Graphs of Sine and Cosine Functions: Properties, Variations, and Transformations

Graphs of Sine and Cosine Functions

Introduction

The sine and cosine functions are fundamental periodic functions in trigonometry. Their graphs, called sinusoidal graphs, are widely used to model periodic phenomena in mathematics, physics, and engineering. This section explores the properties, variations, and transformations of these graphs.

y = sin x: Properties and Graph

Key Properties of y = sin x

• Domain: $(-\infty, \infty)$

• Range: $[-1, 1]$

• Period: $2\pi$

• Odd Function: $\sin(-x) = -\sin(x)$

The sine function completes one full cycle over an interval of $2\pi$ radians. The graph oscillates smoothly between -1 and 1.

Key Points and Behavior

• At $x = 0$, $y = 0$

• At $x = \frac{\pi}{2}$, $y = 1$ (maximum)

• At $x = \pi$, $y = 0$

• At $x = \frac{3\pi}{2}$, $y = -1$ (minimum)

• At $x = 2\pi$, $y = 0$

Graphing Variations of y = sin x

General Form: $y = A \sin(Bx)$

Variations of the sine function involve changes in amplitude and period. The general form is $y = A \sin(Bx)$, where:

• Amplitude: $|A|$ (vertical stretch or shrink)

• Period: $\frac{2\pi}{B}$ (horizontal stretch or shrink)

Example: $y = 3\sin x$

• Amplitude: 3

• Period: $2\pi$

The graph oscillates between -3 and 3, with the same period as $y = \sin x$.

Example: $y = 2\sin\left(\frac{1}{2}x\right)$

• Amplitude: 2

• Period: $\frac{2\pi}{1/2} = 4\pi$

The graph stretches horizontally, completing one cycle over $4\pi$ units.

Phase Shift: $y = A\sin(Bx - C)$

Horizontal Shifts (Phase Shift)

The function $y = A\sin(Bx - C)$ shifts the graph horizontally by $\frac{C}{B}$ units. If $\frac{C}{B} > 0$, the shift is to the right; if $\frac{C}{B} < 0$, the shift is to the left.

Example: $y = 3\sin\left(2x - \frac{\pi}{3}\right)$

• Amplitude: 3

• Period: $\frac{2\pi}{2} = \pi$

• Phase Shift: $\frac{\pi}{6}$ to the right

y = cos x: Properties and Graph

Key Properties of y = cos x

• Domain: $(-\infty, \infty)$

• Range: $[-1, 1]$

• Period: $2\pi$

• Even Function: $\cos(-x) = \cos(x)$

Example: $y = -4\cos(\pi x)$

• Amplitude: 4

• Period: $\frac{2\pi}{\pi} = 2$

Phase Shift: $y = A\cos(Bx - C)$

Horizontal Shifts (Phase Shift)

The function $y = A\cos(Bx - C)$ shifts the graph horizontally by $\frac{C}{B}$ units. The amplitude and period are determined as before.

Example: $y = \frac{3}{2}\cos(2x + \pi)$

• Amplitude: $\frac{3}{2}$

• Period: $\frac{2\pi}{2} = \pi$

• Phase Shift: $-\frac{\pi}{2}$ (to the left)

Vertical Shifts of Sinusoidal Graphs

General Form: $y = A\sin(Bx - C) + D$ or $y = A\cos(Bx - C) + D$

The constant $D$ causes a vertical shift in the graph. The sinusoidal graph oscillates about the line $y = D$ instead of the x-axis.

• Maximum value: $D + |A|$

• Minimum value: $D - |A|$

Example: $y = 2\cos x + 1$

• Amplitude: 2

• Period: $2\pi$

• Vertical shift: 1 unit upward

Summary Table: Properties of Sine and Cosine Functions

Function	Amplitude	Period	Phase Shift	Vertical Shift	Range
$y = A\sin(Bx - C) + D$	$|A|$	$\frac{2\pi}{B}$	$\frac{C}{B}$	$D$	$[D - |A|, D + |A|]$
$y = A\cos(Bx - C) + D$	$|A|$	$\frac{2\pi}{B}$	$\frac{C}{B}$	$D$	$[D - |A|, D + |A|]$

Additional info: Sinusoidal graphs are used to model periodic phenomena such as sound waves, tides, and seasonal temperatures. Understanding their transformations is essential for analyzing real-world periodic behavior.