Find cos(s + t) and cos(s - t).
cos s = - 8/17 and cos t...

Question

Find cos(s + t) and cos(s - t).

cos s = - 8/17 and cos t = - 3/5, s and t in quadrant III

Accepted Answer

Hello, everyone. We are asked to determine the exact value of the cosine of X plus Y and the cosine of X minus Y. We are given that the cosine of X is negative 40 divided by 41. The cosine of Y is negative seven divided by 25 and X and Y are in the third quadrant. Our answer choices are a, the cosine of X plus Y equals 1023 divided by 1025. The cosine of X minus Y equals negative 897 divided by 1025. B, the cosine of X plus Y equals negative 897 divided by 1025. The cosine of X minus Y equals 1023 divided by 1025. See, the cosine of X plus Y equals 496 divided by 1025. The cosine of X minus Y equals 64 divided by 1025 D. The cosine of X plus Y equals 64 divided by 1025. The cosine of X minus Y equals 496 divided by 1025. OK. First, I want to write, what are the identities for the cosine of X plus Y? And I recall that that is the cosine of X multiplied by the cosine of Y minus the sine of X multiplied by the sine of Y. And at the same time, the cosine of X minus Y equals the cosine of X multiplied by the cosine of Y plus the sine of X multiplied by the sine of Y. Now that I know what identities I'm working with, I recognize that I need to find the sign of X. And the sign of Y lucky for us using a Pythagorean identity, we can solve that. The sign of, in this case, we use X will equal plus or minus the square root of one minus the cosine squared of X. And since we're in the third quadrant and S is negative there, we are only going to work with the negative root. So if I want to find the sign of X, I'm gonna say that equals negative square root of one minus. And then our cosine of X is negative 40 divided by 41 squared. So when I square that, that'll become a positive. So we have the sin of X equals the negative square root of one minus 40 squared, which is 1600 divided by 41 squared, which is 1681. So I'm going to change my one under the radical into 1681 divided by 1681. So I will have the sin of X equals the negative square root of 1681 divided by 1681 minus 1600 divided by 1681. And this will be the negative square root of 81 divided by 1681. And when I take the square root of both the numerator and denominator, I get that the sign will equal negative nine divided by 41. So we now know our sign of X, we need to find the sign of why before we can complete the identities. So for the sign of why we again are going to work with a negative because we're in the third quadrant. So it'll be the negative square root of one minus. Let's see our cosine of Y which was negative 7 25th squared. So the sine of Y will equal negative square root of one minus. So a negative squared is a positive. So we'll have 49 divided by 625. So now when I subtract, I'm going to change my one that's under the radical into 625 divided by 625 minus 49 divided by 625. When I combine them into one fraction, I get that sine Y is negative square root of 576 divided by 625. Taking the square root of both the numerator and the denominator. The negative stays out front, the square root of 576 is 24 and the square of 625 is 25. So we now know the sine of X and the sine of Y. So let's go find what is the cosine of X plus Y. So it's the cosine of X. So negative 40 divided by 41 multiplied by the cosine of Y which is negative seven divided by 25 minus the sin of X which we just found to be negative nine divided by 41 multiplied by the sine of Y which we found to be negative 24 divided by 25. So let's multiply when I multiply my first set of fractions, I get 280 divided by 1025. And then we're subtracting. And when I multiply my second set of fraction fractions negative nine divided by 41 and negative 24 divided by 25 I'll get a positive and it's a positive 216 divided by 1025. Subtracting these to make them one single fraction, we get 64 divided by 1025. So that is the cosine of X plus Y. We still need to find the cosine of X minus Y. So we'll do that next and do that right here, cosine of X minus Y. So everything is the same in these identities except for the middle sign. So I can just rewrite 40 sorry, negative 40 divided by 41 multiplied by negative seven divided by 25. Here's where the difference is. It's a plus and then negative nine divided by 41 multiplied by negative 24 divided by 25. All right. So multiplication will come out the same. It'll still be 280 divided by 1025. But now we're adding it to 216 divided by 1025. And when we do so we get that, the cosine of X minus Y equals 496 divided by 1025. And that does not appear to be able to be reduced. So that is the cosine of X minus Y looking in our answer choices to match both the cosine of X plus Y and the cosine of X minus Y. I get answer choice. D cosine of X plus Y is 64 divided by 1025. The cosine of X minus Y is 496 divided by 1025. Have a nice day.

Find cos(s + t) and cos(s - t).
cos s = - 8/17 and cos t = - 3/5, s and t in quadrant III

Key Concepts

Cosine Addition and Subtraction Formulas

Sign and Value of Trigonometric Functions in Quadrant III

Pythagorean Identity

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