SOLUTION: 36. sin(u-v) sin u=5/13 cosv=-3/5 find the exact value

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Question 174442: 36. sin(u-v) sin u=5/13
cosv=-3/5
find the exact value
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!
It is a little difficult to decipher what you are asking here. I think you mean find the exact value of sin%28u-v%29 given that sin%28u%29=5%2F13 and cos%28v%29=-3%2F5

First of all, you need the difference identity: sin%28alpha-beta%29=sin%28alpha%29cos%28beta%29-cos%28alpha%29sin%28beta%29.

Your problem is that you now need to determine cos%28u%29 given that sin%28u%29=5%2F13 and sin%28v%29 given that cos%28v%29=-3%2F5.

Knowing that sin%28u%29%3E0, you are certain that u is an angle in the First or Second Quadrants. Remembering that a 5-12-13 triangle is a right triangle and the fact that the sine function is defined as opposite/hypotenuse, we can see directly that cos%28u%29=12%2F13 (if u is First Quadrant) or cos%28u%29+=+-12%2F13 (if u is Second Quadrant).

Similarly, v must be Quadrant II or III because cos%28v%29%3C0 and 3-4-5 is a right triangle, so sin%28v%29=4%2F5 or sin%28v%29=-4%2F5

All you need to do is plug the values into sin%28alpha-beta%29=sin%28alpha%29cos%28beta%29-cos%28alpha%29sin%28beta%29 or, in the case of this problem sin%28u-v%29=sin%28u%29cos%28v%29-cos%28u%29sin%28v%29, and do the arithmetic. Because there are two possible values for each of the derived single angle functions, you will have 4 different calculations to perform, however you will end up with two pairs of identical results -- hence you will have two answers in the end.