Graphs of Sine and Cosine Functions – Precalculus Study Notes

Chapter 7: The Graphs of Trigonometric Functions

Section 7.1: The Graphs of Sine and Cosine

This section introduces the fundamental properties and graphs of the sine and cosine functions, including their general forms, transformations, and how to determine equations from graphs.

General Angle Definitions of the Trigonometric Functions

Coordinate Definitions

For any angle θ in standard position, if P(x, y) is a point on the terminal side and r = \sqrt{x^2 + y^2} is the distance from the origin to P, then the trigonometric functions are defined as:

• sin θ: $\sin \theta = \frac{y}{r}$

• cos θ: $\cos \theta = \frac{x}{r}$

• tan θ: $\tan \theta = \frac{y}{x},\ x \neq 0$

• csc θ: $\csc \theta = \frac{r}{y},\ y \neq 0$

• sec θ: $\sec \theta = \frac{r}{x},\ x \neq 0$

• cot θ: $\cot \theta = \frac{x}{y},\ y \neq 0$

Graphs of Sine and Cosine Functions

Graph of $y = \sin x$ on $[0, 2\pi]$

The graph of the sine function is a smooth, continuous wave that oscillates between -1 and 1. It starts at the origin, reaches a maximum at $\frac{\pi}{2}$, returns to zero at $\pi$, reaches a minimum at $\frac{3\pi}{2}$, and completes a cycle at $2\pi$.

Periodic Functions

A function f is periodic if there is a positive number P such that $f(x + P) = f(x)$ for all x in the domain. The smallest such P is called the period of f.

• For $y = \sin x$ and $y = \cos x$, the period is $2\pi$.

Characteristics of the Sine Function

Key Properties

• Domain: $(-\infty, \infty)$

• Range: $[-1, 1]$

• Period: $2\pi$

• y-intercept: 0

• x-intercepts (zeros): $x = n\pi$, where $n$ is an integer

• Odd function: $\sin(-x) = -\sin x$ (symmetric about the origin)

Relative Extrema

• Relative maximum: at $x = \frac{\pi}{2} + 2n\pi$, value is 1

• Relative minimum: at $x = \frac{3\pi}{2} + 2n\pi$, value is -1

Example: Finding x-intercepts of $y = \sin x$

On the interval $\left[-3\pi, \frac{7\pi}{4}\right]$, the x-intercepts are the values of $x$ where $y = 0$:

• $x = -3\pi, -2\pi, -\pi, 0, \pi$

Characteristics of the Cosine Function

Key Properties

• Domain: $(-\infty, \infty)$

• Range: $[-1, 1]$

• Period: $2\pi$

• y-intercept: 1

• x-intercepts (zeros): $x = (2n+1)\frac{\pi}{2}$, where $n$ is an integer

• Even function: $\cos(-x) = \cos x$ (symmetric about the y-axis)

Relative Extrema

• Relative maximum: at $x = 2n\pi$, value is 1

• Relative minimum: at $x = \pi + 2n\pi$, value is -1

Amplitude and Period of Sine and Cosine Functions

Amplitude

The amplitude of $y = A\sin x$ or $y = A\cos x$ is $|A|$, representing half the distance between the maximum and minimum values. The range is $[-|A|, |A|]$.

Period of $y = \sin(Bx)$ and $y = \cos(Bx)$

The period of these functions is $\frac{2\pi}{|B|}$, where $B > 0$.

Transformations and Sketching Graphs

General Forms

• $y = A\sin(Bx)$

• $y = A\cos(Bx)$

Where:

• A controls amplitude

• B controls period

Steps for Sketching $y = A\sin(Bx)$ or $y = A\cos(Bx)$

1. If $B < 0$, use the even/odd properties to rewrite with $B > 0$.

2. Determine amplitude $|A|$ and range $[-|A|, |A|]$.

3. Determine period $P = \frac{2\pi}{|B|}$.

4. Divide one period into four equal subintervals to find quarter points.

5. Multiply the y-coordinates of the standard sine or cosine quarter points by $A$.

6. Connect the points smoothly to complete one cycle.

Example: Sketching $y = -2\cos\left(\frac{4}{3}x\right)$

• Amplitude: $|-2| = 2$

• Range: $[-2, 2]$

• Period: $\frac{2\pi}{4/3} = \frac{3\pi}{2}$

• Quarter points: Divide $[0, \frac{3\pi}{2}]$ into four equal parts

• Multiply standard cosine y-values by -2

Determining the Equation from a Graph

Steps

1. Decide if the graph is of the form $y = A\sin(Bx)$ (passes through origin) or $y = A\cos(Bx)$ (does not pass through origin).

2. Find the period $P$ from the graph, then $B = \frac{2\pi}{P}$.

3. Use a known point on the graph to solve for $A$.

Example: Given a graph that completes one cycle on $[0, 2\pi]$ and passes through $(0, 4)$

• Since it does not pass through the origin, use $y = A\cos(Bx)$

• Period $P = 2\pi \implies B = 1$

• At $x = 0$, $y = 4 \implies 4 = A\cos(0) \implies A = 4$

• Equation: $y = 4\cos(x)$

Summary Table: Properties of Sine and Cosine Functions

Property	$y = \sin x$	$y = \cos x$
Domain	$(-\infty, \infty)$	$(-\infty, \infty)$
Range	$[-1, 1]$	$[-1, 1]$
Period	$2\pi$	$2\pi$
y-intercept	0	1
x-intercepts	$n\pi$	$(2n+1)\frac{\pi}{2}$
Symmetry	Odd (origin)	Even (y-axis)