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

Question

Use the given information to find sin(x + y), cos(x - y), tan(x + y), and the quadrant of x + y.

sin x = 3/5, cos y = 24/25, x in quadrant I, y in quadrant IV

Accepted Answer

Hello, everyone. We are asked to find the exact values of the sine of X plus Y, the cosine of X minus Y, the tangent of X plus Y and identify the quadrant in which X plus Y lies using trigonometric identities. We are told that the sine of X is 35 divided by 37. The cosine of Y is 15 divided by 17 X is in the first quadrant Y is in the fourth quadrant. So first, in order to use the identities for our sums and differences, I know I need to find the sine cosine tangent of both X and Y. So I'm gonna start with X. We are given that the sine of X equals 35 divided by 37. I recall that s is the ratio of A Y divided by R. If we were looking at this on a coordinate plane that the cosine of X would be the ratio of X divided by R and that the tangent of X would be the ratio of Y divided by X. So using the Pythagorean theorem, we know that X squared plus Y squared equals R squared. So we are told with sign Y would be 35 X is 37. I'm sorry, R is 37. We have to solve for X. So X squared plus 35 squared equals 37 squared. Doing out the squares, I get X squared plus 120 I'm sorry, 1225 equals 37 squared reaches 1369 to isolate the X squared. I'm subtracting 1225 from both sides, IX squared equals 144. And when I take the square root of both sides, X will equal plus or minus 12. So now going back to our ratios X will now be plus or minus 12, um we are told that X is in the first quadrant. So all of our ratios should be positive. So the cosine would be 12 divided by 37. So X divided by R, the tangent is Y which we know is 35 divided by X which we just found is 12. So we have some of our ratios. Now, I need to find the ratios for why? So using the same idea, we are given the cosine of Y is 15 divided by 17. We need to find the sign of why, which has a ratio of Y divided by R and the tangent of why has a weight ratio of Y divided by X. Again, the ratio for cosine is X divided by R. So once again, I'm using the Pythagorean theorem. Do we know our value for X? We do? It is 15. So 15 squared plus Y squared is what we don't know equals R squared. So 17 squared doing out the squares, 15 squared is 225 plus Y squared equals 17 squared, which is 289 subtracting 225 from both sides to isolate the Y squared, Y squared equals 64. Taking the square root of both sides. Y will equal plus or minus eight. So we're calling that we are in quadrant four, our only positive ratio should be cosine. So we're gonna be using a negative eight here a lot. So for the sine of Y, we have negative Y which is gonna be negative E divided by R which was 17 for tangent. Also that ratio is gonna be negative. So we'll use again negative eight divided by X which is 15. So negative 8/15. All right. So now we know all of our values that we need to be able to use the identities to find the sign of X plus Y cosine of X minus Y and tangent of X plus Y starting with the sign of X plus Y. So the identity here is that the sine of X plus Y is the sine of X multiplied by the cosine of Y plus the cosine of X multiplied by the sine of Y. Luckily, for us, all of our trig ratios are fractions with no radicals in them. So this should make these a little simpler than they could be. So the sine of X 35 divided by 37 multiplied by the cosine of Y. So 15 divided by 17 plus the cosine of X 12 divided by 37 multiplied by the sine of Y which is negative eight divided by 17, multiplying the fractions together the first fraction 35 times 15 or 35 multiplied by 15 will give us 525 in the denominator 37 multiplied by 17 is 629. For our second set of fractions, 12 multiplied by negative eight is negative 96 in the denominator 37 multiplied by 17 is 629. Combining these. Since we have a common denominator of 629 we will get that. We have 429 divided by 629. And that is the sign of X plus Y. So there's part of our answer. Next, we're gonna find the cosine of X minus Y. And that identity states that the cosine of X multiplied by the cosine of Y plus the sine of X multiplied by the sine of Y. So let's plug in our values. The cosine of X is 12 37th multiplied by the cosine of Y which is 15/17 plus the sign of X which was 35 37th multiplied by the sign of Y which is negative 8/17. So let's multiply this out. We know our denominator is gonna be 629 for both of them because we did 17 multiplied by 37 in our previous identity. So one last thing to multiply for our numerators 12 multiplied by 15 is 180. So our first fraction is 180 divided by 629. For our second numerator 35 multiplied by negative eight is negative 280. Again divided by 629. Combining these into one fraction, we get negative 100 divided by 629. And that does not appear to be able to be reduced. So the cosine of X minus Y equals negative 100 divided by 629. So almost there two thirds of the way through the problem one last identity to work with. And that would be the tangent of X plus Y. And this identity states that we have the tangent of X plus the tangent of Y divided by one minus the tangent of X tangent of Y multiplied together. So let's plug in our values. The tangent of X was 35 12th plus the tangent of Y which is negative eight divided by 15. All of that will be divided by one minus 35 divided by 12 multiplied by negative eight divided by 15. So for our numerator, we need to create a common denominator between 12 and 15. And I think that's going to be 60. So to get 12 to be 60 I need to multiply it by five. So I multiply my numerator also by five which will be 175 divided by 60 plus my second fraction to get a denominator of 60. I need to multiply 15 by four. So negative eight, multiplied by four is negative 32. That's my numerator right now in my denominator, I'm going to do the multiplication. So I have one minus and I'm gonna put in parentheses 35 multiplied by negative eight is negative 280. Then the denominator 12 multiplied by 15 is 180. All right, let's see what we can simplify here in the numerator. Now that I have a common denominator, I can rate them as one fraction and that would be 143 divided by 60 in our numerator in the denominator, I recall double negative. So one minus the negative is like adding and I'm gonna think of the number one as 180 divided by 180. So I have a common denominator of 180 and then 180 plus 280 but equal 460. So currently I have a fraction that in the numerator has 143 divided by 60 in the denominator has 460 divided by 180. So I'm gonna rewrite this horizontally. So 143 divided by 60 divided by 460 divided by 180. And I recall when I divide fractions I multiply by the reciprocal. So 143 divided by 60 multiplied by 180 divided by 460. And now I wanna reduce everything I can before I multiply. So in my second fraction, they both end in zeros. So I'm gonna cancel out a zero there just to simplify it a little bit. Looking at the ability to cross reduce, I noticed that six goes into both 60 and into 18. So 18 reduces to 360 reduces to 10. And I'm not sure if there's anything else here that will easily reduce. So I'm gonna multiply at this point and then we might have to simplify one more time when I multiply across 143 multiplied by three is 429. 10, multiplied by 46 is 460. That cannot be reduced any further. So the tangent of X plus Y equals 429 divided by 460. And then the last piece we need to answer here is what quadrant X plus Y lies in. So X plus Y, its sign is positive and its tangent is positive. And the only place that happens is in quadrant one. So this is in the first quadrant, I'm gonna rate it as quadrant one. So that is the final piece of this problem. Have a nice day.

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Use the given information to find sin(x + y), cos(x - y), tan(x + y), and the quadrant of x + y.
sin x = 3/5, cos y = 24/25, x in quadrant I, y in quadrant IV