Ailles rectangle

The 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

$\sin 15^{\circ }=\cos 75^{\circ }=\frac{\sqrt{3}-1}{2\sqrt{2}}=\frac{\sqrt{6}-\sqrt{2}}{4},$

$\sin 75^{\circ }=\cos 15^{\circ }=\frac{\sqrt{3}+1}{2\sqrt{2}}=\frac{\sqrt{6}+\sqrt{2}}{4},$

$\tan 15^{\circ }=\cot 75^{\circ }=\frac{\sqrt{3}-1}{\sqrt{3}+1}=\frac{(\sqrt{3}-1{)}^2}{3-1}=2-\sqrt{3},$

and

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

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.

See also

• Exact trigonometric values