Use the given information to find sin(x + y).
cos x = 2/...

Question

Use the given information to find sin(x + y).

cos x = 2/9, sin y = - 1, x in quadrant IV, y in quadrant III

Accepted Answer

Hello, everyone. We are asked to find the exact value of the sine of X plus Y using a trigonometric identity. We are given that the cosine of X is 5/11. The sine of Y is negative 1/8 X is in the fourth quadrant, Y is in the third quadrant. Our answer choices are a 12 radical 42 divided by 71 B negative five plus 12 radical 42 divided by 88 C negative 12 radical 42 divided by 71 D five minus 12 radical 42 divided by 88. To begin with. If I want to find the exact value of the sine of X plus Y, I wanna find the identity that goes with that. And that identity is that the sine of X multiplied by the cosine of Y plus the cosine of X multiplied by the sine of Y. So we have the cosine of X and the sine of Y. We need to find the sine of X and we also need to find the cosine of why. So let's do that. Looking at what we're given for X, we have the cosine of X equals 5/11. And we are told this is in the fourth quadrant. So in the fourth quadrant, our sign will be negative. And I recall that cosine is the ratio of the X value divided by the R value. And that s is the ratio of the Y value divided by the R value. So I could say the R value is 11. But I need to find the X value I can do this using the Pythagorean theorem. So X squared plus Y squared equals R squared. So X we have five, so five squared plus our unknown Y squared equals R squared. So 11 squared 25 plus Y squared. Well, 121 subtracting 25 from both sides, I get Y squared equals 96. When I take the square to both sides, I get that Y equals plus or minus the square root of 96 which is equivalent to plus or minus four radical six. Since we know sine is negative in quadrant four, the sin of X will be negative four radical six divided by 11. So I know this, I'm gonna move on and find the cosine of Y. So then I can put this all together. So cosine is in quadrant 30 I'm sorry, why is in quadrant three? And in quadrant three, our cosine will be negative. We know that the sine of Y equals negative 1/8. And again, I recall that sign is the ratio of the Y value divided by the R value. So to find the cosine of Y, I need the ratio of the X value divided by the R value, I know our denominator will be eight because that's our R value, I need to find the X value. So again, using a Pythagorean theorem X squared is our unknown plus Y squared. So one squared equals R squared which is eight squared. So X squared plus one equals 64 subtracting one from both sides, X squared equals 63 like the square to both sides X plus or minus the square root of 63. And that can be reduced to plus or minus three radical seven. Since we want our cosine to be negative, I'm going to say this is negative three radical seven divided by eight. Now that I know all of the parts I can use my identity. So the sign of X plus Y is going to equal the sine of X. So negative four radical six divided by 11 multiplied by the cosine of Y which is negative three radical seven divided by eight plus the cosine of X which is 5/11 multiplied by the sine of Y which is negative 1/8. So doing out that multiplication for the first set of fractions negative rad oh sorry, negative four radical six multiplied by negative three radical seven will give me a positive 12 radical 42 divided by 88 plus. My second set of fractions, five multiplied by negative one is a negative five divided by 88. I'm going to combine these into a single fraction. So I'll have negative five plus 12 radical 42 divided by 88. And I don't think there's anything else here that can be reduced. So this is our answer. It's a negative five plus 12 radical 42 divided by 88. And that's answer choice B have a nice day.

Use the given information to find sin(x + y).
cos x = 2/9, sin y = - 1, x in quadrant IV, y in quadrant III

Key Concepts

Trigonometric Identities

Quadrants and Signs of Trigonometric Functions

Finding Missing Trigonometric Values

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