Graphs of Sine and Cosine Functions: Amplitude, Period, and Transformations

Trigonometric Functions

Introduction to Sine and Cosine Functions

The sine and cosine functions are fundamental periodic functions in trigonometry, widely used to model oscillatory phenomena. Their graphs exhibit regular patterns characterized by amplitude, period, and symmetry.

Graphs of the Sine Function

Key Values and Graph of y = sin x

The sine function, y = sin x, oscillates between -1 and 1 and is periodic with period $2\pi$. The table below shows key values for y = sin x:

x	y = sin x	(x, y)
0	0	(0, 0)
$\frac{\pi}{6}$	$\frac{1}{2}$	($\frac{\pi}{6}$, $\frac{1}{2}$)
$\frac{\pi}{2}$	1	($\frac{\pi}{2}$, 1)
$\frac{5\pi}{6}$	$\frac{1}{2}$	($\frac{5\pi}{6}$, $\frac{1}{2}$)
$\pi$	0	($\pi$, 0)
$\frac{7\pi}{6}$	$-\frac{1}{2}$	($\frac{7\pi}{6}$, $-\frac{1}{2}$)
$\frac{3\pi}{2}$	-1	($\frac{3\pi}{2}$, -1)
$\frac{11\pi}{6}$	$-\frac{1}{2}$	($\frac{11\pi}{6}$, $-\frac{1}{2}$)
$2\pi$	0	($2\pi$, 0)

The graph of y = sin x for $0 \leq x \leq 2\pi$ is shown below:

For all real x, the graph repeats every $2\pi$:

Properties of the Sine Function

• Domain: All real numbers.

• Range: $[-1, 1]$.

• Odd Function: Symmetric about the origin; $\sin(-x) = -\sin(x)$.

• Periodicity: Period $2\pi$.

• x-intercepts: $x = 0, \pi, 2\pi, \ldots$

• Maximum: 1 at $x = \frac{\pi}{2}, \frac{5\pi}{2}, \ldots$

• Minimum: -1 at $x = \frac{3\pi}{2}, \frac{7\pi}{2}, \ldots$

Graphs of the Cosine Function

Key Values and Graph of y = cos x

The cosine function, y = cos x, also oscillates between -1 and 1 and is periodic with period $2\pi$. The table below shows key values for y = cos x:

x	y = cos x	(x, y)
0	1	(0, 1)
$\frac{\pi}{3}$	$\frac{1}{2}$	($\frac{\pi}{3}$, $\frac{1}{2}$)
$\frac{\pi}{2}$	0	($\frac{\pi}{2}$, 0)
$\frac{2\pi}{3}$	$-\frac{1}{2}$	($\frac{2\pi}{3}$, $-\frac{1}{2}$)
$\pi$	-1	($\pi$, -1)
$\frac{4\pi}{3}$	$-\frac{1}{2}$	($\frac{4\pi}{3}$, $-\frac{1}{2}$)
$\frac{3\pi}{2}$	0	($\frac{3\pi}{2}$, 0)
$\frac{5\pi}{3}$	$\frac{1}{2}$	($\frac{5\pi}{3}$, $\frac{1}{2}$)
$2\pi$	1	($2\pi$, 1)

The graph of y = cos x for $0 \leq x \leq 2\pi$ is shown below:

For all real x, the graph repeats every $2\pi$:

Properties of the Cosine Function

• Domain: All real numbers.

• Range: $[-1, 1]$.

• Even Function: Symmetric about the y-axis; $\cos(-x) = \cos(x)$.

• Periodicity: Period $2\pi$.

• x-intercepts: $x = \frac{\pi}{2}, \frac{3\pi}{2}, \ldots$

• Maximum: 1 at $x = 0, 2\pi, 4\pi, \ldots$

• Minimum: -1 at $x = \pi, 3\pi, 5\pi, \ldots$

Transformations of Sine and Cosine Functions

Amplitude and Period

For functions of the form y = A sin(\omega x) or y = A cos(\omega x), the amplitude and period are determined by the constants A and $\omega$:

• Amplitude: $|A|$

• Period: $\frac{2\pi}{\omega}$

Vertical Stretch and Horizontal Compression

Multiplying by A stretches or compresses the graph vertically. Replacing x by $\omega x$ compresses or stretches the graph horizontally.

• Vertical Stretch: $y = 2 \cos x$ is twice as tall as $y = \cos x$.

• Horizontal Compression: $y = 2 \cos(3x)$ compresses the period by a factor of $\frac{1}{3}$.

Graphing Sinusoidal Functions Using Key Points

Steps for Graphing y = A sin(\omega x) or y = A cos(\omega x)

To graph a sinusoidal function, follow these steps:

1. Determine the amplitude and period.

2. Divide the interval $[0, \frac{2\pi}{\omega}]$ into four equal subintervals.

3. Use the endpoints to obtain five key points.

4. Plot the points and draw the graph for one cycle. Extend as needed.

Example: Graphing y = 2 sin(2x)

For y = 2 sin(2x):

• Amplitude: 2

• Period: $\frac{2\pi}{2} = \pi$

• Key x-coordinates: $0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \pi$

• Key points: (0,0), ($\frac{\pi}{4}$,2), ($\frac{\pi}{2}$,0), ($\frac{3\pi}{4}$,-2), ($\pi$,0)

Relationship Between Sine and Cosine

Phase Shift

The sine and cosine functions are related by a phase shift:

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

Summary Table: Steps for Graphing Sinusoidal Functions

Step	Description
1	Determine amplitude and period
2	Divide interval $[0, \frac{2\pi}{\omega}]$ into four equal parts
3	Find five key points
4	Plot points and draw one cycle

Conclusion

Understanding the graphs, properties, and transformations of sine and cosine functions is essential for mastering trigonometric concepts in precalculus. These functions model periodic behavior and are foundational for further study in mathematics and science.