# Analytic Expressions for Selected Trigonometric Functions

In what follows we generally will provide expressions that result from evaluating ${\displaystyle \sin \theta }$, with the understanding that ${\displaystyle \cos \theta }$ and ${\displaystyle \tan \theta }$ then also can be evaluated straightforwardly via the familiar relations,

 ${\displaystyle \cos \theta }$ ${\displaystyle =}$ ${\displaystyle \pm (1-\sin ^{2}\theta )^{1/2}\,,}$ and, ${\displaystyle \tan \theta }$ ${\displaystyle =}$ ${\displaystyle {\frac {\sin \theta }{\pm (1-\sin ^{2}\theta )^{1/2}}}\,.}$

## Widely Used Evaluations

 ${\displaystyle 0\leq }$ Angle ${\displaystyle (\theta )\leq {\frac {\pi }{2}}}$ ${\displaystyle \sin \theta }$ Radians Degrees ${\displaystyle 0}$ ${\displaystyle 0^{\circ }}$ ${\displaystyle 0}$ ${\displaystyle {\frac {\pi }{6}}}$ ${\displaystyle 30^{\circ }}$ ${\displaystyle {\frac {1}{2}}}$ ${\displaystyle {\frac {\pi }{4}}}$ ${\displaystyle 45^{\circ }}$ ${\displaystyle {\frac {\sqrt {2}}{2}}}$ ${\displaystyle {\frac {\pi }{3}}}$ ${\displaystyle 60^{\circ }}$ ${\displaystyle {\frac {\sqrt {3}}{2}}}$ ${\displaystyle {\frac {\pi }{2}}}$ ${\displaystyle 90^{\circ }}$ ${\displaystyle 1}$

## Integer-Degree Angles

A PDF-formatted document generated by James T. Parent lists exact values for the sine of all integer-degree angles between zero and ninety degrees, inclusive. An explanation of how the expressions in this document were derived, can be found on the SquareCirclez IntMath blog.

 Examples Extracted from the James T. Parent Document† ${\displaystyle 0\leq }$ Angle ${\displaystyle (\theta )\leq {\frac {\pi }{2}}}$ ${\displaystyle \sin \theta }$ Radians Degrees ${\displaystyle {\frac {\pi }{60}}}$ ${\displaystyle 3^{\circ }}$ ${\displaystyle {\frac {\sqrt {6}}{48}}{\biggl (}{\sqrt {5}}-1{\biggr )}{\biggl (}3+{\sqrt {3}}{\biggr )}-{\frac {\sqrt {3}}{24}}{\biggl (}3-{\sqrt {3}}{\biggr )}{\biggl (}5+{\sqrt {5}}{\biggr )}^{1/2}}$ ${\displaystyle {\frac {\pi }{12}}}$ ${\displaystyle 15^{\circ }}$ ${\displaystyle {\frac {\sqrt {2}}{4}}{\biggl (}{\sqrt {3}}-1{\biggr )}={\frac {1}{2}}{\biggl (}2-{\sqrt {3}}{\biggr )}^{1/2}}$ ${\displaystyle {\frac {\pi }{5}}}$ ${\displaystyle 36^{\circ }}$ ${\displaystyle {\frac {\sqrt {2}}{8}}{\biggl (}{\sqrt {5}}-1{\biggr )}{\biggl (}5+{\sqrt {5}}{\biggr )}^{1/2}={\frac {1}{4}}{\biggl (}10-2{\sqrt {5}}{\biggr )}^{1/2}}$ ${\displaystyle {\frac {5\pi }{12}}}$ ${\displaystyle 75^{\circ }}$ ${\displaystyle {\frac {{\sqrt {3}}+1}{2{\sqrt {2}}}}}$ †James T. Parent has previously taught mathematics at Schenectady County Community College, Schenectady, New York, and at Great Bay Community College, Portsmouth, New Hampshire.

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