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$4 \cos^2{\frac{\pi}{5}} - 2 \cos{\frac{\pi}{5}} -1 = 0$

The hint says "note $\sin{\frac{3\pi}{5}} = \sin{\frac{2\pi}{5}}$" and "use double/triple angle or otherwise"

So I have

$4 \cos^2{\frac{\pi}{5}} - 2 (2 \cos^2{\frac{\pi}{10}} - 1) - 1$

$4 \cos^2{\frac{\pi}{5}} - 4 \cos^2{\frac{\pi}{10}} +1$

Now what? Theres $\frac{\pi}{5}$ and $\frac{\pi}{10}$ and I havent used the tip on $\sin$ so perhaps I am missing something?

4 Answers 4

8

The hint says to use the following:

$\sin 3x = 3 \sin x - 4 \sin^3x$

and

$ \sin 2x = 2 \sin x \cos x$

So if $x = \frac{\pi}{5}$, then we have, using $\sin 3x = \sin 2x$ that

$3 \sin x - 4 \sin^3 x = 2 \sin x \cos x $

Since $\sin x \neq 0$, cancel it, and use $\sin^2 x = 1 - \cos ^2x$.

For a geometric way to find $\cos \frac{\pi}{5}$ see: http://www.cut-the-knot.org/pythagoras/cos36.shtml

8

Here is an alternate approach.

Using de Moivre's Formula, we get for $\theta=\frac{\pi}{5}$ $ 0=(\cos(\theta)+i\sin(\theta))^5+1\tag{1} $ Looking at the real part of $(1)$ yields $ \begin{align} 0 &=\color{red}{\cos^5(\theta)}\color{green}{-10\cos^3(\theta)\sin^2(\theta)}\color{blue}{+5\cos(\theta)\sin^4(\theta)}+1\\ &=\color{red}{\cos^5(\theta)}\color{green}{-10\cos^3(\theta)+10\cos^5(\theta)}\color{blue}{+5\cos(\theta)-10\cos^3(\theta)+5\cos^5(\theta)}+1\\ &=16\cos^5(\theta)-20\cos^3(\theta)+5\cos(\theta)+1\\ &=(4\cos^2(\theta)-2\cos(\theta)-1)^2(\cos(\theta)+1)\tag{2} \end{align} $ Since $\cos(\theta)=-1$ only when $\theta$ is an odd multiple of $\pi$, we are left with $ 4\cos^2\left(\frac{\pi}{5}\right)-2\cos\left(\frac{\pi}{5}\right)-1=0\tag{3} $

5

Use the identities $\sin 2\theta=2\cos\theta\sin\theta$

and $\eqalign{\sin 3\theta &=\sin(2\theta+\theta)\cr &= \sin2\theta \cos\theta+\sin\theta\cos2\theta\cr &= 2\sin\theta\cos^2\theta +\sin\theta(2\cos^2\theta-1)\cr } $ to write the identity $\textstyle\sin{2\pi\over5}=\sin(\pi-{2\pi\over5})=\sin{3\pi\over5}$ as $\tag{1}\textstyle 2\sin{\pi\over5}\cos{\pi\over5} = 2\sin{\pi\over5}\cos^2{\pi\over5} +\sin{\pi\over5}(2\cos^2{\pi\over5}-1). $ Cancelling the term $\sin{\pi\over5}\ne0$ from both sides of equation $(1)$ gives: $\textstyle 2 \cos{\pi\over5} = 2\cos^2{\pi\over5} + (2\cos^2{\pi\over5}-1); $ or, $\textstyle 0 = 4\cos^2{\pi\over5} -2\cos{\pi\over5} -1 , $ as desired.

2

You don't have to use the hint. Let $x = {\pi \over 5}$. Then $\cos(3x) = -\cos(2x)$ since $3x + 2x = \pi$. Since $\cos(3x) = 4\cos^3(x) - 3\cos(x)$ and $\cos(2x) = 2\cos^2(x) - 1$, you get $4\cos^3(x) - 3\cos(x) = - ( 2\cos^2(x) - 1)$ This simplifies to $4\cos^3(x) + 2\cos^2(x) - 3\cos(x) - 1 = 0$ This factors into $(4\cos^2(x) - 2\cos(x) - 1)(\cos(x) + 1) = 0$ Since $\cos(x)$ is not $-1$, you get the desired equation $4\cos^2(x) - 2\cos(x) - 1 = 0$.