How do you simplify this complex number with Sin and Cos

Question

Simplify this expression with complex number $32(\cos\frac34\pi+i\sin\frac34\pi)$.

I want the form $a+bi$ but I do not know well the exact trigonometric values.

Answer 1

You need the identities $\cos(\pi-x)=-\cos x,\,\sin(\pi-x)=\sin x$. Since $\cos\frac{3\pi}{4}=-\cos\frac{\pi}{4}=-\frac{1}{\sqrt{2}}$ while $\sin\frac{3\pi}{4}=\sin\frac{\pi}{4}=\frac{1}{\sqrt{2}}$, the number is $16\sqrt{2}(-1+i)$.