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GCSE Maths · Algebra

Trigonometric graphs GCSE Questions and Worked Answers

Sine and cosine graphs repeat every 360° and stay between −1 and 1. Tangent repeats every 180° and has breaks where it is undefined. Use the graph to find every requested angle.

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Start with the meaning

What you need to know about trigonometric graphs

Imagine a point moving anticlockwise around a circle of radius one, starting at the right-hand edge. As it turns, its height changes: zero at the start, one at the top, zero on the left, and minus one at the bottom. The angle turned is x, measured in degrees. We call this height sin x. Plotting the angle against the height produces the sine graph. For an acute angle this is the same sine ratio, opposite divided by hypotenuse, used in right-angled triangles.

See the idea first

Angle is the input; height is the output

The notation y = sin x means that y is the sine value of angle x. The circle returns to its start after 360°, so the pattern repeats. The point's horizontal coordinate is cos x; that starts at 1. These coordinates can be negative even though ordinary triangle side lengths cannot.

011130°Psin 30° = ½cos 30°One turn = 360°
The radius is 1. At 30°, the point is half a unit above the horizontal axis, so sin 30° = ½. Its horizontal position is cos 30°. On the sine graph below, the same observation is the point (30, ½).
090180270360-1-0.500.51Angle x (degrees)sin x
Height y = sin x is read against angle x. One full wave takes 360°. The smooth curve continues for negative angles and beyond 360°.
From problem to method

Typical problems you need to be able to solve

These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.

Sketch a graph

What the problem asks: Sketch y = cos x for 0° ≤ x ≤ 360°.

How to solve it: Plot (0,1), (90,0), (180,−1), (270,0), (360,1) and join smoothly. Start with a coordinate table, not a guessed wave.

Find every solution

What the problem asks: Solve sin x = 0.5 for 0° ≤ x ≤ 360°.

How to solve it: A horizontal line at height 0.5 crosses the sine wave twice: x = 30° and 150°. One calculator result is not the complete answer.

Handle tangent's breaks

What the problem asks: Explain why tan x is undefined at 90°.

How to solve it: Tangent is sin x divided by cos x. At 90°, cos x = 0, and division by zero is undefined. The graph has separate branches, not a vertical joining line.

A reliable routine

Sketch standard trig graphs and read solutions

Use key points for y = sin x, cos x or tan x in degrees. Repetition and symmetry locate all crossings; this guide does not use calculus.

  1. Mark the angle interval and use degree mode on a calculator.
  2. Plot known values: sine starts at 0; cosine starts at 1.
  3. Draw smooth sine/cosine curves; keep tangent branches separate at 90° + 180°n for integer n.
  4. For an equation, intersect the graph with the required horizontal value and list all angles within the stated interval.

Check: For tangent, zeros are at multiples of 180° and the period is 180°. Sine and cosine have period 360°. Here ‘period’ means the smallest positive horizontal shift that repeats the whole pattern.

Fully worked

Trigonometric graphs GCSE worked examples

Each solution explains the clue, why the method fits and how to check the result.

Example 1

Sine key points

Higher only3 marks
Question

Sketch y = sin x from 0° to 360°.

The five key pairs are (0,0), (90,1), (180,0), (270,−1), (360,0). Join smoothly with no corners. The height is a ratio, with no degree unit.

090180270360-1-0.500.51Angle x (degrees)sin x
Height y = sin x is read against angle x. One full wave takes 360°. The smooth curve continues for negative angles and beyond 360°.
Example 2

Cosine key points

Higher only3 marks
Question

Sketch y = cos x from 0° to 360°.

Cosine records the horizontal coordinate around the unit circle. It starts at 1, falls to zero at 90°, reaches −1 at 180°, and returns through zero to 1.

090180270360-1-0.500.51Angle x (degrees)cos x
Cosine starts at 1, not zero. The horizontal line at −0.5 would cross the curve at 120° and 240°.
Example 3

Tangent branches

Higher only3 marks
Question

State the zeros and undefined angles of tan x for 0° ≤ x ≤ 360°.

Zeros: 0°, 180°, 360°. Undefined: 90°, 270°. Draw separate increasing branches between the breaks.

090180270360-3-2-10123tan xAngle x (degrees)
Tangent has no value at 90° and 270°. Dashed vertical lines mark where values grow without bound; they are not part of the curve. Branches must never be joined across them.
Example 4

Positive sine

Higher only3 marks
Question

Solve sin x = 0.5 for 0° ≤ x ≤ 360°.

The first angle is 30°. Sine has the same positive height on the other side of 90°.

180∘−30∘=150∘180^\circ-30^\circ=150^\circ

Therefore x = 30° or 150°.

Example 5

Negative cosine

Higher only3 marks
Question

Solve cos x = −0.5 for 0° ≤ x ≤ 360°.

Since cos 60° = 0.5, the matching negative horizontal coordinate occurs at 180° − 60° = 120°. Reflection about the horizontal axis of the unit circle keeps the same horizontal coordinate.

360∘−120∘=240∘360^\circ-120^\circ=240^\circ

Answers: 120°, 240°.

Example 6

Repeat beyond one cycle

Higher only3 marks
Question

Solve tan x = 1 for 0° ≤ x ≤ 540°.

Tangent equals one at 45° and repeats after 180°.

45∘+180∘=225∘45^\circ+180^\circ=225^\circ 225∘+180∘=405∘225^\circ+180^\circ=405^\circ

The next is 585°, outside the interval. Answers: 45°, 225°, 405°.

10 original questions · total 20 marks

Trigonometric graphs GCSE exam-style questions

Try each question before opening its fully worked answer. The difficulty rises through the set.

Before you startAllow about 25 minutes · Show your working. Give exact answers unless a question asks you to round, and include units where needed. · answers start collapsed
1

Read sine

Higher only1 mark

Find sin 270°.

Show worked answer

The unit-circle point is at the bottom.

sin⁡270∘=−1\sin270^\circ=-1
2

Read cosine

Higher only1 mark

Find cos 180°.

Show worked answer

The point is on the left of the circle.

cos⁡180∘=−1\cos180^\circ=-1
3

Period

Higher only1 mark

State the period of y = cos x in degrees.

Show worked answer

360°: a whole turn repeats the horizontal coordinate.

4

All zeros

Higher only2 marks

Solve sin x = 0 for 0° ≤ x ≤ 360°.

Show worked answer

The crossings include both endpoints: 0°, 180°, 360°.

5

Cosine zeros

Higher only2 marks

Solve cos x = 0 for 0° ≤ x ≤ 360°.

Show worked answer

The horizontal coordinate is zero at the top and bottom: 90°, 270°.

6

Negative sine

Higher only3 marks

Solve sin x = −0.5 for 0° ≤ x ≤ 360°.

Show worked answer

Use sin 30° = 0.5. A reference angle is the acute angle giving the same size of sine value, ignoring its sign. For a negative height, use the lower half of the circle: the matching angles are

180∘+30∘=210∘180^\circ+30^\circ=210^\circ 360∘−30∘=330∘360^\circ-30^\circ=330^\circ
7

Tangent equation

Higher only3 marks

Solve tan x = −1 for 0° ≤ x ≤ 360°.

Show worked answer

Since tan 45° = 1, the matching negative value in the second quadrant occurs at 180° − 45° = 135°. Add the 180° period: 135°, 315°.

8

Impossible height

Higher only2 marks

Solve sin x = 1.2 for real angles x.

Show worked answer

There are no real solutions: sine lies between −1 and 1.

9

Negative angle

Higher only2 marks

Find sin(−90°).

Show worked answer

Turn clockwise to the bottom of the unit circle.

sin⁡(−90∘)=−1\sin(-90^\circ)=-1
10

Shifted sine

Higher only3 marks

Give the range and first maximum at non-negative x of y = sin x + 2.

Show worked answer

Adding 2 lifts every height.

−1+2≤y≤1+2-1+2\le y\le1+2

Range: 1 ≤ y ≤ 3. The first maximum is (90°, 3).

Examiner-style feedback

Common trigonometric graphs mistakes

Only one calculator angle

Use the graph and interval to find every solution.

Joining tangent across a break

An asymptote is not a part of the curve.

Confusing angle and output

x is in degrees; sin x and cos x are ratios.

30-second recap

Choose, carry out, check

Translate the wording before you reach for a rule.

  1. Start from key values.
  2. Use degrees.
  3. Keep tangent branches separate.
  4. Check interval endpoints.
Quick answers

Trigonometric graphs FAQ

Do I need radians?

This GCSE guide uses degrees; radians are not needed here.

Why are there several answers?

Different angles can have the same circle height or horizontal coordinate, and the patterns repeat.

Build connected skills

What to revise next

Browse all GCSE Maths study guidesReview the worked examplesTry the questions again
Content standards

Curriculum and rights review

A12–A13 Higher: standard trigonometric graphs in degrees, periodicity, interval solutions and a vertical translation. Questions, suggested marks, solutions and diagrams are original practice material, not official past-paper questions.

Official specification references