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nailpolishfanatic's avatar

Math help : how do I solve out sin (u+v) + sin(u-v)?

Asked by nailpolishfanatic (6590 points ) February 15th, 2013

My question pretty much says it all. I’m stranded on this very problem. The answer is supposed to be 2sinu x cosv and I just don’t understand how we get the last answer.
Simplify : Sin ( u + v) + Sin (u – v)

Sin ( u + v ) = sin u * cos v + cos u * sin v
Sin ( u – v ) = sin u * cos v – cos u * sin v
– sin u * cos v + cos u * sin v + sin u * cos v – cos u * sin v
– 2sin u * cos2v * sin2v
this is as far as I can go, I cannont figure out how the cos v comes in the answer. Please help, I’ve got a lot of these types of exercises awaiting me ahead.

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4 Answers

LostInParadise's avatar

Check your algebra. When you substitute into sin(u+v) + sin(u – v), the sin u * cos v terms add together and the cos u * sin v terms cancel.

gasman's avatar

@nailpolishfanatic You wrote…

Simplify : Sin ( u + v) + Sin (u – v)
Sin ( u + v ) = sin u * cos v + cos u * sin v
Sin ( u – v ) = sin u * cos v – cos u * sin v

…which is totally right. Then you made mistakes in the lines after that. It’s probably easier than you think.
If you simplify the notation by substituting x = sin u cos v and y = cos u sin v
then your own formulas read:
Sin ( u + v ) = x + y
Sin ( u – v ) = x – y

So the sum of these quantities on the left yields, on the right, x+y+x-y = 2x
and 2x is shorthand for 2 sin u cos v.

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