Find values of the sine and cosine functions for each angle me...

Question

Find values of the sine and cosine functions for each angle measure.

B, given cos 2B = 1/8 , 540° < 2B < 720°

Accepted Answer

Welcome back. I am so glad you're here. We are asked to determine the exact values of the sine of A and the cosine of A. If the cosine of two A equals 2/7 and two A is greater than 540 degrees, but less than 720 degrees. Our answer choices are answer choice A, the sine of A equals the square root of 51 divided by 26. And the cosine of A equals negative square root 11 divided by 26. Answer choice B the sine of A equals the square root of 22 divided by 15. And the cosine of A equals negative square root seven divided by 15. Answer trace C, the sine of A equals negative square root 70 divided by 14. And the cosine of A equals three square root 14 divided by 14. And answer choice D, the sine of A equals negative square root 34 divided by 21. And the cosine of A equals negative two square root 17 divided by 21. All right. So a few different directions we need to go in here. Let's start off by taking a look at the restrictions of where our angle is, we know that two A is going to be between 540 degrees and 720 degrees. But how about a? Well, we can solve that by dividing everything here by two. And so 540 degrees divided by two is 270 degrees, which is less than two A divided by two, which is a, which is less than 720 degrees divided by two, which is 360 degrees. So we're between 270 degrees and 360 degrees. We recall from previous lessons, if we've got a coordinate plane with a vertical Y axis and a horizontal X axis, they come together at the origin in the middle. The negative part of the Y axis is 270 degrees and the positive part of the X axis is 360 degrees. If we're in between that, we are in quadrant four. Now we're looking for sine and cosine. We recall from previous lessons that in the fourth quadrant sine is going to be negative and cosine is going to be positive. So that'll be important in a little bit. Now, the other thing we wanna take a look at here as we've got the cosine of two a equal to something. Well, there is a double angle identity that we can use to help solve this. We'll use two of them. One for sine and one for cosine. We recall from previous lessons. One of the double angle identities is that the cosine of two A equals one minus two sine squared A. There's one of our double angle identities and the other one we'll use, we'll just have cosine on the right hand side. So the cosine of two A equals two cosine squared A minus one. So here we know that the left hand side, the cosine of two A is equal to 2/7. So we can substitute in for both of these. On the left hand side, we can say that 2/7 is equal to. And then for the s we've got 2/7 equals one minus two S squared A. And for the cosine 2/7 equals to cosine squared A minus one. Now it's just solving for a, so well, for the sign of A and the cosine of A, so let's do the work on the sign first. So we can subtract one from both sides of the equation we'll have on the left 2/7 minus one that's equal to on the right, a negative two sine squared A 2/7 minus one is going to give us a negative 5/7 equals the negative two sine squared A. And now we can divide both sides by a negative two. And then on the left that's going to give us now a positive 5/14 equals on the right sine squared A and we just need to take the square rut of both sides. Oops I'm drawing over my own stuff, the square of both sides in order to get rid of that squared for the sine squared. So this is leaving us with the sine of A equals the plus or minus because we took the square rut of 5/14. Now we can do a little bit of simplification here. One, we know from what we said before that in quadrant four S is going to be negative. So we can disregard the positive result for our plus or minus. And we know this is just going to be a negative square at 5/14 but we do need to rationalize that. And so what we're going to do is multiply both the numerator and the denominator by the square root of 14. And then this one will simplify, we'll have the sign of A equals negative. And then in the numerator, we have the square rut of five multiplied by 14, which is 70 and that's divided by, in the denominator, the square root of 14 multiplied by itself is just 14. And there's our sign. Now we can go through the same process and solve for cosine. So again, it's 2/7 equals two cosine squared A minus one will add one to both sides on the left, 2/7 plus one is nine sevenths that equals two cosine squared A, we can divide both sides by two. And then we'll have on the left 9/14 and that's equal to the cosine squared of A, we'll take the square rut of both sides and then we'll have the cosine of a equals plus or minus the square root of 9/14. Well, we can take the square root of nine, that's going to be a three in the numerator that's still over the square root of 14. How about the plus or minus? Well, we said before that cosine is going to be positive in the fourth quadrant. So here we can disregard the negative and it'll just be a positive three divided by the square root of 14. But again, we want to rationalize this. So we'll multiply both the numerator and the denominator by the square root of 14. And so this will simplify to be the cosine of a equals three square at 14 all divided by wart. There's our cosign. Now we look at our answer choices and this matches with answer choice. C well done. We'll catch you on the next one.

Find values of the sine and cosine functions for each angle measure.
B, given cos 2B = 1/8 , 540° < 2B < 720°

Key Concepts

Double-Angle Identities

Angle Measurement and Quadrants

Inverse Trigonometric Functions and Solving Equations

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