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

Question

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

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

Accepted Answer

Hello, everyone. We are asked to find the exact value of the tangent of X plus Y using a trigonometric identity. Our cosine of X is 5/11. The sine of Y is negative +18, X is in quadrant four, Y as in quadrant three. Our answer choices are a negative 1280 radical six plus 363 radical seven. All of which is divided by 1479 B 11 radical three divided by 57 C negative 1280 radical six minus 363 radical seven of which is divided by 1479 D negative 11 radical three divided by 57. Since we are looking for the exact value of the tangent of X plus Y, I'm gonna rate that identity first and it is that the tangent of X plus Y equals the tangent of X lest, the tangent of lie divided by one minus the tangent of X tangent of Y. So we need to find the values of the tangent of X and the tangent of Y. So starting with X, we are told X is in quadrant four. So we have that the cosine of X is 5/11. And I recall that this is a ratio of the X value divided by the R value. If I were thinking about a coordinate plane and that for a tangent, we'd have the ratio of the Y value divided by the X value. And I also note that in quadrant four tangent is negative. So I know this will be a negative. I know the X value is five, but I need to find the Y value. I'm going to use the Pythagorean theorem. So X squared plus Y squared equals R squared. So we'll have five squared plus Y squared equals 11 squared. So 25 plus Y squared equals 121 isolating Y squared will equal 96. And if I take the square root of both sides, I get that Y equals plus or minus four radical six. So that's my numerator for my tangent. And since we know it's negative, it'll be a negative four radical six divided by five. And I'm going to add that up here just so I have it later. So the tangent of X is negative four radical six divided by five. Next, I need to find the tangent of why. So thinking about Yy as in quadrant three, we are given that the sign of why is negative 18. And again, thinking about those ratios, I know the ratio for sine was Y divided by R when I'm thinking about a coordinate plane. So tangent of Y again is gonna be Y divided by X. So I know the Y value here is one, I don't know the X value yet. And in quadrant three, I recall tangent is positive. So we can start solving our Pythagorean theorem X squared plus Y squared, Y is one squared equals R squared. So eight squared so X squared plus one equals 64 subtracting one from both sides, X squared equals 63 taking the square root of both sides. I get the X equals plus or minus three radical seven. So again, tangents positive. So I'm going to put a positive three radical seven in my denominator for tangent and then I'm going to bring it down to the bottom so I can rationalize it. So the tangent of why? Right now we have one divided by three radical seven when to multiply numerator and denominator by the square root of seven. And I get that the tangent of Y will be the square root of seven divided by 21. So I'm going to add that up here again. So we have it all in one place. So tangent of Y equals the square to seven divided by 21. So now I could start using that identity. So we have the tangent of acts so negative four radical six divided by five plus the tangent of Y. So the square root of seven divided by 21 divided by one minus the tangent of X. So negative four radical six divided by five multiplied by the tangent of Y. So the square root of seven divided by 21 in my numerator, I know I need a common denominator and it looks like our common denominator will be 105. So multiplying my first fraction numerator and denominator by 21 gives me negative 84 radical six divided by 105. The second one, I must multiply numerator and denominator by five. So it's plus five radical seven divided by 105 in the denominator. I'm gonna leave the one as is for now we've got one minus, it'll be negative four radical 42 divided by 105. I'm gonna combine everything. So I have a single fraction in my numerator and my denominator. So in the numerator, I have negative 84 radical six plus five radical seven divided by 105 in the denominator. I'm gonna treat one as if it's 105 divided by 105. I will have 105 plus four radical seven. sorry radical 42 divided by 105. Since both denominators in this complex fraction are 105 I can cancel them out. This leaves me with negative 84 radical six plus five radical seven divided by 105 plus four radical 42. So I need to rationalize that denominator, which means I need to multiply numerator and denominator here by the conjugate of 105 plus four radical 42. So that's 105 minus four radical 42. So when I do that, I'm going to foil my numerator and I'm gonna start by doing it over here. So that way I have a little bit more space. So first, I have negative 84 radical six multiplied by 105 which will give me negative 8820 radical six. So that was first outer, I'll have negative 84 radical six multiplied by negative four radical 42. And that will give me a positive 336 radical 252. So that was first outer inner, I have five radical seven multiplied by 105 and those are positive. So I have a positive 525 radical seven. And last, I have five radical seven multiplied by negative four radical seven radical 42. So that'd be a negative 20 radical 294. In my denominator, I have the product of a sum in a difference. So I know I can square the first term um and square the last term and put a minus sign between them. So 105 square is 11,025 and then minus and let's see four times or multiplied by 416. And then we're going to multiply that by 42 because radical 42 multiplied by radical 42 would be 42. So let's see what we can simplify in my numerator. I'm gonna say I probably can simplify the square root of 252 into something else. Now, I'm gonna see if it's divisible by six or seven since those are the other radicals I have in there and it is divisible by seven. And so we can factor out or reduce a square root of 36. So I'm gonna go down here for a little space. The first term isn't gonna change negative 8820 radical six. My second term, I am factoring out the square root of 36. So I need to do 336 multiplied by six. And that gives me 2016 radical seven and then we have plus 525 radical seven. And then looking at my last term here, I wanna know is 294 divisible by six. And if so, is it a perfect square? So let's find out it is, it'll leave us with radical six and we're gonna reduce the square to 49 out of it, which is seven. So seven multiplied by 120 is 140. So our numerator is down to having two separate kinds of radicals, which is great. Now, let's simplify our denominator when I do so the 11,025 minus 16 multiplied by 42 leaves us with 10,353. I want to combine my like radicals in my numerator. So when I do, I have negative 8960 radical six. And then for radical seven, we have a positive 2541. So radical seven divided by 10,353. So since this doesn't look like any of my answer choices yet, I'm gonna see if I can reduce the values that are in front of the radicals with the denominator when I do so, I don't think they're divisible by nine. So I'm gonna check seven because that's one of those numbers that sometimes sneaks up on me and it looks like seven works. So when I reduce this all by seven, I have negative 1280 radical six plus 363 radical seven divided by 1479. And I don't believe that can be reduced any further. And it actually does match one of our answer choices. So I'm gonna say we are done at this point and the answer is answer choice. A have a nice day.

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

Key Concepts

Trigonometric Ratios and Quadrants

Sum of Angles Formula for Tangent

Finding Missing Trigonometric Values

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