Special Angle Families and Trigonometric Functions in the Coordinate Plane

Special Angle Families in Trigonometry

Quadrantal Family of Angles

The Quadrantal Family of Angles consists of angles in standard position that are coterminal with $0$,$\frac{\pi}{2}$,$\pi$, or$\frac{3\pi}{2}$. These angles correspond to the axes on the coordinate plane and are fundamental in trigonometric analysis.

• Definition: Quadrantal angles have their terminal sides lying along the x-axis or y-axis.

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

• Application: Used to determine the values of 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 $5\frac{\pi}{3}$. These angles are multiples of $60^\circ$ and are important for evaluating trigonometric functions at special values.

• Definition: Angles whose reference angle is $60^\circ$ ($\frac{\pi}{3}$ radians).

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

• Application: Used in problems involving equilateral triangles and hexagonal symmetry.

$\frac{\pi}{6}$ 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 $11\frac{\pi}{6}$. These are multiples of $30^\circ$ and are frequently used in trigonometric calculations.

• Definition: Angles whose reference angle is $30^\circ$ ($\frac{\pi}{6}$ radians).

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

• Application: Common in problems involving 30-60-90 triangles.

$\frac{\pi}{4}$ 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 $7\frac{\pi}{4}$. These are multiples of $45^\circ$ and are essential for evaluating trigonometric functions at these special values.

• Definition: Angles whose reference angle is $45^\circ$ ($\frac{\pi}{4}$ radians).

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

• Application: Used in problems involving squares and octagonal symmetry.

General Angle Definition of Trigonometric Functions

Coordinates and Distance Formula

For any angle $\theta$ in standard position, let $P(x, y)$ be a point on the terminal side of $\theta$. The distance from the origin to $P$ is given by:

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

Trigonometric Functions in Terms of Coordinates

The six trigonometric functions can be defined using the coordinates $(x, y)$ and the distance $r$:

• $\sin \theta = \frac{y}{r}$

• $\cos \theta = \frac{x}{r}$

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

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

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

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

Right Triangle Definitions

For acute angles, the trigonometric functions are defined as ratios of sides in a right triangle:

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

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

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

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

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

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

Values of Trigonometric Functions for Quadrantal Angles

Table of Trigonometric 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.

Mnemonic: "All Students Take Calculus" (ASTC) helps remember which functions are positive in each quadrant.

Reference Angles and Their Use

Definition of Reference Angle

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

• Reference angle symbol: $\theta_R$

• Application: The value of a trigonometric function for any angle is the same as for its reference angle, up to sign.

Reference Angle Cases by Quadrant

• Quadrant I: $\theta_C = \theta_R$

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

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

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

Steps for Evaluating Trigonometric Functions of Special Angles

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

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

3. Determine if the reference angle $\theta_R$ is $\frac{\pi}{6}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$, or $\frac{\pi}{2}$.

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

Example

Find $\sin(150^\circ)$:

• $150^\circ$ is in Quadrant II.

• Reference angle: $180^\circ - 150^\circ = 30^\circ$.

• $\sin(30^\circ) = \frac{1}{2}$, and sine is positive in Quadrant II.

• Therefore, $\sin(150^\circ) = \frac{1}{2}$.

Additional info: The notes provide a systematic approach to evaluating trigonometric functions for any angle using quadrant, sign, reference angle, and special triangle values.