Cosine Proof

**harold** · November 1st 2006, 06:50 PM

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).

**ThePerfectHacker** · November 1st 2006, 06:59 PM

Originally Posted by harold

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).

$\cos (3x)=\cos (x+2x)$
$\cos x\cos 2x-\sin x\sin 2x$
$\cos x(\cos ^2 x-\sin^2 x)-\sin x (2\sin x\cos x)$
$\cos^3 x-3\cos x \sin^2 x$
And,
$\sin^2 x=1-\cos ^2x$
So if $a=\cos x$ is rational
Then,
$a^3-3 a(1-a^2)$ is also rational.
---
If $\cos (5^o)$ is rational.
Then,
$\cos (15^o)$ is rational.
Then,
$\cos (45^o)=\frac{\sqrt{2}}{2}$ is rational.
But that is not true.
Contradiction.

**harold** · November 1st 2006, 07:46 PM

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...?

**ThePerfectHacker** · November 1st 2006, 07:53 PM

Originally Posted by harold

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...?

Yes it does.

If assume $\cos 5^o$
Is rational and you arrived at a contradiction.

If $a=\cos x$ is a rational number then,
$\cos 3x=a^3-3a(1-a^2)$ is a rational number.
Because when you add or multiply rationals you get a rational.

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