Trigonometric Graphs
This free Higher GCSE Maths worksheet on trigonometric graphs helps you revise recognising and transforming trigonometric graphs. Questions build from straightforward to exam standard, with full worked answers in the mark scheme — ideal for revision or homework. Suitable for AQA, Edexcel and OCR. Sine and cosine repeat every 360°; tan repeats every 180°.
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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.
This is a Higher tier only topic, so there's no Foundation paper — only the Higher worksheet and mark scheme below.
Topic overview
Trigonometric graphs show how sine, cosine and tangent vary as the angle changes. They repeat, which makes them periodic, and recognising their shapes is what most questions test.
The sine curve starts at zero, rises to \(1\) at \(90^\circ\), returns to zero at \(180^\circ\), falls to \(-1\) at \(270^\circ\) and returns to zero at \(360^\circ\). The cosine curve is the same shape but starts at \(1\).
The tangent graph behaves quite differently. It repeats every \(180^\circ\) rather than \(360^\circ\), and it has vertical asymptotes at \(90^\circ\) and \(270^\circ\), where the value is undefined.
Revision notes
The sine and cosine curves
Both oscillate between \(-1\) and \(1\) and repeat every \(360^\circ\).
Sine starts at \(0\) and cosine starts at \(1\). Learning the key values at \(0\), \(90\), \(180\), \(270\) and \(360\) degrees lets you sketch either curve quickly.
The tangent curve
Tangent repeats every \(180^\circ\) and is not bounded, taking every value from very large negative to very large positive.
It has asymptotes where cosine is zero, at \(90^\circ\) and \(270^\circ\), because tangent is sine divided by cosine.
Using symmetry to find other solutions
Because the curves repeat, an equation such as \(\sin x = 0.5\) has more than one solution in a given range.
Sketching the curve and the horizontal line shows how many solutions there are and roughly where they lie.
Key points
- Sine and cosine oscillate between \(-1\) and \(1\).
- Both repeat every \(360^\circ\).
- Sine starts at 0; cosine starts at 1.
- Tangent repeats every \(180^\circ\).
- Tangent has asymptotes at \(90^\circ\) and \(270^\circ\).
- Use the curve to find all solutions in a range.
Worked examples
Example 1
State the value of \(\sin 90^\circ\) from the graph.
Working
Example 2
State the value of \(\cos 180^\circ\).
Working
Example 3
How many solutions does \(\sin x = 0.5\) have between \(0^\circ\) and \(360^\circ\)?
Working
Common mistakes
Confusing the sine and cosine starting points.
Sine starts at 0 and cosine at 1. Learn the two shapes separately.
Giving the tangent graph a maximum of 1.
Tangent is unbounded and has no maximum or minimum.
Forgetting the second solution.
Within 0 to 360 degrees, sin x = k usually has two solutions.
Drawing the tangent graph as continuous.
It breaks at the asymptotes, where the value is undefined.
Exam tips
- Learn the key values at 0, 90, 180, 270 and 360 degrees.
- Sketch the curve before answering, even roughly.
- Draw the horizontal line to count solutions.
- Remember tangent has a period of 180 degrees, not 360.
Key terms
- Periodic
- Repeating at regular intervals.
- Amplitude
- The maximum distance from the centre line, which is 1 for sine and cosine.
- Asymptote
- A line the curve approaches but never meets.
- Period
- The interval after which a curve repeats.
Related topics
Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.