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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

user100478
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    $\pi^4$ in radians is $\pi^4 \cdot \frac{180}{\pi} = 180 \pi^3 \approx 5581.1298$ in degrees ... – Martin R Oct 30 '19 at 08:42
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    You have to be more careful. It is indeed true that $$ \pi~\text{rad} = 180~\text{deg} $$ But if we put both sides to the fourth power, we get $$ \pi^4~\text{rad}^4 = 180^4~\text{deg}^4 $$ The units are messed up and we cannot calculate with this. On the other hand, if you MULTIPLY both sides by $\pi^3$, we get $$ \pi^4~\text{rad} = 180 \pi^3~\text{deg} $$ – Matti P. Oct 30 '19 at 08:43

2 Answers2

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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.

Ninad Munshi
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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 $-\cos 0=-1.$

Z Ahmed
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