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Trigonometric graphs for AQA GCSE Maths

Trigonometric graphs

This page covers the following topics:

1. Graphs of trigonometric functions

The graphs for sine and cosine are almost identical, with the only difference being that the cosine graph is the sine graph shifted by 90ยฐ to the right. The graphs for sine and cosine both have an amplitude of 1, meaning that the maximum value that the two trigonometric functions can take is 1. They are both 360ยฐ periodic, meaning that the graph will be repeated after every 360ยฐ.

The cosine graph is symmetric over the y-axis, since when it is flipped across it, the two halves of the graph match up. On the other hand, the graph of sine is symmetric about the origin, since the graph is flipped over both the x-axis and the y-axis. This means that for every x value with a corresponding value of y, โˆ’x matches to โˆ’y.

Tangent graph is significantly different to both sine and cosine graphs. The graph of tangent is 180ยฐ periodic. The graph passes through the origin and tends to infinity both in the positive and the negative direction, and thus its amplitude is undefined. Since tangent is undefined for odd multiples of 90ยฐ, at these values of x, the graph of tangent has asymptotes.

Graphs of trigonometric functions

1

Explain why the graph of tangent has asymptotes.

Tangent is undefined for odd multiples of 90ยฐ, therefore for those x values, the graph will not have a point, thus it has asymptotes there.

Tangent is undefined at odd multiples of 90ยฐ.

Explain why the graph of tangent has asymptotes.

2

Sketch a graph of y = sin(x).

The graphs for sine and cosine both have an amplitude of 1, meaning that the maximum value that the two trigonometric functions can take is 1. They are both 360ยฐ periodic, meaning that the graph will be repeated after every 360ยฐ. The graph of sine is symmetric about the origin, since the graph is flipped over both the x-axis and the y-axis. This means that for every x value with a corresponding value of y, โˆ’x matches to โˆ’y.

image

Sketch a graph of y = sin(x).

3

Cho has sketched the following graph. State which trigonometric ratio it represents.

The graph of tangent is 180ยฐ periodic. The graph passes through the origin and tends to infinity both in the positive and the negative direction, and thus its amplitude is undefined. Since tangent is undefined for odd multiples of 90ยฐ, at these values of x, the graph of tangent has asymptotes.

tangent

Cho has sketched the following graph. State which trigonometric ratio it represents.

4

Waves emitted by a transmitter are modelled by a cosine curve. Sketch the graph of this function.

The graphs for sine and cosine both have an amplitude of 1, meaning that the maximum value that the two trigonometric functions can take is 1. They are both 360ยฐ periodic, meaning that the graph will be repeated after every 360ยฐ. The cosine graph is symmetric over the y-axis, since when it is flipped across it, the two halves of the graph match up.

image

Waves emitted by a transmitter are modelled by a cosine curve. Sketch the graph of this function.

5

Describe the periodicity of the sine graph.

The sine graph is 180ยฐ periodic, meaning that it repeats itself when 180ยฐ passes.

180ยฐ periodic

Describe the periodicity of the sine graph.

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