20 min read

Graphs of Sine and Cosine

Plotting sin(θ) and cos(θ) against θ produces a repeating wave.

The Wave Shape

Plotting y = sin(θ) for θ from 0 ° to 360 ° produces a smooth wave: starting at 0, rising to a peak of 1 at 90 °, back to 0 at 180 °, down to a trough of - 1 at 270 °, and back to 0 at 360 °. y = cos(θ) has the exact same wave shape, just shifted: it starts at its peak (1) when θ = 0 °.

Key points on the sine and cosine graphs
Key points on the sine and cosine graphs

Example 1 — reading the graph

At what angle between 0 ° and 360 ° does cos(θ) first equal - 1 ? 1 cos starts at 1 (θ= 0 °), falls to 0 at 90 °, and continues falling. 2 The minimum value of - 1 is reached at θ = 180 °. Answer: 180 °

Practice

What is the amplitude of y = sin(θ)?

The Wave Shape

Plotting y = sin(θ) for θ from 0 ° to 360 ° produces a smooth wave: starting at 0, rising to a peak of 1 at 90 °, back to 0 at 180 °, down to a trough of - 1 at 270 °, and back to 0 at 360 °. y = cos(θ) has the exact same wave shape, just shifted: it starts at its peak (1) when θ = 0 °.

Key points on the sine and cosine graphs
Key points on the sine and cosine graphs

Example 1 — reading the graph

At what angle between 0 ° and 360 ° does cos(θ) first equal - 1 ? 1 cos starts at 1 (θ= 0 °), falls to 0 at 90 °, and continues falling. 2 The minimum value of - 1 is reached at θ = 180 °. Answer: 180 °

Practice

What is the amplitude of y = sin(θ)?