Trigonometric Functions

Graphs of Sine and Cosine: Sinusoids

The sine and cosine functions are fundamental in precalculus, especially for modeling periodic phenomena. Their graphs, known as sinusoids, exhibit regular, wave-like patterns and are used to describe cycles in mathematics, physics, and engineering.

The Sine Function

• Domain: All real numbers ()

• Range:

• Continuity: The function is continuous everywhere.

• Periodicity: The sine function repeats every units.

• Symmetry: Odd function; symmetric with respect to the origin.

• Boundedness: Maximum value is 1, minimum value is -1.

• Inflection Points: Occur at all integer multiples of .

• End Behavior: Does not approach any horizontal or vertical asymptotes.

The Cosine Function

• Domain: All real numbers ()

• Range:

• Continuity: The function is continuous everywhere.

• Periodicity: The cosine function repeats every units.

• Symmetry: Even function; symmetric with respect to the y-axis.

• Boundedness: Maximum value is 1, minimum value is -1.

• Inflection Points: Occur at all odd integer multiples of .

• End Behavior: Does not approach any horizontal or vertical asymptotes.

Sinusoid Definition

A sinusoid is any function that can be written in the form:

or

where a, b, c, and d are constants, and neither a nor b is zero.

Amplitude of a Sinusoid

The amplitude of a sinusoid is the absolute value of the coefficient a:

Graphically, amplitude is half the height of the wave from its maximum to minimum.

Period of a Sinusoid

The period is the length of one complete cycle of the wave. For a function or :

Frequency of a Sinusoid

The frequency is the number of cycles completed in a unit interval. It is the reciprocal of the period:

Phase Shift and Vertical Translation

• Phase Shift: The horizontal shift of the graph, determined by in .

• Vertical Translation: The vertical shift, determined by .

Constructing a Sinusoidal Model

To construct a sinusoidal model for periodic behavior:

1. Determine the maximum () and minimum () values.

2. Calculate amplitude:

3. Calculate vertical shift:

4. Determine the period (): the time interval of a single cycle.

5. Calculate horizontal stretch/shrink:

6. Choose the appropriate sinusoid based on initial conditions (maximum, minimum, or midpoint at a given time).

Simple Harmonic Motion

A point exhibits simple harmonic motion if its distance from the origin is given by:

or

where and are real numbers, and $b$ determines the frequency of oscillation.

• Frequency:

• Period:

Example: Modeling Harmonic Motion

A mass on a spring oscillates with maximum displacement 3 cm and completes one cycle in 0.5 seconds. The equation modeling this motion is:

Here, amplitude , period , so . Frequency is $2$ cycles per second.

Summary Table: Sinusoidal Function Properties

Additional info: Academic context and formulas have been expanded for clarity and completeness. Only image_2 is included as it directly illustrates amplitude and period of a sinusoidal graph.