Sine and Cosine Graphs

Understanding Sine and Cosine Graphs

Sine and cosine functions are fundamental in trigonometry, and their graphs are wave-like patterns that repeat at regular intervals. These functions are periodic, meaning they repeat their values in regular cycles.

Key Characteristics

• Amplitude: The height from the center line to the peak (or trough) of the graph. For y = a \sin x or y = b \cos x, the amplitude is the absolute value of a or b.
• Period: The length of one complete cycle of the graph. For sine and cosine functions, the standard period is 2\pi.
• Frequency: The number of cycles the function completes in a given interval. Frequency is the reciprocal of the period.
• Phase Shift: A horizontal shift left or right in the graph.
• Vertical Shift: A shift up or down in the graph.

Graphing Sine and Cosine Functions

To graph y = a \sin x or y = b \cos x:

• Identify the amplitude, period, and any shifts.
• Plot key points: maximum, minimum, and intercepts.
• Draw a smooth curve through these points.

Examples

Example 1: Graph y = 2 \sin x

Example 2: Graph y = 3 \cos x