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