Use the given information to find the quadrant of x + y.

Question

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 quadrant in which X plus Y lies using trigonometric identities. We are given the cosine of X equals 5/11. The sine of Y equals negative 1/8 X is in the fourth quadrant, Y as in the third quadrant, our answer choices are a quadrant one, B quadrant two C quadrant 3d quadrant four to begin with. I'm gonna think about what identities I wanna use. So since I'm given sign and cosine, I think I'm gonna work with the identities um that are the sum identities for X and Y. So the sign of the sum of X and Y is the sine of X multiplied by the cosine of Y plus, the cosine of X multiplied by the sine of Y. And I'm also going to work with the cosine of X plus Y and that's the cosine of X multiplied by the cosine of Y minus the sine of X multiplied by the sine of Y. And I'm going to do both because that way I can find out where X plus Y is positive versus negative and use that to figure out which quadrant. But before I can use those. I need to know the cosine of Y and the sine of X. So I'm gonna start with X and I recall we were told that the cosine of X is 5/11 and that X as in quadrant four and that this has a ratio of the X value divided by the R value. If I were to think of this on a coordinate plane using that same logic, the sin of X has a ratio of the Y value divided by the R value. So here we already know the R value is 11, but we need to find the numerator for our sign. Luckily, we can use the Pythagorean theorem to do this, we can say X squared plus Y squared equals R squared. So we are looking for Y so X is five, so five squared plus Y squared equals 11 squared. So 25 plus Y squared equals 121 subtracting 25 from both sides Y squared equals 90 six. And I will take the square root of both sides and I will have that Y equals plus or minus the square root of 96. And I can rewrite that or simplify it as plus or minus four radical six. So this is our Y value and since we were in quadrant four sine is negative, so this will be negative four radical six divided by 11. So I'm gonna write this up here. So I have it when I need it. So the sign of X is negative four radical six divided by 11. Next, I need to find the cosine of Y. So I'm going to do the same thing why we are told is in the third quadrant, we are told that the sine of Y equals negative 1/8. And again, that's the ratio of Y divided by R. So if I want to find the cosine of Y, that's the ratio of X divided by R. And we already know our R is eight. We're gonna use the Pythagorean theorem again. So X squared plus Y squared equals R squared. And we're looking for X so X squared is our unknown plus Y squared, which is one squared equals R squared. So eight squared, so X squared, X squared plus, sorry about that one equals 64 subtracting one X squared equals 63. I take the square root of both sides X squared. Oh I'm sorry, just X X equals plus or minus the square root of 63. And let's see if we can reduce that because I think we can. So that'd be plus or minus three radical seven. So when I put that into my cosine, um again, we're in quadrant three and cosine is negative in quadrant three. So this will be negative three radical seven divided by eight. So I'm gonna put that up here at the top too just so I have it easily accessible So the cosine of Y is negative three radical seven divided by eight. So now that I know all of that I can use these identities. So starting with the sine of X plus Y. So I have the sign of X which is negative four radical six divided by 11 multiplied by the cosine of Y. So negative three radical seven divided by eight plus the cosine of X which we were told was 5/11 multiplied by the sine of Y which is negative 1/8. So doing this multiplication in my first product, I have a denominator of 88. And in my numerator four multiplied by three positive 12. And then we have the square root of this should be 42. So we have 12 radical 42 divided by 88 plus. I now have 5/11 multiplied by negative 1/8 to give me negative 5 88th. And if I rewrite this as a single value or single fraction, I get 12 radical 42 minus five divided by 88. So this appears to be a positive value. So I know for sign the sign of X plus Y is positive. So this means if sin is positive, we are in either quadrant one or quadrant two. So let's see what we get when we do cosign. So the cosine of X plus Y is the cosine of X. So 5/11 multiplied by the cosine of Y which is negative three radical seven divided by eight minus the sine of X which is negative four radical six divided by 11 multiplied by the sine of Y which is negative 1/8. All right, let's see what happens when we multiply this together in the numerator, I get negative 15 radical seven and then divided by 88. My second set of fractions when I multiply I get a positive four radical six divided by 88. And I'm gonna use a calculator to estimate what these values are. This looks negative to me, but I'm gonna find an approximation so I can be certain. So when I approximate this negative 15 radical seven divided by 88 is approximately negative 0.451. And then we subtract the approximation for radical. I'm sorry for radical six divided by 88 which is 0.111. And when I subtract those, this is gonna be a negative. So the cosine of X plus Y is negative. So which quadrant is sine positive and cosine negative? Well, that is quadrant two. So that is our answer and it is answer choice B quadrant two have a nice day.

Use the given information to find the quadrant of 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 and Quadrant Determination

Using Pythagorean Identity to Find Missing Ratios

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