Factsheet: Trigonometric identities (radians)
The main study guide for this factsheet is Guide: Trigonometric identities (radians). If you would like to know more about these, please read the guide.
This factsheet measures angles in radians. For the associated factsheet measuring angles in degrees, please go to Factsheet: Trigonometric identities (degrees).
Trigonometric identities
Periodicity and parity
For all angles \(\theta\) and for all whole numbers \(k\in\mathbb{Z}\):
\[\begin{aligned} \cos(-\theta) &= \cos(\theta) \\[0.25em] \sin(-\theta) &= -\sin(\theta) \\[0.25em] \tan(-\theta) &= -\tan(\theta) \\[0.25em] \cos(\theta + 2k\pi) &= \cos(\theta) \\[0.25em] \sin(\theta + 2k\pi) &= \sin(\theta) \\[0.25em] \tan(\theta + k\pi) &= \tan(\theta) \end{aligned}\]
Pythagorean formulas
For all angles \(\theta\) \[\begin{aligned} \cos^2(\theta) + \sin^2(\theta) &= 1 \\[0.5em] 1 + \tan^2(\theta) &= \sec^2(\theta) \\[0.5em] \cot^2(\theta) + 1 &= \csc^2(\theta) \end{aligned}\]
Sum and difference formulas
For all angles \(\alpha,\beta\):
\[\begin{aligned} \cos(\alpha + \beta) &= \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta) \\[0.5em] \cos(\alpha - \beta) &= \cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta) \\[0.5em] \sin(\alpha + \beta) &= \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta) \\[0.5em] \sin(\alpha - \beta) &= \sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta) \\[0.5em] \tan(\alpha + \beta) &= \frac{\tan(\alpha) + \tan(\beta)}{1 - \tan(\alpha)\tan(\beta)}\\[0.5em] \tan(\alpha - \beta) &= \frac{\tan(\alpha) - \tan(\beta)}{1 + \tan(\alpha)\tan(\beta)} \end{aligned}\]
Double angle formulas
For all angles \(\theta\):
\[\begin{aligned} \cos(2\theta) &= \cos^2(\theta) - \sin^2(\theta) \\[0.5em] \sin(2\theta) &= 2\sin(\theta)\cos(\theta) \\[0.5em] \tan(2\theta) &= \frac{2\tan(\theta)}{1 - \tan^2(\theta)} \end{aligned}\]
Shift formulas
For all angles \(\theta\):
\[\begin{aligned} \cos\left(\theta + \frac{\pi}{2}\right) &= -\sin(\theta) \\[0.5em] \cos\left(\theta - \frac{\pi}{2}\right) &= \sin(\theta) \\[0.5em] \sin\left(\theta + \frac{\pi}{2}\right) &= \cos(\theta) \\[0.5em] \sin\left(\theta - \frac{\pi}{2}\right) &= -\cos(\theta) \\[0.5em] \cos\left(\theta \pm \pi\right) &= -\cos(\theta) \\[0.5em] \sin\left(\theta \pm \pi\right) &= -\sin(\theta) \\[0.5em] \sin\left(\pi - \theta\right) &= \sin(\theta) \\[0.5em] \cos\left(\pi - \theta\right) &= -\cos(\theta) \end{aligned}\]
Sine and cosine rules
For a triangle with corners \(A,B,C\), angles \(\alpha\), \(\beta\), \(\gamma\) respectively at those corners, and sides \(a,b,c\) opposite their respective corners, the sine rule is
\[\frac{\sin(\alpha)}{a} = \frac{\sin(\beta)}{b} = \frac{\sin(\gamma)}{c}\]
and the cosine rule is \[a^2 = b^2 + c^2 - 2bc\cos(\alpha).\]
Common values of trigonometric functions
| Angle \(\theta\) | \(0\) | \(\dfrac{\pi}{6}\) | \(\dfrac{\pi}{4}\) | \(\dfrac{\pi}{3}\) | \(\dfrac{\pi}{2}\) | \(\dfrac{2\pi}{3}\) | \(\dfrac{3\pi}{4}\) | \(\dfrac{5\pi}{6}\) | \(\pi\) |
|---|---|---|---|---|---|---|---|---|---|
| \(\sin\theta\) | \(0\) | \(\dfrac{1}{2}\) | \(\dfrac{\sqrt{2}}{2}\) | \(\dfrac{\sqrt{3}}{2}\) | \(1\) | \(\dfrac{\sqrt{3}}{2}\) | \(\dfrac{\sqrt{2}}{2}\) | \(\dfrac{1}{2}\) | \(0\) |
| \(\cos\theta\) | \(1\) | \(\dfrac{\sqrt{3}}{2}\) | \(\dfrac{\sqrt{2}}{2}\) | \(\dfrac{1}{2}\) | \(0\) | \(-\dfrac{1}{2}\) | \(-\dfrac{\sqrt{2}}{2}\) | \(-\dfrac{\sqrt{3}}{2}\) | \(-1\) |
| \(\tan\theta\) | \(0\) | \(\dfrac{1}{\sqrt{3}}\) | \(1\) | \(\sqrt{3}\) | undef. | \(-\sqrt{3}\) | \(-1\) | \(-\dfrac{1}{\sqrt{3}}\) | \(0\) |
Version history
v1.0: created in 08/25 by tdhc.