Ailles rectangle

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

From the construction of the rectangle, it follows that

: $\backslash sin\; 15^\backslash circ\; =\; \backslash cos\; 75^\backslash circ\; =\; \backslash frac\{\backslash sqrt3\; -\; 1\}\{2\backslash sqrt2\}\; =\; \backslash frac\{\backslash sqrt6\; -\; \backslash sqrt2\}\; 4,$

: $\backslash sin\; 75^\backslash circ\; =\; \backslash cos\; 15^\backslash circ\; =\; \backslash frac\{\backslash sqrt3\; +\; 1\}\{2\backslash sqrt2\}\; =\; \backslash frac\{\backslash sqrt6\; +\; \backslash sqrt2\}\; 4,$

: $\backslash tan\; 15^\backslash circ\; =\; \backslash cot\; 75^\backslash circ\; =\; \backslash frac\{\backslash sqrt3\; -\; 1\}\{\backslash sqrt3\; +\; 1\}\; =\; \backslash frac\{(\backslash sqrt3\; -\; 1)^2\}\{3\; -\; 1\}\; =\; 2\; -\; \backslash sqrt3,$

and

: $\backslash tan\; 75^\backslash circ\; =\; \backslash cot\; 15^\backslash circ\; =\; \backslash frac\{\backslash sqrt3\; +\; 1\}\{\backslash sqrt3\; -\; 1\}\; =\; \backslash frac\{(\backslash sqrt3\; +\; 1)^2\}\{3\; -\; 1\}\; =\; 2\; +\; \backslash sqrt3.$

An alternative construction (also by Ailles) places a 30°–60°–90° triangle in the middle with sidelengths of $\backslash sqrt\{2\}$, $\backslash sqrt\{6\}$, and $2\backslash sqrt\{2\}$. Its legs are each the hypotenuse of a 45°–45°–90° triangle, one with legs of length $1$ and one with legs of length $\backslash sqrt\{3\}$.{{cite web

|title=Third Ailles Rectangle

|url=https://math.stackexchange.com/q/1651208

|date=11 February 2016

|work=Stack Exchange

|access-date=2017-11-01

}}{{cite web

|title=The Mathematical Ninja and Ailles' Rectangle

|author=Colin Beveridge

|url=http://www.flyingcoloursmaths.co.uk/the-mathematical-ninja-and-ailles-rectangle/

|date=31 August 2015

|website=Flying Colours Maths

|access-date=2017-11-01

}} The 15°–75°–90° triangle is the same as above.

• Exact trigonometric values

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Category:Triangle geometry

Category:Types of quadrilaterals