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

Question

Use the given information to find tan(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 tangent of X plus Y and use a trigonometric identity. We are provided that the sine of Y equals negative 3/7 the cosine of X equals negative 2/9 X as in quadrant two and Y as in quadrant three, our answer choices are a nine radical 10 divided by eight B two multiplied by the parentheses negative 49 radical 77 plus 243 radical 10. All divided by 533 C negative nine radical 10 divided by eight D negative two multiplied by the parentheses, negative 49 radical 77 plus 243 radical 10, all of which is divided by 533. First, let's write the tangent of X plus Ys identity. And that is the tangent of X plus the tangent of Y divided by one minus the tangent of X tangent of Y. So we are given the sign and cosines of both X and Y. So let's find the tangents starting with X I know X is in quadrant two and that means our tangent will be negative. So if the cosine of X is negative 2/9. This means we have a ratio of an X value divided by an R value when I think about the coordinate plane. So the tangent of X will be the ratio of the Y value divided by the X value. So we know our X value is two. We don't know our Y value yet. We can use the Pythagorean theorem to do this though. So X squared plus Y squared equal R squared. We know X is two. So two squared plus our unknown Y squared equals R squared. So nine squared four plus Y squared will equal 81. Solving for Y squared, we get Y squared equals 77 and I take the square root of both sides, Y will equal plus or minus the square root of 77. And as we stated, tangent is negative here. So this would be negative radical 77 divided by two. I'm gonna place that up top. So I have it when I need it. So the tangent of X is negative radical 77 divided by two. Next I want to find or Y value. So for why we know we are in quadrant three and tangent is positive in quadrant three, we are given the sign of why is negative 3/7. And just like before we're going to use ratios. So this is Y divided by R. So the tangent of Y again is still Y divided by X here, we know our Y value is three, but we need to find our X value. So we'll do X squared plus three squared equals seven squared. So X squared plus nine equals 49. Solving for X squared, we have X squared equals 40 taking the square root of both sides X will equal plus or minus two radical 10. So plugging that in we know tangent is positive, so we have three divided by two radical 10. I want to rationalize that by multiplying numerator and denominator by 10. So we get that the tangent of Y is three radical 10 divided by 20. So again, I'm gonna add that up top. So I have it for my identity. So the tangent of Y equals three radical 10 divided by 20. So now let's work on plugging those things into this identity. So the tangent of X plus Y is gonna equal the tangent of X. So negative radical 77 divided by two plus the tangent of Y. So three radical 10 divided by 20 divided by one minus the tangent of X. So negative radical 77 divided by two multiplied by the tangent of Y. So three radical 10 divided by 20 in my numerator, I notice that I have an easy common denominator of 20. So I'm multiplying my first fraction top and bottom by 10. So I have negative 10 radical 77 divided by 20 plus three radical 10 divided by 20 in the denominator, we leave the one as is so one minus and then multiplying these together, I get negative three radical 770 divided by 40 gonna combine the numerator into one fraction. So I have negative 10 radical 77 plus three radical 10 divided by 20. In my denominator, I'm going to treat one as 40 divided by 40. So I can make a single fraction that is 40 plus three radical 770 divided by 40. Now, I want to reduce the denominators. Um I'm gonna do that by rewriting this as a division problem horizontally. So I have negative 10 radical 77 plus three radical 10 divided by 20. And I'm multiplying by the reciprocal of that bottom fraction. So multiplied by 40 divided by 40 plus three radical 770. Cross reducing 40 becomes a 2, 20 becomes a one. So I have two multiplied by the parentheses. Negative 10 radical 77 plus three radical 10, close parentheses divided by 40 plus three radical 770. So now I need to rationalize my denominator. So I'm gonna multiply numerator and denominator by the conjugate which is 40 minus three radical 770 in our numerator, there's not really a better way to do this than to foil it out. Um I'm gonna leave the two outside of the parentheses. So I have two open parentheses and then using foil first gives me negative 400 radical 77 outer gives me a positive 30 radical, 59,290 inner gives me a positive 120 radical 10. And last gives me a negative nine radical 7700 closed parentheses in my denominator, it's the product of a sum and a difference. So 40 squared is 1600 minus and we have nine from three, multiplied by three. And then we multiply that by 770 because a radical times itself leaves us what's under the radical. So let's see what we can simplify. So we have two and then the parentheses negative 400 radical 77 that can't be simplified any further. In my next term, 30 radical 59,290 77 squared actually goes evenly into this as it is 5929. So that means we will multiply 30 by 77 and get 2310 outside of our radical. And under all we're left with is 10, which is kind of nice plus 120 radical 10. And then for our last term, we have a minus and I noticed that the perfect square of 100 will go into 7700. So that means I'm multiplying my nine by 10. So there'll be a negative 90 radical 77 closed parentheses divided by, when I do 1600 minus nine times multiplied by 770 I get a negative 5000 330. So now I want to see is there anything I can reduce? And it looks like all the terms outside the radicals end in zero, my denominator ends in zero. So I can say I'm factoring a 10 out and canceling it. And when I do, I'm going to pull that negative sign out front. So I have negative two multiplied by, we will have negative 40 radical 77 plus 231 radical 10 plus 12 radical 10 minus nine radical 77. Close parentheses divided by 533. And last thing I'm doing is combining my lake radicals and I will have negative and then two in parentheses, negative 49 radical 77 plus 243 radical 10 all which is divided by 533. And I think we got there, we found the end. So the answer is negative and then two times the parentheses negative 49 radical 77 plus 243 radical 10 all divided by 533. And that is answer choice. D have a nice day.

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

Key Concepts

Trigonometric Identities for Sum of Angles

Determining Signs of Trigonometric Functions in Quadrants

Finding Missing Trigonometric Ratios Using Pythagorean Identity

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