GCSE-readyGraphs of sine, cosine and tangent
The trigonometric graphs show at a glance how many solutions an equation has, and how numbers in the equation stretch or shift the graph.
The three graphs
sin x starts at 0; cos x starts at 1.
y = tan x repeats every 180° (π radians), takes every value, and has asymptotes at x = 90° and 270° (π2 and 3π2).
| x | 0 | π2 | π | 3π2 | 2π |
|---|---|---|---|---|---|
| sin x | 0 | 1 | 0 | −1 | 0 |
| cos x | 1 | 0 | −1 | 0 | 1 |
a sin bx
y = 3 cos 2x: maximum 3, period 180°.
y = 2 cos x + 1: the graph moves up 1, so it goes from −1 to 3 and crosses the y-axis at 3.
Counting solutions
So in 0° ≤ x ≤ 720° there are 4 solutions.
Sketching the graph first stops you missing solutions or giving ones outside the interval.
How many solutions does sin x = 0.3 have for 0° ≤ x ≤ 720°?
- Each 360° of y = sin x crosses y = 0.3 twice.
- 720° is two full cycles.
Answer: 4 solutions
sin and cos: period 360° (2π), range −1 to 1. tan: period 180° (π), asymptotes at 90° and 270°. y = a sin bx has amplitude a and period 360°/b.
Try two quick questions
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2 more questions
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Find the period of y = sin 5x, in degrees.
Show the answer
Answer: 72 °
The period of sin is 360°, so the period is 360 ÷ 5 = 72°.
Find the maximum value of y = 5 sin 4x.
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Answer: 5
The largest value of sin 4x is 1, so the maximum is 5.
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Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.