Graphs of Trigonometric Functions

Sine Function: $y = \sin x$

Key Features

• Domain: All real numbers
• Range: $[-1, 1]$
• Period: $2\pi$
• Amplitude: $1$
• x-intercepts: $0, \pm\pi, \pm 2\pi, \ldots$ (multiples of $\pi$)
• Maximum: $1$ at $x = \frac{\pi}{2} + 2\pi n$
• Minimum: $-1$ at $x = \frac{3\pi}{2} + 2\pi n$

Key Points in One Period $[0, 2\pi]$

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

Cosine Function: $y = \cos x$

Key Features

• Domain: All real numbers
• Range: $[-1, 1]$
• Period: $2\pi$
• Amplitude: $1$
• x-intercepts: $\pm\frac{\pi}{2}, \pm\frac{3\pi}{2}, \ldots$ (odd multiples of $\frac{\pi}{2}$)
• Maximum: $1$ at $x = 2\pi n$
• Minimum: $-1$ at $x = \pi + 2\pi n$

Key Points in One Period $[0, 2\pi]$

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

Tangent Function: $y = \tan x$

Key Features

• Domain: $x \neq \frac{\pi}{2} + n\pi$
• Range: All real numbers
• Period: $\pi$
• Vertical asymptotes: $x = \frac{\pi}{2} + n\pi$
• x-intercepts: $0, \pm\pi, \pm 2\pi, \ldots$ (multiples of $\pi$)

Key Points in One Period $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$

$x$	$-\frac{\pi}{4}$	$0$	$\frac{\pi}{4}$
$\tan x$	$-1$	$0$	$1$

Cotangent Function: $y = \cot x$

Key Features

• Domain: $x \neq n\pi$
• Range: All real numbers
• Period: $\pi$
• Vertical asymptotes: $x = n\pi$
• x-intercepts: $\frac{\pi}{2} + n\pi$ (odd multiples of $\frac{\pi}{2}$)

Secant Function: $y = \sec x$

Key Features

• Domain: $x \neq \frac{\pi}{2} + n\pi$
• Range: $(-\infty, -1] \cup [1, \infty)$
• Period: $2\pi$
• Vertical asymptotes: $x = \frac{\pi}{2} + n\pi$
• No x-intercepts

Cosecant Function: $y = \csc x$

Key Features

• Domain: $x \neq n\pi$
• Range: $(-\infty, -1] \cup [1, \infty)$
• Period: $2\pi$
• Vertical asymptotes: $x = n\pi$
• No x-intercepts

Transformed Trig Functions

General Form

$y = A \sin(B(x - C)) + D$

$y = A \cos(B(x - C)) + D$

Parameter	Effect
$A$	Amplitude $=
$B$	Period $= \frac{2\pi}{
$C$	Phase shift (horizontal shift right by $C$)
$D$	Vertical shift (midline $y = D$)

For Tangent/Cotangent

$\text{Period} = \frac{\pi}{|B|}$

Summary Table

Function	Period	Amplitude	Domain Restrictions	Range
$\sin x$	$2\pi$	$1$	None	$[-1, 1]$
$\cos x$	$2\pi$	$1$	None	$[-1, 1]$
$\tan x$	$\pi$	N/A	$x \neq \frac{\pi}{2} + n\pi$	$\mathbb{R}$
$\cot x$	$\pi$	N/A	$x \neq n\pi$	$\mathbb{R}$
$\sec x$	$2\pi$	N/A	$x \neq \frac{\pi}{2} + n\pi$	$\lvert y \rvert \geq 1$
$\csc x$	$2\pi$	N/A	$x \neq n\pi$	$\lvert y \rvert \geq 1$