Graphs of sine, cosine and tangent

Sine and cosine

y = sin x and y = cos x have period 360° (2π) and lie between -1 and 1. sin x crosses zero at 0°, 180°, 360°; cos x starts at 1 when x = 0. The cos graph is the sin graph translated 90° to the left.

Tangent

y = tan x has period 180° (π). It has vertical asymptotes at x = 90°, 270°, ... where cos x = 0, and takes every real value.

Exact values and symmetry

Learn the table. Symmetry gives more angles: sin(180° - x) = sin x, cos(360° - x) = cos x, tan(180° + x) = tan x.

Exact values to learn.

Transformed trig graphs

• y = 3 sin x: amplitude 3
• y = sin 2x: period 360°/2 = 180°
• y = sin(x + 30°): translated 30° to the left

Worked example

Find all x with 0 ≤ x ≤ 360° such that sin x = 1/2.

1. sin 30° = 1/2
2. sin(180° - 30°) = sin 30°, so 150° also works

Answer: x = 30° or x = 150°

Check yourself

1. What is the exact value of tan 60°?
   • sqrt(3)
   • 1/sqrt(3)
   • sqrt(3)/2
   • 1/2
   Show answer

   sqrt(3). tan 60° = sin 60° / cos 60° = (sqrt(3)/2)/(1/2) = sqrt(3).

2. What is the period of y = sin 2x, in degrees?
   Show answer

   180 °. Doubling x squashes the graph by a factor of 2: 360°/2 = 180°.

3. What is the minimum value of y = 2 + 3 sin x?
   Show answer

   -1. The smallest value of sin x is -1, so y = 2 - 3 = -1.

4. Where are the asymptotes of y = tan x for 0 ≤ x ≤ 360°?
   • x = 0° and x = 180°
   • x = 180° only
   • x = 45° and x = 225°
   • x = 90° and x = 270°
   Show answer

   x = 90° and x = 270°. tan x = sin x / cos x is undefined where cos x = 0: at 90° and 270°.

5. How is y = cos(x + 60°) related to y = cos x?
   • Translated 60° to the right
   • Translated 60° to the left
   • Translated 60 units up
   • Stretched by scale factor 60
   Show answer

   Translated 60° to the left. Adding inside the bracket moves the graph left.

6. Which of these is equal to sin 150°?
   • -sin 30°
   • sin 60°
   • cos 30°
   • sin 30°
   Show answer

   sin 30°. sin(180° - x) = sin x, so sin 150° = sin 30° = 1/2.

7. How many times does y = sin x meet the x-axis for 0 ≤ x ≤ 360°?
   Show answer

   3. sin x = 0 at x = 0°, 180° and 360°: three times.

Practise with instant feedback Print this lesson

Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3)
• Pearson Edexcel IAL Pure Mathematics 1 scheme of work

Last updated 7 November 2026. Spotted a mistake? Tell us.