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Trigonometric identities and table of values

The identities that keep coming back in calculus, Fourier series and trigonometric integrals, with the values of the common angles.

Table of values

$α$	$sin α$	$cos α$	$tan α$
$0$	$0$	$1$	$0$
$\frac{π}{6} = 3 0^{\circ}$	$\frac{1}{2}$	$\frac{3}{2}$	$\frac{3}{3}$
$\frac{π}{4} = 4 5^{\circ}$	$\frac{2}{2}$	$\frac{2}{2}$	$1$
$\frac{π}{3} = 6 0^{\circ}$	$\frac{3}{2}$	$\frac{1}{2}$	$3$
$\frac{π}{2} = 9 0^{\circ}$	$1$	$0$	undefined

Basic identities

Expression	Equals
$sin^{2} α + cos^{2} α$	$1$
$1 + tan^{2} α$	$\frac{1}{cos ^{2} α}$
$tan α$	$\frac{sin α}{cos α}$
$sin (- α)$	$- sin α$
$cos (- α)$	$cos α$

Angle sum and difference

Expression	Equals
$sin (α \pm β)$	$sin α cos β \pm cos α sin β$
$cos (α \pm β)$	$cos α cos β \mp sin α sin β$
$tan (α \pm β)$	$\frac{tan α \pm tan β}{1 \mp tan α tan β}$

Double angle and power reduction

Expression	Equals
$sin 2 α$	$2 sin α cos α$
$cos 2 α$	$cos^{2} α - sin^{2} α = 2 cos^{2} α - 1 = 1 - 2 sin^{2} α$
$tan 2 α$	$\frac{2 tan α}{1 - tan ^{2} α}$
$sin^{2} α$	$\frac{1 - cos 2 α}{2}$
$cos^{2} α$	$\frac{1 + cos 2 α}{2}$

Product to sum and sum to product

Expression	Equals
$sin α sin β$	$\frac{1}{2} [cos (α - β) - cos (α + β)]$
$cos α cos β$	$\frac{1}{2} [cos (α - β) + cos (α + β)]$
$sin α cos β$	$\frac{1}{2} [sin (α + β) + sin (α - β)]$
$sin α + sin β$	$2 sin \frac{α + β}{2} cos \frac{α - β}{2}$
$cos α + cos β$	$2 cos \frac{α + β}{2} cos \frac{α - β}{2}$

Notes

Power reduction, sin²α = (1 − cos 2α)/2, is what makes an integral of sin²x easy: after it, only standard integrals remain.

To switch between degrees and radians: 180° = π. Calculus and physics almost always work in radians.