Families of Special Angles and Trigonometric Functions in the Coordinate Plane

Families of Special Angles

Quadrantal Family of Angles

The Quadrantal Family of Angles consists of angles in standard position whose terminal sides lie along the axes. These angles are coterminal with $0$,$\frac{\pi}{2}$,$\pi$, or$\frac{3\pi}{2}$radians (or $0^\circ$, $90^\circ$, $180^\circ$, $270^\circ$).

• Definition: Quadrantal angles have terminal sides on the x- or y-axis.

• Examples: $0^\circ$, $90^\circ$, $180^\circ$, $270^\circ$

• Application: Used to evaluate trigonometric functions at axis-aligned positions.

$\frac{\pi}{3}$ Family of Angles

The $\frac{\pi}{3}$ Family of Angles consists of angles coterminal with $\frac{\pi}{3}$, $2\frac{\pi}{3}$, $4\frac{\pi}{3}$, and $5\frac{\pi}{3}$ radians (or $60^\circ$, $120^\circ$, $240^\circ$, $300^\circ$).

• Definition: Angles whose reference angle is $\frac{\pi}{3}$.

• Examples: $60^\circ$, $120^\circ$, $240^\circ$, $300^\circ$

$\frac{\pi}{6}$ Family of Angles

The $\frac{\pi}{6}$ Family of Angles consists of angles coterminal with $\frac{\pi}{6}$, $5\frac{\pi}{6}$, $7\frac{\pi}{6}$, and $11\frac{\pi}{6}$ radians (or $30^\circ$, $150^\circ$, $210^\circ$, $330^\circ$).

• Definition: Angles whose reference angle is $\frac{\pi}{6}$.

• Examples: $30^\circ$, $150^\circ$, $210^\circ$, $330^\circ$

$\frac{\pi}{4}$ Family of Angles

The $\frac{\pi}{4}$ Family of Angles consists of angles coterminal with $\frac{\pi}{4}$, $3\frac{\pi}{4}$, $5\frac{\pi}{4}$, and $7\frac{\pi}{4}$ radians (or $45^\circ$, $135^\circ$, $225^\circ$, $315^\circ$).

• Definition: Angles whose reference angle is $\frac{\pi}{4}$.

• Examples: $45^\circ$, $135^\circ$, $225^\circ$, $315^\circ$

General Angle Definition and Trigonometric Functions

Coordinates and Distance Formula

For any angle $\theta$ in standard position, a point $P(x, y)$ on its terminal side is used to define trigonometric functions. The distance from the origin to $P$ is:

• Distance formula: $r = \sqrt{x^2 + y^2}$

Right Triangle Definitions of Trigonometric Functions

Trigonometric functions can be defined using the sides of a right triangle:

• $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$

• $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$

• $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

• $\csc \theta = \frac{\text{hypotenuse}}{\text{opposite}}$

• $\sec \theta = \frac{\text{hypotenuse}}{\text{adjacent}}$

• $\cot \theta = \frac{\text{adjacent}}{\text{opposite}}$

General Angle Definitions (Coordinate Plane)

For a point $P(x, y)$ on the terminal side of $\theta$ in standard position, the six trigonometric functions are defined as:

• $\sin \theta = \frac{y}{r}$

• $\cos \theta = \frac{x}{r}$

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

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

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

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

Values of Trigonometric Functions for Quadrantal Angles

Table of Trigonometric Function Values

The following table lists the values of the six trigonometric functions for the quadrantal angles $0$,$\frac{\pi}{2}$,$\pi$, and$\frac{3\pi}{2}$:

$\theta$	$\sin \theta$	$\cos \theta$	$\tan \theta$	$\csc \theta$	$\sec \theta$	$\cot \theta$
$0$	0	1	0	Undefined	1	Undefined
$\frac{\pi}{2}$	1	0	Undefined	1	Undefined	0
$\pi$	0	-1	0	Undefined	-1	Undefined
$\frac{3\pi}{2}$	-1	0	Undefined	-1	Undefined	0

Signs of Trigonometric Functions in Each Quadrant

Quadrant Sign Rules

The sign of each trigonometric function depends on the quadrant in which the terminal side of the angle lies:

• Quadrant I: All trigonometric functions are positive.

• Quadrant II: Sine and cosecant are positive.

• Quadrant III: Tangent and cotangent are positive.

• Quadrant IV: Cosine and secant are positive.

Reference Angles

Definition and Cases

A reference angle is the acute angle formed by the terminal side of $\theta$ and the nearest x-axis. Reference angles are used to simplify the evaluation of trigonometric functions for any angle.

• Quadrant I: $\theta_R = \theta_C$

• Quadrant II: $\theta_R = \pi - \theta_C$ (or $180^\circ - \theta_C$)

• Quadrant III: $\theta_R = \theta_C - \pi$ (or $\theta_C - 180^\circ$)

• Quadrant IV: $\theta_R = 2\pi - \theta_C$ (or $360^\circ - \theta_C$)

Steps for Evaluating Trigonometric Functions

Procedure

To evaluate a trigonometric function for a given angle:

1. Draw the angle and determine the quadrant in which the terminal side lies.

2. Determine if the sign of the function is positive or negative in that quadrant.

3. Determine the reference angle: $\frac{\pi}{6}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$, or a quadrantal angle.

4. Use the appropriate special right triangle to determine the value of the trigonometric function.

Additional info: These notes cover material from Precalculus Chapter 6 (An Introduction to Trigonometric Functions) and Chapter 7 (The Graphs of Trigonometric Functions), including angle families, reference angles, and the coordinate definitions of trigonometric functions.