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.