Graphs of sin x, cos x and tan x

The sine, cosine and tangent graphs repeat forever. Their symmetry lets you find every angle that solves an equation.

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y = sin x

Starts at (0, 0), rises to a maximum of 1 at 90°, back to 0 at 180°, down to −1 at 270° and back to 0 at 360°.
It repeats every 360° and always stays between −1 and 1.
x0°90°180°270°360°
sin x010−10
cos x10−101
90° 180° 270° 360° −1 1 x y (90°, 1) (270°, −1) y = sin x
y = sin x: maximum 1 at 90°, minimum −1 at 270°, repeats every 360°.

y = cos x and y = tan x

cos x starts at its maximum 1 at 0°, falls to −1 at 180° and is back to 1 at 360°. It repeats every 360°.
tan x passes through (0, 0), (180°, 0) and (360°, 0). It repeats every 180°.
tan x has asymptotes at 90° and 270°, where it is undefined.
90° 180° 270° 360° −1 1 x y (180°, −1) y = cos x 90° 180° 270° 360° −4 −2 2 4 x y asymptote asymptote y = tan x
y = cos x repeats every 360°; y = tan x repeats every 180° with asymptotes at 90° and 270°.

Using symmetry

The sine graph is symmetrical about x = 90°, so sin x = sin(180° − x).
sin x = 0.5 for 0° ≤ x ≤ 360°: x = 30° or 180° − 30° = 150°.
The cosine graph is symmetrical about x = 180°, so cos x = cos(360° − x).
cos x = −0.5: x = 120° or 360° − 120° = 240°.
90° 180° 270° 360° −1 1 x y symmetry x = 90° y = 0.5 30° 150°
sin x = 0.5 has two solutions from 0° to 360°: 30° and 150°.
Worked example

Solve sin x = 0.5 for 0° ≤ x ≤ 360°.

  1. Calculator: sin−1(0.5) = 30°
  2. Symmetry about 90°: 180° − 30° = 150°

Answer: x = 30° or 150°

Key idea

sin and cos repeat every 360° and stay between −1 and 1. tan repeats every 180° with asymptotes at 90° and 270°. Use sin x = sin(180° − x) and cos x = cos(360° − x) for the second solution.

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Extra practice

2 more questions

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  1. After how many degrees does the graph of y = cos x repeat?

    Show the answer

    Answer: 360 °

    cos x repeats every 360°.

  2. For 0° ≤ x ≤ 360°, where does y = sin x reach its minimum value of −1?

    1. x = 270°
    2. x = 90°
    3. x = 360°
    4. x = 180°
    Show the answer

    Answer: x = 270°

    sin 270° = −1.

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Sources and references

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.

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