Expert: Paul Klarreich - 3/29/2009

Question
I have puzzled over this question for several days and everything I've tried has gotten me nowhere. I know you say to post  how far I've gotten, but I haven't really been able to do anything that would lead to any where but a dead end. I'm in a high school precalculus class, and we are supposed to be verifying identities. My teacher wants us to use the formula sin(u+v)=sinucosv+cosusinv for the left side, and break sin 3x down into sin(2x+x).

Here's the problem:

sin3x=4sinxcos^2x-sinx

Thanks for the help!

Answer
Questioner:   Mckenzie
Country:  United States
Category:  Advanced Math
Private:  No
 
Subject:  Verifying identities
Question:  I have puzzled over this question for several days and everything I've tried has gotten me nowhere. I know you say to post  how far I've gotten, but I haven't really been able to do anything that would lead to any where but a dead end. I'm in a high school precalculus class, and we are supposed to be verifying identities. My teacher wants us to use the formula sin(u+v)=sinucosv+cosusinv for the left side, and break sin 3x down into sin(2x+x).

Here's the problem:

sin3x = 4 sin x cos^2(x) - sinx

Thanks for the help!
....................................
Teacher says do something, then do it:(who's the boss?)

sin(3x) = sin(2x + x) =

sin(2x) cos x + cos(2x) sin x  

Now you can apply your double-angle identities:

sin(2x) cos x       + cos(2x) sin x  =

2 sin x cos x cos x + (2 cos^2(x) - 1) sin x

Simplify a bit:

2 s c^2 + 2 c^2 s - s

Re-alphabetize:

2 c^2 s + 2 c^2 s - s

Simplify a bit more:

4 c^2 s  - s

Aha!