Trigonometric Functions: Sine and Cosine

Objectives

• Discuss properties of the sine and cosine functions.

• Graph the sine and cosine functions.

• Find the amplitude and period of sinusoidal functions.

• Find phase shifts and graph sinusoidal functions of the forms and .

Domain and Range of Sine and Cosine

Unit Circle Definition

For any real number t, the point lies on the unit circle . This means both sine and cosine functions are defined for all real numbers.

• Domain:

• Range:

• Pythagorean Identity: For every real number :

Periodic Functions

Definition

A function f is periodic if there exists a positive number p such that:

• for every in the domain of .

• The smallest value of for which this holds is called the period of .

• The graph of over any interval of length is called one cycle of the graph.

Period of Sine and Cosine

• For every real number :

• Both sine and cosine functions have period .

Properties of Sine and Cosine

Summary Table

Examples and Applications

• Example: The graph of starts at , reaches a maximum at , returns to $0t = \pit = \frac{3\pi}{2}t = 2\pi$.

• Application: Sine and cosine functions are used to model periodic phenomena such as sound waves, light waves, and seasonal temperature variations.