Trigonometric Functions¶

Definition¶

Let $x \in \mathbb{R}\text{.}$ Consider the point $(\cos x, \sin x)$ on the unit circle $\{(u,v)\in\mathbb{R}^2 \colon u^2+v^2=1\}$ obtained by rotating the point $(1,0)$ counterclockwise by an angle $x$ (measured in radians). The functions $\sin x$ and $\cos x$ are defined as the second and first coordinates of this point, respectively.

For all $x \in \mathbb{R}$ such that $\cos x \neq 0\text{,}$ the tangent function is defined by $\displaystyle \tan x = \frac{\sin x}{\cos x}\text{.}$

Interpretation¶

The trigonometric functions turn an angle into coordinates: $\cos x$ and $\sin x$ are the horizontal and vertical coordinates of the point reached by turning through the angle $x$ on the unit circle. Because a full turn of $2\pi$ brings the point back to where it started, they provide periodic models for oscillatory behaviour.

A unit circle with a point at angle x whose coordinates are cos x and sin x, next to the graphs of sine, cosine and tangent over one period, with vertical asymptotes of the tangent.

The point at angle $x$ on the unit circle is $(\cos x,\sin x)$ : $\cos x$ is its horizontal coordinate (red dashed), $\sin x$ its vertical one (green dashed), and $\tan x=\sin x/\cos x$ $sin x / cos x$ . On the right the same coordinates are plotted against $x$ . The tangent has no value at $\tfrac{\pi}{2}$ and $\tfrac{3\pi}{2}$ , where $\cos x=0$ $0$ ; near these dashed lines its graph shoots up or down.

Standard facts can be read off the circle. The point lies on $u^2+v^2=1$ $1$ , so $\cos^2x+\sin^2x=1$ and both values lie in $[-1,1]$ . Turning by $2\pi$ more gives the same point, so $\sin(x+2\pi)=\sin x$ and $\cos(x+2\pi)=\cos x$ $cos x$ . Turning by $-x$ instead of $x$ reflects the point in the horizontal axis, so $\cos(-x)=\cos x$ and $\sin(-x)=-\sin x$ $- sin x$ . The tangent is the slope of the line from the origin through the point; turning by a further $\pi$ gives the opposite point on the same line, which is why $\tan$ has period $\pi$ rather than $2\pi$ .

A common slip is to measure the angle in degrees. Here $x$ is in radians, the length of the arc travelled on the unit circle, so a full turn is $2\pi$ , and $\sin 1$ is the sine of one radian, about $57^\circ$ . In Math II, Euler's formula links sine and cosine to the exponential function.

Key Results¶

Sine and cosine are defined for all real numbers.
Trigonometric functions are periodic.
Tangent is defined only where cosine is nonzero.

Examples¶

$\sin 0=0$ and $\cos 0=1\text{.}$
$\tan x$ is undefined at $x=\dfrac{\pi}{2}+\pi k\text{.}$

Prerequisites¶

Function

Sources¶

Math I: Calculus · Lecture 4

Aliases: trig functions, circular functions Keywords: sine, cosine, tangent Symbols: element of, pi, real numbers

Last updated: 20 October 2026 03:02