Properties of Cosine Functions - Algebra-2

1. Fundamental Concepts

Definition: The cosine function, denoted as , is a periodic function with a period of . It represents the x-coordinate of a point on the unit circle corresponding to an angle .
Domain and Range: The domain of is all real numbers, , and its range is .
Symmetry: The cosine function is even, meaning for all in its domain.

Period: Repeats every radians (period = ), so for any real .

Amplitude: The maximum distance from the midline (y = 0) to the peak/trough, equal to (no vertical stretch/compression in the standard function).

Symmetry: Even function, satisfying (symmetric about the y-axis).

Key Points: Critical points in one period ( to ):

(starting point, maximum)

(midline)

(minimum, trough)

(midline)

(end of one period, maximum)

Easy

Find the values of and .

Solution: Use key points of the cosine function:

(maximum), (minimum).

Medium

Verify if .

Solution:

1. Use even function property: .

2. From key points: .

3. Thus, (true).

Hard

For , find all in where .

Solution:

1. In one period ( ), at and (key points).

2. Extend to : add the period ( ) to , getting .

3. Final solutions: .

Reference Unit Circle: Always refer to the unit circle for standard values and symmetries.
Use Identities: Apply trigonometric identities such as to solve complex equations.
Graphical Interpretation: Use graphs to visualize solutions and understand periodicity and symmetry.