   SEARCH HOME Math Central Quandaries & Queries  hi, i...indeed,to..need..your..help how..cn..i..prove..this, formula??? sin3a=3sina-4sin^3a cos3a=4cos^3-3cosa tan3a=3tan-tan^3a /1-3tan^2a the..symbol...^is..the..expOnent We have two responses for you.

Mailene,

If you know expressions for sin(p + q), cos(p + q) and tan(p + q) you can use these. For example

sin(p+q) = sin(p)cos(q) + cos(p)sin(q) sin(3a)

and hence for sin(3a) write

sin(3a) = sin(2a + a) = sin(2a)cos(a) + cos(2a)sin(b)

Now expand sin(2a) and use the expression for cos(p + q) to expand cos(2a). Simplify the final expression to get the form you want.

I hope this helps,
Harley

Hi mailene,

I'll give you a strategy for the first one, and then see if you can figure out the others, since they are similar.

Required to prove: sin(3a)=3sin(a)-4sin^3(a)

Strategy:
Start with sin(3a) = sin(2a + a) and expand it using the addition formula for sine:
sin(x + y) = sin(x)cos(y) + sin(y)cos(x) (use x = 2a, and y = a)

Following this, apply double angle formulae. For cos(2a) remember that you are trying to get a final expression in terms of sin(a), so choose the appropriate double angle formula.

sin(2a) = 2sin(a)cos(a)
cos(2a) = cos2(a) - sin2(a)
cos(2a) = 2cos2(a) - 1
cos(2a) = 1 - 2sin2(a)

You may also need to use the identity cos2(a) + sin2(a) = 1 to get everything in terms of sin(a).

Hope this helps,
Haley     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.