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 $\sqrt$. When two such triangles are placed in the positions shown in the illustration, the smallest rectangle that can enclose them has width $1+\sqrt$ and height $\sqrt$. 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) $2\sqrt$. The remaining space at the top of the rectangle is a right triangle with acute angles of 15° and 75° and sides of $\sqrt-1$, $\sqrt+1$, and $2\sqrt$.

Derived trigonometric formulas

From the construction of the rectangle, it follows that

: $\sin 15^\circ = \cos 75^\circ = \frac = \frac 4,$
: $\sin 75^\circ = \cos 15^\circ = \frac = \frac 4,$
: $\tan 15^\circ = \cot 75^\circ = \frac = \frac = 2 - \sqrt3,$
and
: $\tan 75^\circ = \cot 15^\circ = \frac = \frac = 2 + \sqrt3.$

Variant

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

See also

* Exact trigonometric values

References

{{reflist

Triangle geometry
Types of quadrilaterals