Exact Angles

Pentagrams. Click to zoom in/out!

The exact values of sine and cosine for the angles \(30^\circ, 45^\circ\) and \(60^\circ \); can be derived from the triangles in the picture below.


Apart from these well know angles, there are also other angles that yield exact values of sine and cosine. In his book Almagest, Ptolemy gives the chords for the angles \(36^\circ \) and \(72^\circ \).

The golden triangle

The numbers \(36, 72\) and \(108\), have the properties that \(2\cdot 36 = 72\) and \(3\cdot 36 = 108\). Furthermore: \[72+108=72+72+36=108+36+36=180\]

Starting with the isosceles triangle to the left in the picture below, you can construct two isosceles triangles by bisecting one of the \(72^\circ \) angles.


Considering the interior angles of a regular pentagon


one can form a variety of self-similar patterns using pentagons and pentagrams that contain the angles \(36^\circ, 72^\circ, 108^\circ\) - and no other angles.


The triangle is called the golden triangle since the ratio between the sides is the golden ratio.



Exercise 1

Use similarity to set up a quadratic equation for x (using the notation from the picture to the right). By solving the equation, show that the ratio of the sides in the golden triangle is the golden ratio.


Exercise 2

Find the exact values of \(\cos 36^\circ \) and \(\cos 72^\circ \).

animated gifs:

Pentagrams on tumblr.

by Malin Christersson under a Creative Commons Attribution-Noncommercial-Share Alike 2.5 Sweden License