The graphs of the trigonometric functions repeat in a regular pattern. You should be able to sketch y = sin x, y = cos x and y = tan x for 0° ≤ x ≤ 360° and read key values from them.
Sine and cosine
Both y = sin x and y = cos x are smooth waves with a period of 360° and an amplitude of 1, so their values lie between −1 and 1. The cosine curve is the sine curve shifted 90° to the left. Multiplying by a number changes the amplitude: y = a sin x has amplitude a and swings between −a and a, but its period stays 360°.
Key idea
y = sin x: 0 at 0°, maximum 1 at 90°, 0 at 180°, minimum −1 at 270°. y = cos x: 1 at 0°, 0 at 90°, minimum −1 at 180°.
The tangent graph
y = tan x has a period of 180° and no maximum or minimum. It has vertical asymptotes at 90° and 270°, where the value is undefined.
Worked example
Solve sin x = 0.5 for 0° ≤ x ≤ 360°.
The first solution is x = 30°. The sine curve is symmetric about 90°, so the second solution is 180° − 30° = 150°. There are two solutions: x = 30° and x = 150°. To find every solution in a range, always sketch the curve and use its symmetry rather than relying on the calculator's single answer.
Remember
- sin and cos: period 360°, range −1 to 1.
- tan: period 180°, undefined at 90° and 270°.
- For y = a sin x the amplitude is a.