I have to answer the following and support it with a full proof.

Is cos(5°) rational? (Given hint: show that if cos(x) is rational, then cos(3x) is rational).

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- Nov 1st 2006, 06:50 PMharoldCosine Proof
I have to answer the following and support it with a full proof.

Is cos(5°) rational? (Given hint: show that if cos(x) is rational, then cos(3x) is rational). - Nov 1st 2006, 06:59 PMThePerfectHacker
$\displaystyle \cos (3x)=\cos (x+2x)$

$\displaystyle \cos x\cos 2x-\sin x\sin 2x$

$\displaystyle \cos x(\cos ^2 x-\sin^2 x)-\sin x (2\sin x\cos x)$

$\displaystyle \cos^3 x-3\cos x \sin^2 x$

And,

$\displaystyle \sin^2 x=1-\cos ^2x$

So if $\displaystyle a=\cos x $ is rational

Then,

$\displaystyle a^3-3 a(1-a^2)$ is also rational.

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If $\displaystyle \cos (5^o)$ is rational.

Then,

$\displaystyle \cos (15^o)$ is rational.

Then,

$\displaystyle \cos (45^o)=\frac{\sqrt{2}}{2}$ is rational.

But that is not true.

Contradiction. - Nov 1st 2006, 07:46 PMharold
Hi THP,

I was thinking the exact same thing by showing cos(45°) but it still doesn't__prove__it since we need to show if cos x is rational, then cos 3x is rational....then show cos 5° is irrat. based on the previous two proofs...? - Nov 1st 2006, 07:53 PMThePerfectHacker
Yes it does.

If assume $\displaystyle \cos 5^o$

Is rational and you arrived at a contradiction.

If $\displaystyle a=\cos x$ is a rational number then,

$\displaystyle \cos 3x=a^3-3a(1-a^2)$ is a rational number.

Because when you add or multiply rationals you get a rational.