Year 10: Sketching trigonometric functions

Trigonometric graphs show how sine, cosine, and tangent values change as the angle changes. In Year 10, the main goal is to recognise the basic shape, key points, and repeating pattern.

Sine and cosine repeat every

360 °

. Tangent also repeats, but its graph has breaks where the function is undefined.

Work through the sections in order, then try the linked Year 10 skill quizzes after each section.

The sine graph

The sine graph starts at $(0 °, 0)$ , rises to $1$ at $90 °$ , returns to $0$ at $180 °$ , falls to $- 1$ at $270 °$ , and comes back to $0$ at $360 °$ .

Example 1: Sketch

y = \sin x

from

0 °

360 °

. Plot the five key points:

(0, 0)

(90, 1)

(180, 0)

(270, - 1)

(360, 0)

, then draw a smooth curve through them.

Skill quiz: Sketch sine from its key points

Plot the zeros, maximum and minimum before drawing a smooth sine curve.

Sketch trigonometric functions

The cosine graph

Cosine begins at $1$ when $x = 0 °$ . That is the main visual difference from the sine graph.

Example 2: Key points for

y = \cos x

are

(0, 1)

(90, 0)

(180, - 1)

(270, 0)

(360, 1)

Skill quiz: Compare cosine with sine

Check the starting value and the positions of the maxima, minima and zeros.

Sketch trigonometric functions

The tangent graph

Tangent passes through $(0 °, 0)$ and $(180 °, 0)$ , but it is undefined at $90 °$ and $270 °$ . Those become vertical asymptote lines on the sketch.

Example 3: When sketching

y = \tan x

, first mark

x = 90 °

and

x = 270 °

as undefined, then draw each branch rising through the intercepts.

Skill quiz: Mark tangent asymptotes

Identify where tangent is undefined and draw each branch separately.

Sketch trigonometric functions

Common mistakes

Plotting the right points but joining them with straight lines instead of smooth curves.
Forgetting that cosine starts at $1$ , not $0$ .
Drawing tangent through its undefined points.

Skill quiz: Check curves and undefined points

Use smooth curves for sine and cosine, and keep tangent branches separate at the asymptotes.

Sketch trigonometric functions