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

Question

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

sin y = - 2/3 , cos x = - 1/5, x in quadrant II, y in quadrant III

Accepted Answer

Hello, everyone. We are asked to find the exact value of the sign of X plus Y using a trigonometric identity. We are provided that the sine of Y equals negative 3/7 and the cosine of X equals negative 2/9. We are also told that X is in the second quadrant and Y is in the third quadrant. Our answer choices are a six minus two radical 770 divided by 63 B two radical 115 divided by 63 C negative six minus two radical 770 divided by 63 D negative two radical 115 divided by 63. First, I want to write down what is the sum identity for S so the sign of X plus Y will equal the sine of X multiplied by the cosine of Y plus the cosine of X multiplied by the sine of Y. So at this point, we are given the sine of Y and the cosine of X. But we also need to find the cosine of Y and the sine of X. So I'm gonna start with X again. I know the cosine of X is negative 2/9. And I recall that this is the ratio of the values X to R X divided by R. And that for the sin of X, I can look for the ratio Y divided by R. And that will help me find the sign. I recall that these have a Pythagorean identity that says YX squared plus Y squared equals R squared. So we use that to find our Y value to find the value of sine. So X is going to be two. So two squared plus Y squared equal R squared, which is nine squared. So four plus Y squared equals 81 subtracting four from both sides, I get Y squared equals and it looks like it'll be 77 taking the square root of both sides Y will equal plus or minus the square root of 77. So now looking back at our original problem, we are told we're in quadrant two for X and in quadrant two, S is positive. So I'm gonna use a positive seven Ds um sorry, positive square to 77 divided by nine. So I now know the sign of X, I now need to find the cosine of Y. So I'm gonna use the same process. We were told that the sign of Y is negative 3/7. And again, I'm going to use the same ratios where Y is divided by R for S and then in cosine, we're going to look for X divided by R and using the Pythagorean theorem, I don't know X so X squared is my unknown plus Y squared. So three squared equals seven squared. So R squared, which is seven squared, so X squared plus nine equals 49 subtracting nine from both sides, X squared is gonna equal 40 taking the square root of both sides. X will equal plus or minus the square root of 40 which I can rewrite as plus or minus two radical 10 when I reduce it. So now back up to our cosine, we're told Y is in the third quadrant and in the third quadrant cosine is negative. So this will be a negative two radical 10 divided by R which is seven. So now that we have all of our values for the sine of X, the cosine of X, the sine of Y, the cosine of Y. I can use the identity. I wrote to begin with that the sine of X plus Y is equal to the sin of X cosine of Y plus the cosine of X sine of Y. So I'm gonna plug in the values. So the sine of X we said was the square root of 77 divided by nine, multiplied by the cosine of Y which is negative two radical 10 divided by seven. And that will be added to the cosine of X was negative 2/9 multiplied by the sine of Y which was negative 3/7. All right, let's multiply our groups of fractions. So my first product, when I do negative two radical 10 multiplied by the square root of 77 I will get negative two radical 770 divided by 63 because nine multiplied by seven is 63. My second pair of fractions that need to be multiplied negative 2/9 multiplied by negative 3/7 will give me a positive six divided by 63. And now I'm going to rewrite this as one fraction and we'll have the sine of X plus Y equals. And I'm gonna put the six first then minus two square root of 770 all of which is divided by 63. And that is our exact value. It is not the nicest looking value because I don't want to work with the square root of 770 but it is the exact value and it is answer choice. A six minus two radical 770 divided by 63 have a nice day.

Use the given information to find sin(x + y).
sin y = - 2/3 , cos x = - 1/5, x in quadrant II, y in quadrant III

Key Concepts

Sum of Angles Formula for Sine

Determining Signs of Trigonometric Functions by Quadrant

Using Pythagorean Identity to Find Missing Values

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