Advanced Sine and Cosine Graphs

Understanding Sine and Cosine Graphs

Sine and cosine graphs are fundamental in trigonometry. They help us visualize how the sine and cosine functions behave over different angles.

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 |a| or |b|.
• Period: The distance (along the x-axis) required for the function to complete one full cycle. For sine and cosine, the period is 360° or 2\pi radians.
• Phase Shift: A horizontal shift left or right. This occurs if there is an addition or subtraction inside the function, like y = \sin(x - c).
• Vertical Shift: Moving the graph up or down, such as in y = \sin x + d.

Graphing y = a sin x and y = b cos x

Let's explore how to graph these functions:

 

Examples

Example 1: Graph y = 4\sin x

Solution:

• Amplitude = 4
• Period = 360° or 2\pi radians
• Key points: (0, 0), (90°, 4), (180°, 0), (270°, -4), (360°, 0)

Plot these points and draw a smooth curve through them.