Circular Functions

Definition

Circular functions define trigonometric functions using the unit circle, where the input is a real number representing arc length (or angle in radians).

The Unit Circle

A circle with center at the origin and radius 1:

$x^2 + y^2 = 1$

Wrapping Function

The wrapping function $W(t)$ maps a real number $t$ to a point $(x, y)$ on the unit circle:

• Start at $(1, 0)$
• Move counterclockwise for $t > 0$
• Move clockwise for $t < 0$
• Arc length traveled $= |t|$

$W(t) = (\cos t, \sin t)$

Circular Function Definitions

For any real number $t$ and point $P(x, y) = W(t)$ on the unit circle:

Function	Definition
$\cos t$	x-coordinate of $P$
$\sin t$	y-coordinate of $P$
$\tan t$	$\frac{y}{x} = \frac{\sin t}{\cos t}$
$\cot t$	$\frac{x}{y} = \frac{\cos t}{\sin t}$
$\sec t$	$\frac{1}{x} = \frac{1}{\cos t}$
$\csc t$	$\frac{1}{y} = \frac{1}{\sin t}$

The Sine Function: $\sin t$

Graph Characteristics

• Domain: All real numbers
• Range: $[-1, 1]$
• Period: $2\pi$
• Amplitude: $1$
• Zeros: $t = n\pi$ ($n$ is any integer)
• Maximum: $1$ at $t = \frac{\pi}{2} + 2\pi n$
• Minimum: $-1$ at $t = \frac{3\pi}{2} + 2\pi n$

Key Points (one period)

$t$	$0$	$\frac{\pi}{2}$	$\pi$	$\frac{3\pi}{2}$	$2\pi$
$\sin t$	$0$	$1$	$0$	$-1$	$0$

The Cosine Function: $\cos t$

Graph Characteristics

• Domain: All real numbers
• Range: $[-1, 1]$
• Period: $2\pi$
• Amplitude: $1$
• Zeros: $t = \frac{\pi}{2} + n\pi$
• Maximum: $1$ at $t = 2\pi n$
• Minimum: $-1$ at $t = \pi + 2\pi n$

Key Points (one period)

$t$	$0$	$\frac{\pi}{2}$	$\pi$	$\frac{3\pi}{2}$	$2\pi$
$\cos t$	$1$	$0$	$-1$	$0$	$1$

Relationship Between Sine and Cosine

$\cos t = \sin\left(t + \frac{\pi}{2}\right)$

$\sin t = \cos\left(t - \frac{\pi}{2}\right)$

The cosine graph is the sine graph shifted left by $\frac{\pi}{2}$.

Fundamental Identity

For any point $(x, y)$ on the unit circle:

$x^2 + y^2 = 1$

$\cos^2 t + \sin^2 t = 1$

This is the Pythagorean Identity.

Even and Odd Properties

Cosine is even:

$\cos(-t) = \cos t$

Sine is odd:

$\sin(-t) = -\sin t$

Period and Periodicity

Both sine and cosine repeat every $2\pi$:

$\sin(t + 2\pi) = \sin t$

$\cos(t + 2\pi) = \cos t$

For any function $f$ with period $P$:

$f(t + P) = f(t)$

Special Values from Unit Circle

$t$	$(\cos t, \sin t)$
$0$	$(1, 0)$
$\frac{\pi}{6}$	$\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$
$\frac{\pi}{4}$	$\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$
$\frac{\pi}{3}$	$\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$
$\frac{\pi}{2}$	$(0, 1)$
$\pi$	$(-1, 0)$
$\frac{3\pi}{2}$	$(0, -1)$
$2\pi$	$(1, 0)$

Applications

1. Simple Harmonic Motion: Position of oscillating object

$y = A \sin(\omega t + \varphi)$

2. Circular Motion: Projection onto axes

$x(t) = r \cos(\omega t)$

$y(t) = r \sin(\omega t)$

3. Waves: Modeling periodic phenomena

$y = A \sin(kx - \omega t)$