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# Trigonometric graphs
- URL: https://www.esheets.io/trigonometric-graphs/
- Published: 2026-10-09T17:39:50.000Z
- Updated: 2026-10-09T17:39:50.000Z
- Author: Richard Linnington
- Tags: Maths, Geometry and Shape

Practise recognising, sketching and interpreting the standard sine, cosine and tangent graphs in degrees. You will use key points, periods and tangent asymptotes before reading values and finding every solution in a stated interval. [Jump to the questions](#practise-now)

## Practise now

Worksheet preview and key skills 

### Worksheet preview

This eight-question GCSE Higher worksheet focuses on the standard graphs of *y* \= sin *x*, *y* \= cos *x* and *y* \= tan *x*, with all angles in degrees.

Questions progress from recognising and plotting the graphs to reading values and finding complete solution sets in extended intervals.

#### What you’ll practise

- Recognising graph shapes, key points, periods and asymptotes.
- Plotting the quarter-turn points for sine and cosine.
- Reading values and solving equations graphically, including with periodicity.

Use the interactive worksheet below, or read the [Topic guide](#topic-guide) for the method and worked example.

# Trigonometric graphs

Answer all eight questions. All angles are in degrees; use each supplied graph rather than inverse-trig calculator buttons.

## Topic guide

The standard trigonometric graphs show how sine, cosine and tangent change as an angle changes. In this topic, *x* is measured in degrees.

### Recognise the three graphs

- ***y* \= sin *x*** starts at 0 when *x* \= 0°, rises first, and stays between −1 and 1.
- ***y* \= cos *x*** starts at 1 when *x* \= 0°, falls first, and stays between −1 and 1.
- ***y* \= tan *x*** has separate increasing branches. It is undefined at 90° + 180°*n*, so dashed vertical asymptotes separate the branches.

Sine and cosine each have period 360°. Tangent has period 180°. A period is the horizontal distance before the graph repeats.

### Sketch from key features

For sine over 0° to 360°, plot (0°, 0), (90°, 1), (180°, 0), (270°, −1) and (360°, 0), then join them with a smooth curve.

For cosine, plot (0°, 1), (90°, 0), (180°, −1), (270°, 0) and (360°, 1). Do not swap the starting values: sine starts at 0, but cosine starts at 1.

For tangent, mark the zeros at 0°, 180° and 360° and draw dashed asymptotes at 90° and 270°. Sketch each increasing branch separately; never join a curve through an asymptote.

### Read values and solve equations

To estimate a trig value, move vertically from the given angle to the curve, then horizontally to the *y*\-axis. Read the scale carefully and give only the requested accuracy.

To solve an equation such as sin *x* \= 0.5, draw or imagine the horizontal line *y* \= 0.5\. Read every *x*\-coordinate where it meets the sine graph. For 0° ≤ *x* ≤ 360°, the solutions are 30° and 150°.

### Use the stated interval

A horizontal line can meet a graph more than once. List every solution inside the interval, including endpoints when they are valid. For an extended or negative interval, repeat sine and cosine solutions every 360°, and tangent solutions every 180°, keeping only angles in the stated interval.

### Common mistakes

- Swapping sine and cosine, or forgetting that cosine starts at 1.
- Drawing tangent continuously through an asymptote or giving it a 360° period.
- Giving sine or cosine values outside the range −1 to 1.
- Missing another intersection, a repeated solution or an allowed negative angle.
- Including an angle outside the stated interval or reading the axes in the wrong direction.
- Reporting excessive decimal places when the graph supports only an estimate.