Trigonometric Functions

Definition

The cosine $\cos x$ and sine $\sin x$ are defined using radians. An angle in radians equals arc length divided by radius; a full revolution is $2\pi$ radians ($360°$). Arguments of trigonometric functions are dimensionless.

Intuition

Place a point on the unit circle at angle $\theta$ from the positive $x$-axis. The $x$-coordinate of that point is $\cos \theta$ and the $y$-coordinate is $\sin \theta$. As $\theta$ increases, the point travels around the circle, producing the familiar oscillating waves of cosine and sine.

Formal Description

Periodicity (period $2\pi$):

$$\cos (x+2\pi n)=\cos x,\sin (x+2\pi n)=\sin x,n\in \mathbb{Z}.$$

Phase shift:

$$\cos \left(x-\frac{\pi }{2}\right)=\sin x,\sin \left(x+\frac{\pi }{2}\right)=\cos x.$$

Pythagorean identity:

$${\cos }^{2}x+{\sin }^{2}x=1.$$

Addition formulas:

$$\sin (x+y)=\sin x\cos y+\cos x\sin y,\cos (x+y)=\cos x\cos y-\sin x\sin y.$$

Other trigonometric functions:

$$\tan x=\frac{\sin x}{\cos x},\cot x=\frac{\cos x}{\sin x},\sec x=\frac{1}{\cos x},\csc x=\frac{1}{\sin x}.$$

Polar parametrisation of a circle of radius $r$:

$$x=r\cos \theta ,y=r\sin \theta ,0\le \theta \le 2\pi .$$

Inverse functions are defined by restricting domains to ensure bijectivity:

Function	Restricted domain	Range
$\arccos x$	$[0,\pi ]$	$[0,\pi ]$
$\arcsin x$	$[-\frac{\pi }{2},\frac{\pi }{2}]$	$[-\frac{\pi }{2},\frac{\pi }{2}]$
$\arctan x$	$(-\frac{\pi }{2},\frac{\pi }{2})$	$\mathbb{R}$

The arctangent has horizontal asymptotes: $\arctan x\to \frac{\pi }{2}$ as $x\to +\infty$ and $\arctan x\to -\frac{\pi }{2}$ as $x\to -\infty$.

Composite inverse-trig expressions (e.g. $\sin (\arctan x)$, $\cos (\arctan x)$) can be simplified using a right-triangle argument together with ${\cos }^{2}x+{\sin }^{2}x=1$.

Applications

Trigonometric functions model periodic phenomena: oscillations in mechanics ($x(t)=A\cos \omega t$), electromagnetic waves, and audio signals. The addition formulas are the foundation of Fourier analysis, which decomposes arbitrary periodic functions into sinusoidal components.

Trade-offs

Inverse trigonometric functions are only single-valued after domain restriction; different branches give different answers for the “same” inverse problem. Arguments must be in radians for calculus identities (derivatives, Taylor series) to hold without correction factors.

Links

• Complex Numbers — Euler’s formula connects $e^{i\theta }$ to $\cos \theta$ and $\sin \theta$
• Growth, Decay and Oscillation — sinusoidal solutions to $\ddot{x}+{\omega }^{2}x=0$