Ailles rectangle

A 30°–60°–90° triangle has sides of length 1, 2, and ${\displaystyle {\sqrt {3}}}$. When two such triangles are placed in the positions shown in the illustration, the smallest rectangle that can enclose them has width ${\displaystyle 1+{\sqrt {3}}}$ and height ${\displaystyle {\sqrt {3}}}$. 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) ${\displaystyle 2{\sqrt {2}}}$. The remaining space at the top of the rectangle is a right triangle with acute angles of 15° and 75° and sides of ${\displaystyle {\sqrt {3}}-1}$, ${\displaystyle {\sqrt {3}}+1}$, and ${\displaystyle 2{\sqrt {2}}}$.

From the construction of the rectangle, it follows that

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

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

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

and

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

An alternative construction (also by Ailles) places a 30°–60°–90° triangle in the middle with sidelengths of ${\displaystyle {\sqrt {2}}}$, ${\displaystyle {\sqrt {6}}}$, and ${\displaystyle 2{\sqrt {2}}}$. Its legs are each the hypotenuse of a 45°–45°–90° triangle, one with legs of length ${\displaystyle 1}$ and one with legs of length ${\displaystyle {\sqrt {3}}}$.^[4][5] The 15°–75°–90° triangle is the same as above.

• Exact trigonometric values