Trigonometric Functions and the Unit Circle

Definition of Trigonometric Functions Using the Unit Circle

The six trigonometric functions can be defined using the unit circle, which is a circle of radius 1 centered at the origin in the coordinate plane. For a real number t, let P = (a, b) be the point on the unit circle corresponding to t (where t is the length of the arc from (1, 0) to P, measured in radians).

• Sine: (the y-coordinate of P)

• Cosine: (the x-coordinate of P)

• Tangent: , provided

• Cosecant: , provided

• Secant: , provided

• Cotangent: , provided

Trigonometric Functions of an Angle

If radians, then the six trigonometric functions of the angle are defined as:

Finding Trigonometric Function Values Using the Unit Circle

Example: Using a Point on the Circle

Given a point on the terminal side of an angle in standard position, and the circle , the trigonometric functions are:

• ,

• ,

• ,

• ,

Example: If is on the circle , then .

Special Triangles and the Unit Circle

45-45-90 and 30-60-90 Triangles

Special right triangles are used to find exact values of trigonometric functions for common angles. The 45-45-90 triangle has side ratios 1:1:, and the 30-60-90 triangle has side ratios 1::2. These triangles correspond to points on the unit circle for angles such as , , and .

Domain and Range of Trigonometric Functions

Domain and Range

The domain and range of the six trigonometric functions are as follows:

• Sine: Domain: all real numbers; Range:

• Cosine: Domain: all real numbers; Range:

• Tangent: Domain: all real numbers except odd multiples of ; Range:

• Cosecant: Domain: all real numbers except integer multiples of ; Range:

• Secant: Domain: all real numbers except odd multiples of ; Range:

• Cotangent: Domain: all real numbers except integer multiples of ; Range:

Periodic Properties of Trigonometric Functions

Definition of Periodic Function and Fundamental Period

A function is periodic if there is a positive number such that for all in the domain of . The smallest such $p$ is called the fundamental period.

• Sine and Cosine: Period is

• Tangent and Cotangent: Period is

Even-Odd Properties of Trigonometric Functions

Theorem: Even-Odd Properties

Trigonometric functions have specific symmetry properties:

• Even functions: ,

• Odd functions: , , ,

Using Symmetry and the Unit Circle to Find Exact Values

Examples with the Unit Circle

By using the symmetry of the unit circle and the coordinates of special points, you can find the exact values of trigonometric functions for common angles such as , , , , , etc.

Additional info: The images reinforce the geometric interpretation of trigonometric functions and their properties on the unit circle, as well as the use of symmetry and periodicity in evaluating trigonometric expressions.