# Math Help - Deduce the value in surd form

1. ## Deduce the value in surd form

If $A=36^o$, show that $\sin 3A= \sin 2A$, and deduce that $\cos 36^o=\frac{\sqrt{5}+1}{4}$

I have already shown $\sin 3A= \sin 2A$ when $A=36^o$, since $108^o=180^o-72^o$.
But I don't know how to deduce that $\cos 36^o=\frac{\sqrt{5}+1}{4}$.
Thanks!

2. Hello arze
Originally Posted by arze
If $A=36^o$, show that $\sin 3A= \sin 2A$, and deduce that $\cos 36^o=\frac{\sqrt{5}+1}{4}$

I have already shown $\sin 3A= \sin 2A$ when $A=36^o$, since $108^o=180^o-72^o$.
But I don't know how to deduce that $\cos 36^o=\frac{\sqrt{5}+1}{4}$.
Thanks!
Four steps:

• Use the identities $\sin2A=2\sin A \cos A$ and $\sin3A=3\sin A -4\sin^3A$ in the equation $\sin 3A= \sin 2A$

• Divide through by $\sin A$ (it's non-zero, so that's OK)

• Replace $\sin^2A$ by $1 - \cos^2A$

• Solve the quadratic for $\cos A$, taking the positive root.

Can you complete it now?