This page reviews basic trigonometric functions and identities.
Disclaimer: Always verify formulas from another source before using them.
Basic Trigonometric Functions
The six basic trigonometric functions are defined in terms of a right triangle or the unit circle. For an angle $\theta$
in a right triangle with opposite side $a$, adjacent side $b$, and hypotenuse $c$:
Right Triangle Definitions
\begin{aligned}
\sin(\theta) &= \frac{\text{opposite}}{\text{hypotenuse}} = \frac{a}{c} \\
\cos(\theta) &= \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{b}{c} \\
\tan(\theta) &= \frac{\text{opposite}}{\text{adjacent}} = \frac{a}{b} = \frac{\sin(\theta)}{\cos(\theta)} \\
\csc(\theta) &= \frac{1}{\sin(\theta)} = \frac{c}{a} \\
\sec(\theta) &= \frac{1}{\cos(\theta)} = \frac{c}{b} \\
\cot(\theta) &= \frac{1}{\tan(\theta)} = \frac{b}{a} = \frac{\cos(\theta)}{\sin(\theta)}
\end{aligned}
On the unit circle (circle of radius 1 centered at the origin), if a point on the circle makes an angle $\theta$
with the positive $x$-axis, then the coordinates of that point are $(\cos(\theta), \sin(\theta))$.
Fundamental Identities
Pythagorean Identities
Pythagorean Identities
\begin{aligned}
\sin^2(\theta) + \cos^2(\theta) &= 1 \\
1 + \tan^2(\theta) &= \sec^2(\theta) \\
1 + \cot^2(\theta) &= \csc^2(\theta)
\end{aligned}
Example: Verify $\sin^2(\theta) + \cos^2(\theta) = 1$
Using the right triangle definitions with hypotenuse $c$:
\begin{aligned}
\sin^2(\theta) + \cos^2(\theta) &= \left(\frac{a}{c}\right)^2 + \left(\frac{b}{c}\right)^2 \\
&= \frac{a^2 + b^2}{c^2} \\
&= \frac{c^2}{c^2} = 1
\end{aligned}
where the last step uses the Pythagorean theorem: $a^2 + b^2 = c^2$.
Angle Sum and Difference Identities
Angle Sum Identities
\begin{aligned}
\sin(\alpha \pm \beta) &= \sin(\alpha)\cos(\beta) \pm \cos(\alpha)\sin(\beta) \\
\cos(\alpha \pm \beta) &= \cos(\alpha)\cos(\beta) \mp \sin(\alpha)\sin(\beta) \\
\tan(\alpha \pm \beta) &= \frac{\tan(\alpha) \pm \tan(\beta)}{1 \mp \tan(\alpha)\tan(\beta)}
\end{aligned}
Double Angle Identities
Double Angle Identities
\begin{aligned}
\sin(2\theta) &= 2\sin(\theta)\cos(\theta) \\
\cos(2\theta) &= \cos^2(\theta) - \sin^2(\theta) \\
&= 2\cos^2(\theta) - 1 \\
&= 1 - 2\sin^2(\theta) \\
\tan(2\theta) &= \frac{2\tan(\theta)}{1 - \tan^2(\theta)}
\end{aligned}
Half Angle Identities
Half Angle Identities
\begin{aligned}
\sin^2\left(\frac{\theta}{2}\right) &= \frac{1 - \cos(\theta)}{2} \\
\cos^2\left(\frac{\theta}{2}\right) &= \frac{1 + \cos(\theta)}{2} \\
\tan\left(\frac{\theta}{2}\right) &= \frac{\sin(\theta)}{1 + \cos(\theta)} = \frac{1 - \cos(\theta)}{\sin(\theta)}
\end{aligned}
Product-to-Sum and Sum-to-Product Identities
Product-to-Sum Identities
\begin{aligned}
\sin(\alpha)\sin(\beta) &= \frac{1}{2}[\cos(\alpha - \beta) - \cos(\alpha + \beta)] \\
\cos(\alpha)\cos(\beta) &= \frac{1}{2}[\cos(\alpha - \beta) + \cos(\alpha + \beta)] \\
\sin(\alpha)\cos(\beta) &= \frac{1}{2}[\sin(\alpha + \beta) + \sin(\alpha - \beta)] \\
\cos(\alpha)\sin(\beta) &= \frac{1}{2}[\sin(\alpha + \beta) - \sin(\alpha - \beta)]
\end{aligned}
Sum-to-Product Identities
\begin{aligned}
\sin(\alpha) + \sin(\beta) &= 2\sin\left(\frac{\alpha + \beta}{2}\right)\cos\left(\frac{\alpha - \beta}{2}\right) \\
\sin(\alpha) - \sin(\beta) &= 2\cos\left(\frac{\alpha + \beta}{2}\right)\sin\left(\frac{\alpha - \beta}{2}\right) \\
\cos(\alpha) + \cos(\beta) &= 2\cos\left(\frac{\alpha + \beta}{2}\right)\cos\left(\frac{\alpha - \beta}{2}\right) \\
\cos(\alpha) - \cos(\beta) &= -2\sin\left(\frac{\alpha + \beta}{2}\right)\sin\left(\frac{\alpha - \beta}{2}\right)
\end{aligned}
Special Angles
The values of trigonometric functions at special angles are frequently used in physics problems:
| Angle |
$\sin$ |
$\cos$ |
$\tan$ |
| $0$ |
$0$ |
$1$ |
$0$ |
| $\pi/6$ ($30°$) |
$1/2$ |
$\sqrt{3}/2$ |
$1/\sqrt{3}$ |
| $\pi/4$ ($45°$) |
$\sqrt{2}/2$ |
$\sqrt{2}/2$ |
$1$ |
| $\pi/3$ ($60°$) |
$\sqrt{3}/2$ |
$1/2$ |
$\sqrt{3}$ |
| $\pi/2$ ($90°$) |
$1$ |
$0$ |
$\infty$ |
| $\pi$ ($180°$) |
$0$ |
$-1$ |
$0$ |
Even and Odd Properties
Parity of Trigonometric Functions
\begin{aligned}
\sin(-\theta) &= -\sin(\theta) \quad \text{(odd function)} \\
\cos(-\theta) &= \cos(\theta) \quad \text{(even function)} \\
\tan(-\theta) &= -\tan(\theta) \quad \text{(odd function)}
\end{aligned}
Periodicity
Periodic Properties
\begin{aligned}
\sin(\theta + 2\pi n) &= \sin(\theta), \quad n \in \mathbb{Z} \\
\cos(\theta + 2\pi n) &= \cos(\theta), \quad n \in \mathbb{Z} \\
\tan(\theta + \pi n) &= \tan(\theta), \quad n \in \mathbb{Z}
\end{aligned}
The sine and cosine functions have period $2\pi$, while the tangent function has period $\pi$.
Inverse Trigonometric Functions
The inverse trigonometric functions are defined with restricted domains to ensure they are functions:
Inverse Trigonometric Functions
\begin{aligned}
\arcsin(x) &: [-1, 1] \to [-\pi/2, \pi/2] \\
\arccos(x) &: [-1, 1] \to [0, \pi] \\
\arctan(x) &: (-\infty, \infty) \to (-\pi/2, \pi/2)
\end{aligned}
These satisfy: $\sin(\arcsin(x)) = x$ for $x \in [-1, 1]$, $\cos(\arccos(x)) = x$ for $x \in [-1, 1]$,
and $\tan(\arctan(x)) = x$ for all $x$.
Example: Small Angle Approximations
For small angles $\theta$ (in radians), the following approximations are frequently used in physics:
\begin{aligned}
\sin(\theta) &\approx \theta \\
\cos(\theta) &\approx 1 - \frac{\theta^2}{2} \\
\tan(\theta) &\approx \theta
\end{aligned}
These approximations are valid when $|\theta| \ll 1$ (in radians). They are derived from the Taylor series
expansions and are essential for analyzing small oscillations, pendulums, and other systems near equilibrium.