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Trigonometric graphs for AQA A-level Maths

Trigonometric graphs

This page covers the following topics:

1. Graphs of trigonometric functions
2. Graphs of inverse trigonometric functions
3. Graphs of reciprocal 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

Both arcsin and arccos have a domain between โˆ’1 and 1 inclusive, whereas arctan has an infinite domain. The ranges of arcsin, arccos and arctan are between โˆ’ฯ€/2 and ฯ€/2 inclusive, between 0 and ฯ€ inclusive and โˆ’ฯ€/2 and ฯ€/2 respectively. The domains and ranges of arcsin and arccos come from the range and domain of sine and cosine respectively. Since tangent has an infinite range, arctan has an infinite domain.

Graphs of inverse trigonometric functions

The graphs for the reciprocal trigonometric functions have asymptotes where the standard trigonometric functions have roots. This is because taking the reciprocal of 0 is undefined.

Graphs of reciprocal trigonometric functions

1

Explain the magnitude of the minimum value that secant can take.

Secant is given by taking the reciprocal of cosine. The reciprocal gets bigger as the value gets smaller, therefore the minimum reciprocal value is attained at the maximum value. the magnitude of the maximum value of cosine is 1 and the reciprocal of this is 1, therefore the magnitude of the minimum value that secant attains is 1.

Minimum reciprocal is attained at the maximum value.

Explain the magnitude of the minimum value that secant can take.

2

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.

3

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).

4

Explain the positions of the asymptotes in the graphs of the reciprocal trigonometric functions.

The graphs for the reciprocal trigonometric functions have asymptotes where the standard trigonometric functions have roots. This is because taking the reciprocal of 0 is undefined.

Asymptotes where the standard trigonometric functions have roots, since dividing by 0 is undefined.

Explain the positions of the asymptotes in the graphs of the reciprocal trigonometric functions.

5

Provide the domain of the arctan graph.

The domain of arctan is infinite, since the range of tangent is infinite.

infinite

Provide the domain of the arctan graph.

End of page

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