Question

Express cos12a in terms of cos3a and in terms of sin3a

Answer 1

So far in terms of cos3a, I got 2(cos^23a-1)^2-1
and
in terms of sin3a:
1-2(2sin3acos3a)^2

Answer 2

@ganeshie8 @zepdrix

Answer 3

\[\begin{align}\cos(12a)&=\cos(2\cdot 6a)\\&=2\cos^2(6a)-1\\&=2[\cos(2\cdot 3a)]^2-1\\&=2[\color{red}{2}\cos^2(3a)-1]^2-1\end{align}\]

Answer 4

right ?

Answer 5

Oh I see. I wrote the identity wrong. Oops

Answer 6

for sin(3a) part, you may start with above result and simply replace \(\cos^2(3a)\) by \(1-\sin^2(3a)\)

Answer 7

cos(12a) in terms of sin(3a) : \[\begin{align}\cos(12a)&=\cos(2\cdot 6a)\\&=2\cos^2(6a)-1\\&=2[\cos(2\cdot 3a)]^2-1\\&=2[1-\color{red}{2}\sin^2(3a)]^2-1\end{align}\]

Answer 8

you need to know the double angle identity :
\[\cos(2t) ~=~ 2\cos^2(t)-1~=~1-2\sin^2t\]

Answer 9

Thank you @ganeshie8! I got it now! :D