Expert: Josh - 4/17/2007

Question
Derive?
1. cos 3x
2. sin 4x
3. cos 4x

Answer
Let me show you one, you can attempt the rest yourself.

Remember that sin(a+b)=sin(a)cos(b)+cos(a)sin(b),
also, cos(a+b)=cos(a)cos(b)-sin(a)sin(b).

Rewrite cos(3x) as cos(2x+x)
Let a=2x, b=x in the formula above.
We have cos(2x+x)=cos(2x)cos(x)-sin(2x)sin(x) ...[#1]

Now, we need to find cos(2x) and sin(2x)
cos(2x)=cos(x)cos(x)-sin(x)sin(x)
      =[cos(x)]^2-[sin(x)]^2, using [sin(x)]^2=1-[cos(x)]^2
      =2[cos(x)]^2-1 ....[#2]

Similarly,
sin(2x)=sin(x)cos(x)+cos(x)sin(x)
      =2*sin(x)cos(x) ....[#3]

Substituting [#2],[#3] into [#1],
cos(2x+x)=(2[cos(x)]^2-1)*cos(x)-2cos(x)[sin(x)]^2
        =2[cos(x)]^3-cos(x)-2cos(x)*(1-[cos(x)]^2)
        =4[cos(x)]^3-3*cos(x)

Q2 and Q3 are done in a similar way. You must do these yourself.