Trigonometric graphs

The sine graph

y = sin x (degrees)

Plot and join smoothly: a wave that starts at 0, rises to 1, falls through 0 to −1, and returns to 0. Then it repeats.

Features of y = sin x

The cosine graph

y = cos x

Same shape, same period and amplitude, but it starts at a maximum. The cosine graph is the sine graph shifted 90° left: cos x = sin(x + 90°).

The tangent graph

y = tan x

• Period: 180° (π)
• Vertical asymptotes at 90°, 270°, … (where cos x = 0)
• Range: all real numbers
• No amplitude (it has no max or min)

Sinusoidal graphs

Sine and cosine are sinusoidal graphs: smooth, repeating waves. One complete repetition is a cycle.

Key-point method

To sketch one cycle quickly, use five key points: start, quarter, half, three-quarters and end of the period.

For sin x: (0, 0), (90°, 1), (180°, 0), (270°, −1), (360°, 0).

Reading values

From the graph of y = sin x, sin x = 0.5 at x = 30° and 150° (in one cycle). The graph shows that trig equations usually have more than one solution.

Radians on the axis

In radians, the key points for sin x are at 0, π/2, π, 3π/2, 2π.

Worked examples

Example 1: Where is y = cos x at its minimum between 0° and 360°? 180°.

Example 2: Zeros of y = sin x between 0 and 2π? 0, π, 2π.

Example 3: Asymptotes of y = tan x between 0° and 360°? 90° and 270°.

Common mistakes

• Starting cos x at 0 instead of 1.
• Drawing pointed peaks instead of smooth curves.
• Giving tan x an amplitude.

Try it yourself

1. Period of y = cos x in radians?
2. Range of y = sin x?
3. At what x (0°–360°) is sin x = −1?
4. How many cycles of y = sin x fit between 0° and 720°?

Answers: 1) 2π 2) −1 ≤ y ≤ 1 3) 270° 4) 2

Quick recap

• Sine and cosine are sinusoidal graphs with period 360°, amplitude 1, midline y = 0.
• Cosine is sine shifted 90° left.
• Tangent has period 180° and vertical asymptotes.