Find $\cos(\pi^4)$ This is what I did.
$\pi^4 = 1049760000\deg \quad$ since $\pi = 180\deg$
and $1049760000$ is divisible by $360$ so this is equivalent to finding $\cos(0\deg) = 1$. But the answer is not correct.
Can someone explain to me what am I missing.
Thank you
 A: To calculate $\cos(\pi^4)$, we will series expand about a nice point near it. $\pi^4 \approx 31\pi$ so we will find the Taylor expansion there. We don't have to do any extra work, the Taylor expansion for cosine about $0$ is 
$$\cos(x) = 1 - \frac{1}{2}x^2 + \frac{1}{24}x^4 - \cdots$$
and by symmetry, cosine around an odd multiple of $\pi$ looks like negative cosine, so we have that 
$$\cos(x) = -1 + \frac{1}{2}(x-31\pi)^2 - \frac{1}{24}(x-31\pi)^4 + \cdots$$
Then plugging in $x=\pi^4$ we can get better successive approximation by taking more terms of the series:
$$\cos(\pi^4) \approx -1$$
$$\cos(\pi^4) \approx -1 + \frac{(\pi^4-31\pi)^2}{2} \approx -0.9998$$
and that is in fact the exact answer to 4 decimal places.
A: In $\cos (x)$ $x$ is always understood to be in radian unless degrees are mentioned . So $$\pi^4=97.4091... \implies \frac{97.0063}{\pi}= 31.0063 \pi... \implies \pi^4= (31.0063...) ~\pi =  \implies 31\pi+0.0063 \pi...$$ $$\implies \cos {\pi^4}= \cos (31 \pi+0.0063 \pi) =-\cos(0.0063 \pi..)= -\cos (0.019792...)=-0.99980...$$ Which is almost $-\cos 0=-1.$
