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.

2y, given sec y = -5/3, sin y > 0

Accepted Answer

Welcome back. I am so glad you're here. We are asked to determine the exact values of the sine of two X and the cosine of two X. If the C can't of X equals negative 25 7th and sine of X is greater than zero. Our answer choices are answer choice. A sine of two X equals 172 divided by 525. And the cosine of two X equals negative 323 divided by 525. Answer choice B the sine of two X equals negative 363 divided by 625. And the cosine of two X equals negative 572 divided by 625. Answer choice C the sine of two X equals negative 172 divided by 525. And the cosine of two X equals 323 divided by 525. And answer choice D the sine of two X equals negative 336 divided by 625. And the cosine of two X equals negative 527 divided by 625. OK. So we know what the C can of X is. In order to find the sine of two X and the cosine of two X, we're going to use double angle identities for the sine of two X. We recall from previous lessons that that is going to be equal to two multiplied by the sine of X multiplied by the cosine of X. And then for the cosine of two X, we recall the double angle identity from previous lessons that the cosine of two X equals two cosine squared X minus one. So we need to know what the cosine and the sign are before we can plug those in. And since we're given the C can't of X, we recall from previous lessons that the C can't of X is equal to one divided by the cosine of X. Its reciprocal functions with the cosine. So we can just take our C can of X and we'll flip that around to its reciprocal. It'll be the cosine of X is equals to a negative 7 25th. All right. So our cosine is figured out, we just need to figure out s now we're going to switch gears and we're going to use a Pythagorean identity. We recall from previous lessons that the sine squared of X plus the cosine squared of X equals one. Well, we know the cosine of X. So we can solve for the sine squared of X. We'll plug this and we'll start with the sine squared of X plus. Now, our cosine of X is equal to negative 7 25th, but we need to square that and that's equal to one. Now, it's simplifying and solving for the sine of X. We'll bring down sine squared X for now, then plus negative 7 25th squared is going to be 49 625th that equals one. We want to subtract 49 625th from both sides. So on the left, we'll just have the sign squared of X equal to and on the right one minus 49 625th is going to simplify to be 576 625th. Now to get that sine of X by itself, we're going to take the square root of both sides and we'll have on the left, the sine of X, which is one thing we're looking for is equal to. And then on the right, it's plus or minus the square root of 576 625th simplifies to be 24 25th. But we were told back in the instructions that the sign of X is greater than zero. So we can disregard the negative answer and it's just going to be the sign of X equals 24 25th. OK? Now that we know both sine and cosine, we can plug those in to solve for the sine of two X and the cosine of two X. So we can start with the sine of two X sine of two X equals two, multiplied by the sine of X is 24 25th, multiplied by the cosine of X is negative 7 25th. So if you take all of those and multiply, you're going to get the sine of two X equals a negative 336 divided by 625. So we've got our sine of two X great. Now let's figure out the cosine of two X for the cosine of two X that equals two multiplied by. And our cosine of X is negative 7 25th. We need to square that and then we'll be subtracting one. So we'll start off squaring negative 7 25th. We know that that's a positive 49 625th that's still multiplied by two. And then we subtract one multiplying that by two, we'll get 98 625th minus one. And then doing that subtraction, we get that the cosine of two X equals a negative 527 625th. And there are our answer answers. We look at our answer choices and this matches with answer choice. D well done. We'll catch you on the next one.

Find values of the sine and cosine functions for each angle measure.
2y, given sec y = -5/3, sin y > 0

Key Concepts

Reciprocal Trigonometric Functions

Sign of Trigonometric Functions in Quadrants

Double-Angle Formulas

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