## Friday, August 29, 2014

### Cosine of 72 degrees (and constructing a regular pentagon)

By using (say) DeMoivre's theorem, we have that $$\left( \cos\frac{2 \pi}{5}+ i \sin\frac{2\pi}{5} \right)^5=1$$
Expanding the left side as the fifth power of a binomial, equating the imaginary parts on both sides of the equation, and then replacing  $$\sin^2 72^\circ$$ with  $$1- \cos^2 72^\circ$$
$$i\sin 72^\circ \left(5 \cos^4 72^\circ+10 \cos^2 72^\circ i^2 \sin ^2 72 ^\circ+ i^4 \sin^4 72^\circ \right)=0i$$
$$16 \cos^4 72^\circ - 12 \cos^2 72^\circ + 1=0$$
Solving this quadratic in $$\cos^2 72^\circ$$, we get
$\cos^2 72^\circ = \frac{12 \pm \sqrt{80} } {32}$
so
$\cos^2 72^\circ = \frac{6 \pm 2 \sqrt{5} } {16}=\frac{\left( \sqrt{5}\pm 1 \right)^2}{4^2}$
$\cos 72^\circ = \pm \frac{\sqrt{5} \pm 1}{4}$
where we can choose the correct value of the four possible values by noting that, because 72° is between 45° and 90°,  $$\cos 72^\circ$$ must lie between  $$1 / \sqrt{2}$$ and 0. Because  $$\cos 72^\circ$$ is positive, we choose the "+" before the fraction, and because  $$\cos 72^\circ$$ is less than $$1 / \sqrt{2}$$, which in turn is less than $$\frac{\sqrt{5}+1}{4}$$, we choose the "-" in the numerator:
$\cos 72^\circ = \frac{\sqrt{5}-1}{4}$

## Constructing a regular pentagon

So we can construct $$\cos 72^\circ$$. For example, the diagonal of a 1-by-2 rectangle is $$\sqrt{5}$$. We could cut off one unit from a segment of length $$\sqrt{5}$$, then divide the segment of length $$\sqrt{5}-1$$ into four pieces of length $$\frac{\sqrt{5} -1} {4}$$. (Or we could construct the appropriate solution to the equation $$4x^2 +2x -1 = 0$$. See my post on Solving quadratic equations via geometric construction.)

Construct a unit circle centered at O, and construct a radius $$\overline{OA}$$.  Construct the point B on $$\overline{OA}$$ so that $$\overline{OB}$$ has length  $$\cos 72^\circ$$. If C is a point on the circle so that $$\overline{BC}$$ is perpendicular to $$\overline{OA}$$, then $$\angle COA$$ is a 72° angle, and both A and C are vertices of a regular pentagon inscribed in the circle.

#### 1 comment:

Bruce Yoshiwara said...

This method of evaluating cos(2pi/5) was shown to me by my then-UCLA office mate Larry Miller in my first year of grad school.