Question

How are the intervals similar where the graphs of y = sin x and y = cos x are increasing and where the graphs for y = sin x and y = cos x are decreasing? How are they different?

Answer 1

The graphs of y = sin x and y = cos x both show periodic behavior with the same amplitude and period, yet they have different intervals of increase and decrease due to a phase shift; overall, they are more similar than different due to their trigonometric properties.

Step-by-step explanation:

The graphs of y = sin x and y = cos x have several similarities and differences in terms of where they are increasing and decreasing. First, let's explore the similarities:

• Both sin x and cos x graphs are periodic functions, repeating their values in regular intervals.
• They both have the same amplitude (the maximum distance from the axis of the graph to its peak or trough is 1) and period (they complete a cycle every 2π radians).

Now for the differences:

• The sin x graph is increasing in the interval (-π/2, π/2), while the cos x graph is increasing in the interval (0, π).
• The sin x graph is decreasing in the interval (π/2, 3π/2), whereas the cos x graph is decreasing in the interval (π, 2π).

Although the intervals differ due to a phase shift, the overall patterns are quite similar because of the same amplitude and period. When comparing sin x and cos x, one could say they are more similar than different due to the fundamental nature of their trigonometric properties, despite having different intervals of increase and decrease.