Graphs of Sine and Cosine: Sinusoids – Precalculus Study Notes

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 ($-\infty < x < \infty$)

• Range: $[-1, 1]$

• Continuity: The function is continuous everywhere.

• Periodicity: The sine function repeats every $2\pi$ 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 $\pi$.

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

The Cosine Function

• Domain: All real numbers ($-\infty < x < \infty$)

• Range: $[-1, 1]$

• Continuity: The function is continuous everywhere.

• Periodicity: The cosine function repeats every $2\pi$ 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 $\pi$.

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

Sinusoid Definition

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

$y = a \sin(bx + c) + d$ or $y = a \cos(bx + c) + d$

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:

$\text{Amplitude} = |a|$

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 $y = a \sin(bx + c) + d$ or $y = a \cos(bx + c) + d$:

$\text{Period} = \frac{2\pi}{|b|}$

Frequency of a Sinusoid

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

$\text{Frequency} = \frac{|b|}{2\pi}$

Phase Shift and Vertical Translation

• Phase Shift: The horizontal shift of the graph, determined by $\frac{-c}{b}$ in $y = a \sin(bx + c) + d$.

• Vertical Translation: The vertical shift, determined by $d$.

Constructing a Sinusoidal Model

To construct a sinusoidal model for periodic behavior:

1. Determine the maximum ($M$) and minimum ($m$) values.

2. Calculate amplitude: $A = \frac{M - m}{2}$

3. Calculate vertical shift: $k = \frac{M + m}{2}$

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

5. Calculate horizontal stretch/shrink: $b = \frac{2\pi}{p}$

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:

$d = a \sin(bt)$ or $d = a \cos(bt)$

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

• Frequency: $\text{Frequency} = \frac{b}{2\pi}$

• Period: $\text{Period} = \frac{2\pi}{b}$

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:

$d = 3 \cos(4\pi t)$

Here, amplitude $a = 3$, period $p = 0.5$, so $b = \frac{2\pi}{0.5} = 4\pi$. Frequency is $2$ cycles per second.

Summary Table: Sinusoidal Function Properties

Property	Sine Function	Cosine Function
Domain	All real numbers	All real numbers
Range	[-1, 1]	[-1, 1]
Period	$2\pi$	$2\pi$
Symmetry	Odd (origin)	Even (y-axis)
Amplitude	1	1
Inflection Points	Integer multiples of $\pi$	Odd integer multiples of $\frac{\pi}{2}$

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.