the fact that X is in between pi divided by two and pi. However, we're not talking about X, we're talking about X divided by two. Our angle is X divided by two. So what we want to do is take our X is greater than pi divided by two, but less than pie. But since we're talking about X divided by two, we're going to divide pi divided by two by two and pi by two. So this simplifies to be that X divided by two is greater than pi divided by two. Divided by two is pi divided by four, but X divided by two is less than pi divided by two. So where are we? Well, if we recall from previous lessons, we've got a vertical Y axis, a horizontal X axis, they come together the origin in the middle pi divided by two is the positive part of the Y axis. And pi divided by four is midway in between the positive part of the X axis and the positive part of the Y axis. So we're talking about an area that is in quadrant one and in quadrant one, all of our trigonometric functions including sine are going to be positive. So that means we can disregard this negative value that we've been carrying around. And the sign of X divided by two is only equal to a positive square at 399 divided by 21. And there is our exact value. We look at our answer choices and this matches with answer choice. A well done. We'll catch you on the next one.

Use the given information to find each of the following.
sin x/2 , given cos x = - 5/8, with π/2 < x < π

Key Concepts

Trigonometric Identities

Quadrants and Sign of Trigonometric Functions

Cosine Function and Its Range

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