Expert: Paul Klarreich - 10/13/2010

Question
Please show me how to prove sin ^3 (x) = 3/4sin x - 1/4sin 3x. Thanks in
advance

Answer
Questioner:   Dan
Country:  United States
Category:  Advanced Math
Private:  No
 
Subject:  Trig
Question:  Please show me how to prove sin ^3 (x) = 3/4sin x - 1/4sin 3x. Thanks in
advance

You have:  
sin ^3 (x) = 3/4sin x - 1/4sin 3x

4 sin ^3 (x) = 3sin x - sin 3x  << mult out the 4

Now work out sin 3x using standard stuff:

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

2 sin x cos x cos x + (2 cos^2 x - 1) sin x =
2 sin x cos^2 x + 2 cos^2 x sin x - sin x =
4 sin x cos^2 x - sin x =
4 sin x (1 - sin^2 x) - sin x =
4 sin x  - 4 sin^3 x - sin x =
3 sin x  - 4 sin^3 x

Now you can finish up.