Use the given information to find each of the following.

Question

sin A/2, given cos A/2 = - 3, 90° < A < 180°

Accepted Answer

Welcome back. I am so glad you're here. We are asked to determine the exact value of the sine of A divided by two. If the cosine of A equals negative 8/15 and A is greater than 90 degrees, but less than 180 degrees. Our answer choices are answer choice. A square root 690 divided by 30 answer choice B two square root 10 divided by 15 answer choice C negative square root 690 divided by 30 answer choice D negative 10 square root or excuse me, negative two square root 10 divided by 15. All right. So a couple things to note here, we're looking for the s of half of A. So this is going to use a half angle identity. We recall from previous lessons that the sine of A divided by two is equal to plus or minus the square root of. And then in the numerator, we have one minus the cosine of A, all divided by two. And that's all within the radical. And we know that A is going to be in between 90 degrees and 180 degrees. But we're looking for half of A and what we can do is we can take our 90 degrees, our A and our 180 degrees and divide those all by 2, 90 degrees divided by two is 45 degrees, which is less than a divided by two, which is less than 180 degrees divided by two, which is 90 degrees. So there we can see 45 degrees to 90 degrees. We're in quadrant one and our A divided by two is going to be positive for all of the trigonometric functions. So we know what our cosine of A is. We can plug that in and we can simplify one step further knowing that our answer, our sine of A divided by two is going to be positive. And so we can disregard that plus or minus. It's just going to be a positive answer. So let's take the square root of in the numerator, one minus cosine of A is negative 8/15. That's all divided by two. Bring over the radical in the numerator, one minus 815, 1 minus negative 8/15 is one plus 8/15 and one plus 8/15 is going to be 23 15th, that's divided by two, taking 23 15th divided by two. That's going to be 23/30. So we have the square of 23/30. But let's rationalize this denominator multiply both the numerator and the denominator by the square of 30 in the numerator the square root of 23 multiplied by the square root of 30 is the square root of 690. That's divided by the square root of 30 multiplied by itself is 30. And that's it. That's our sign of a divided by two. 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 A/2, given cos A/2 = - 3, 90° < A < 180°

Key Concepts

Half-Angle Identities

Sign Determination in Quadrants

Pythagorean Identity

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