Ailles rectangle

The Ailles rectangle is a rectangle constructed from four right-angled triangles which is commonly used in geometry classes to find the values of trigonometric functions of 15° and 75°. It is named after Douglas S. Ailles who was a high school teacher at Kipling Collegiate Institute in Toronto.

Construction
A 30°–60°–90° triangle has sides of length 1, 2, and $$\sqrt{3}$$. When two such triangles are placed in the positions shown in the illustration, the smallest rectangle that can enclose them has width $$1+\sqrt{3}$$ and height $$\sqrt{3}$$. Drawing a line connecting the original triangles' top corners creates a 45°–45°–90° triangle between the two, with sides of lengths 2, 2, and (by the Pythagorean theorem) $$2\sqrt{2}$$. The remaining space at the top of the rectangle is a right triangle with acute angles of 15° and 75° and sides of $$\sqrt{3}-1$$, $$\sqrt{3}+1$$, and $$2\sqrt{2}$$.

Derived trigonometric formulas
From the construction of the rectangle, it follows that


 * $$ \sin 15^\circ = \cos 75^\circ = \frac{\sqrt3 - 1}{2\sqrt2} = \frac{\sqrt6 - \sqrt2} 4, $$
 * $$ \sin 75^\circ = \cos 15^\circ = \frac{\sqrt3 + 1}{2\sqrt2} = \frac{\sqrt6 + \sqrt2} 4, $$
 * $$ \tan 15^\circ = \cot 75^\circ = \frac{\sqrt3 - 1}{\sqrt3 + 1} = \frac{(\sqrt3 - 1)^2}{3 - 1} = 2 - \sqrt3, $$

and
 * $$ \tan 75^\circ = \cot 15^\circ = \frac{\sqrt3 + 1}{\sqrt3 - 1} = \frac{(\sqrt3 + 1)^2}{3 - 1} = 2 + \sqrt3. $$

Variant
An alternative construction (also by Ailles) places a 30°–60°–90° triangle in the middle with sidelengths of $$\sqrt{2}$$, $$\sqrt{6}$$, and $$2\sqrt{2}$$. Its legs are each the hypotenuse of a 45°–45°–90° triangle, one with legs of length $$1$$ and one with legs of length $$\sqrt{3}$$. The 15°–75°–90° triangle is the same as above.