Sine

Sine is a trigonometric function describing the ratio between the opposite side and the hypotenuse in a right triangle. As a function it is usually abbreviated sin.

The sine of something is generally calculated in radians, a means of measuring angles determined by the measure of the arc of a circle cut by such an angle, divided by the length of the radius of the circle, so that a full 360° angle would be 2π radians, 180° would be π, 90° would be ${\frac {\pi }{2}}$ radians, and so on.

Sine Graph

The basic sine graph, $f(x)=sin(x)$, where x is in radians, is as follows:

x values y values 0 0 π/2 1 π 0 3π/2 -1 2π 0

That would be a full period of the graph - as 2π radians is equivalent to one circle. The graph continues in that pattern.

Famous Sines

Because of the known relationship of the sides of a right triangle with the angles 30°, 45°, or 60°, certain values of sine are calculable using exact values in the form of square roots and fractions involving square roots. These include:

$\sin {\frac {\pi }{60}}=\sin 3^{\circ }={\frac {(2-{\sqrt {12}}){\sqrt {5+{\sqrt {5}}}}+({\sqrt {10}}-{\sqrt {2}})({\sqrt {3}}+1)}{16}}\,$

$\sin {\frac {\pi }{30}}=\sin 6^{\circ }={\frac {{\sqrt {30-{\sqrt {180}}}}-{\sqrt {5}}-1}{8}}\,$

$\sin {\frac {\pi }{20}}=\sin 9^{\circ }={\frac {{\sqrt {10}}+{\sqrt {2}}-{\sqrt {20-{\sqrt {80}}}}}{8}}\,$

$\sin {\frac {\pi }{15}}=\sin 12^{\circ }={\frac {{\sqrt {10+{\sqrt {20}}}}+{\sqrt {3}}-{\sqrt {15}}}{8}}\,$

$\sin {\frac {\pi }{12}}=\sin 15^{\circ }={\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}$

$\sin {\frac {\pi }{10}}=\sin 18^{\circ }={\frac {{\sqrt {5}}-1}{4}}={\tfrac {1}{2}}\varphi ^{-1}\,$

$\sin {\frac {7\pi }{60}}=\sin 21^{\circ }={\frac {(2+{\sqrt {12}}){\sqrt {5-{\sqrt {5}}}}-({\sqrt {10}}+{\sqrt {2}})({\sqrt {3}}-1)}{16}}\,$

$\sin {\frac {\pi }{8}}=\sin 22.5^{\circ }={\frac {\sqrt {2-{\sqrt {2}}}}{2}}$

$\sin {\frac {2\pi }{15}}=\sin 24^{\circ }={\frac {{\sqrt {3}}+{\sqrt {15}}-{\sqrt {10-{\sqrt {20}}}}}{8}}\,$

$\sin {\frac {3\pi }{20}}=\sin 27^{\circ }={\frac {{\sqrt {20+{\sqrt {80}}}}-{\sqrt {10}}+{\sqrt {2}}}{8}}\,$

$\sin {\frac {\pi }{6}}=\sin 30^{\circ }={\frac {1}{2}}$

$\sin {\frac {11\pi }{60}}=\sin 33^{\circ }={\frac {({\sqrt {12}}-2){\sqrt {5+{\sqrt {5}}}}+({\sqrt {10}}-{\sqrt {2}})({\sqrt {3}}+1)}{16}}\,$

$\sin {\frac {\pi }{5}}=\sin 36^{\circ }={\frac {\sqrt {10-{\sqrt {20}}}}{4}}\,$

$\sin {\frac {13\pi }{60}}=\sin 39^{\circ }={\frac {(2-{\sqrt {12}}){\sqrt {5-{\sqrt {5}}}}+({\sqrt {10}}+{\sqrt {2}})({\sqrt {3}}+1)}{16}}\,$

$\sin {\frac {7\pi }{30}}=\sin 42^{\circ }={\frac {{\sqrt {30+{\sqrt {180}}}}-{\sqrt {5}}+1}{8}}\,$

$\sin {\frac {\pi }{4}}=\sin 45^{\circ }={\frac {\sqrt {2}}{2}}$

Relationship to cosine and cosecant

A mathematical identity involving sine is:

$\sin \theta =\cos \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\csc \theta }}\,$