infimum and supremum of $(sin(a))^n and (cos(a))^n$

Question

infimum and supremum of $(sin(a))^n$ and $(cos(a))^n$ with $a\in \mathbb{R}, n\in \mathbb{N}$

How do you show that $(sin(a))^n$ and $(cos(a))^n $ have supreme and infimum?

Answer 1

Since $\sin a,\,\cos a$ each have images $[-1,\,1]$ on $\Bbb R$, $\sin^na,\,\cos^na$ each have images $[0,\,1]$ for even $n$ and $[-1,\,1]$ for odd $n$ on $\Bbb R$, because these are the images of $x^n$ on $[-1,\,1]$. So the supremum is $1$, and the infimum is $0$ for even $n$ but $-1$ for odd $n$.