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

Question

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

Accepted Answer

Hello, everyone. We are asked to identify the quadrant in which X plus Y lies using trigonometric identities. We are told the sine of Y equals negative 3/7. The cosine of X equals negative 2/9 X is in the second quadrant Y is in the third quadrant. Our answer choices are a quadrant one, B quadrant two C quadrant three D quadrant four. So first, I'm going to use the trig identities that are the sign of a sum. So the sign of X plus Y equals the sine of X cosine of Y plus the cosine of X sine of Y. I'm also going to find the cosine of X plus Y and that has the identity, the cosine of X cosine of Y minus the sine of X sine of Y. And I think once I find what is positive and negative between sine and cosine, I'll have enough information to decide what quadrant this falls into. If not, I will then do the tangent. Also, before I can do that, I need to find the cosine of Y and the sign of X. So I'm gonna start by trying to find the sign of X I recall that we have a Pythagorean identity that states that the sine of X equals plus or minus the square root of one minus the cosine squared of X. Since X is in quadrant two and sin is positive in quadrant two, we only have to worry about the positive root here. So the sine of X is going to equal the square root of one minus our cosine of X which is negative 2/9 squared. So the sin of X will equal the square root of one minus and negative 2/9 squared will give me a positive 4, 80 firsts or four divided by 81. I'm going to rewrite this again. So the sin of X equals the square root of 81 divided by 81 minus four divided by 81. So I have the square root of 73 divided by 81. So the sin of X when I take the square root of the numerator and the denominator be the squared is 73 divided by nine, not my favorite number, but we'll work with it. Next, I want to find the cosine of why. And luckily we have a very similar identity. So the cosine of Y can equal plus or minus the square root of one minus the sine squared of Y. So the cosine of Yy is in the third quadrant. So this will be a negative route. So the cosine of Y equals negative radical one minus. What was our sign of Y negative 3/7? So negative 3/7 squared. So this means that the cosine of Y equals negative radical one minus 3/7 or negative 3/7 squared will give me 9 49th. I'm going to rewrite this. So I have the cosine of Y equals negative radical 49 49th minus 9 49th. So this would be negative radical 40 49th. When I take the square root of both the numerator and the denominator, I have negative radical 40 divided by 40. 0 sorry, the square of 49 is seven and I can reduce the square root of 40. So I'd have negative, let's say it's two radical five divided by seven. So now I have all the information I need to be able to use these identities to find out if sine plus, I'm sorry if find out if X plus Y is positive or negative in sine and cosine and determine the quadrant. So starting with the sign of X plus Y, the sine of X we said is the square root of 73 divided by nine multiplied by the cosine of Y which we just found to be negative two radical five divided by seven plus the cosine of X which is negative 2/9 multiplied by the sine of Y which is negative 3/7. So multiplying all of this together in our first fraction, the negative two will stay out front and then we will do radical five multiplied by radical 73 should be the square root of 365 divided by nine multiplied by seven, which is 63 plus negative 2/9 multiplied by negative 3/7 will give me 6 63rd. And this would be the same as saying six minus two radical 365 divided by 63. This appears to be positive, but maybe it's not. So I'm going to use a calculator to come up with an estimate. So I can make sure that this is either a positive or a negative. And when I do, I get an estimate of negative 0.511. So I guess it's good that I checked. So now we will do the cosine of X plus Y and that is the cosine of X which is negative 2/9 multiplied by the cosine of Y which is negative two radical five divided by seven minus the sine of X which is the squared is 73 divided by nine multiplied by the sine of Y which is negative 3/7. Let's multiply that out for my first set of fractions negative two multiplied by negative two radical five will give me four radical five divided by 63 in the second set I have minus and then we'll have a negative three radical 73 divided by 63 combining this into one fraction. My double negatives are like adding So I have four radical five plus three radical 73 divided by 63. And in this case, everything is positive. So I am comfortable saying that cosign is positive. So thinking about our coordinates, our coordinate plane and what quadrants have, what values I recall that quadrant one, all the signs are positive quadrant two only sign and Kent are positive quadrant three, tangent and cotangent are positive quadrant four cosine and C can are positive. So here we have a negative sine. So that means it is either quadrants three or four, but our cosine is positive. So this means this must be in quadrant four. So our answer choice is D quadrant four. Have a nice day.

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

Key Concepts

Trigonometric Ratios and Signs in Quadrants

Determining Angle Quadrants from Trigonometric Values

Sum of Angles and Quadrant Determination

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