Английская Википедия:Ailles rectangle

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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 <math>\sqrt{3}</math>. When two such triangles are placed in the positions shown in the illustration, the smallest rectangle that can enclose them has width <math>1+\sqrt{3}</math> and height <math>\sqrt{3}</math>. 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) <math>2\sqrt{2}</math>. The remaining space at the top of the rectangle is a right triangle with acute angles of 15° and 75° and sides of <math>\sqrt{3}-1</math>, <math>\sqrt{3}+1</math>, and <math>2\sqrt{2}</math>.

Derived trigonometric formulas

From the construction of the rectangle, it follows that

<math> \sin 15^\circ = \cos 75^\circ = \frac{\sqrt3 - 1}{2\sqrt2} = \frac{\sqrt6 - \sqrt2} 4, </math>

<math> \sin 75^\circ = \cos 15^\circ = \frac{\sqrt3 + 1}{2\sqrt2} = \frac{\sqrt6 + \sqrt2} 4, </math>

<math> \tan 15^\circ = \cot 75^\circ = \frac{\sqrt3 - 1}{\sqrt3 + 1} = \frac{(\sqrt3 - 1)^2}{3 - 1} = 2 - \sqrt3, </math>

and

<math> \tan 75^\circ = \cot 15^\circ = \frac{\sqrt3 + 1}{\sqrt3 - 1} = \frac{(\sqrt3 + 1)^2}{3 - 1} = 2 + \sqrt3. </math>

Variant

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

See also

• Exact trigonometric values