Use the given information to find each of the following.

Question

sin y, given cos 2y = -1/3 , π/2 < y < π

Accepted Answer

Welcome back. I am so glad you're here. We're asked to determine the exact value of the sine of X. If the cosine of two X equals negative to ninth and X is greater than pi divided by two, but less than pi. Our answer choices are answer choice. A five square root five divided by six. Answer choice B three square root 22 divided by 11 answer choice C two square root six divided by five and answer choice D square root 22 divided by six. All right. So we've got to go in a couple different directions here. Let's start off taking a look at our domain restriction. We're told that X is going to be between pi divided by two and pi. If we draw a quick sketch of a coordinate plane, we've got a vertical Y axis and a horizontal X axis. They come together 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. So between pi divided by two and pi, that means we're in quadrant two and we're looking for the exact value of the sine of X. So we recall from previous lessons that sine is positive in quadrant two, that's something important we'll need in a little bit. Now, let's look at the cosine of two X equals negative two divided by nine. We want to get from the cosine of two multiplied by something to the sign of that same something. And in order to do that, we're going to use a double angle identity, we recall from previous lessons that the cosine of two multiplied by. We use X equals one minus two sine squared of that same something that X. So here this will get us from cosine of two X to sine. And we can set our one minus two sine squared X equal to the same thing that the cosine of two X is equal to which is negative 2/9. And then we solve for that sin of X. So we'll start off solving for the sine of X by subtracting one from both sides on the left, we've got negative two sine squared X equals on the right, negative 2/9 minus one equals negative 11 9th. Now we're going to divide both sides by negative two on the left, we just have sine squared X that's equal to on the right negative 11 9th divided by negative two is going to be a positive 11/18. Now we just need to take that square rut of both sides. And when we do that on the left. Now, we've got the sine of X, which is what we're looking for a yay that equals on the right. Normally we take a square rut and we do a plus or minus. But we already talked about how sine is positive in the second quarter. So we can disregard the negative. And we know that we're only going to have that positive square rut of 11/18 for our answer. And speaking of the positive square rut of 11/18 we want to rationalize that denominator. So we'll multiply both the numerator and the denominator by the square root of 18. And now we'll simplify the square root of 11 multiplied by the square root of 18 is going to simplify to be three square root 22 divided by, in the denominator square root 18 multiply by itself is 18 and there are 3/18 is going to simplify to be 1/6 1 6th. So this is going to be the square rut of 22 divided by six. There's our sign of X, we look at our answer choices and this matches with answer choice. D well done. We'll catch you on the next one.

Use the given information to find each of the following.
sin y, given cos 2y = -1/3 , π/2 < y < π

Key Concepts

Double-Angle Identity for Cosine

Sign of Trigonometric Functions in Quadrants

Pythagorean Identity

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