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