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Trigonometric graphs - Higher

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Tutor: Toby

Summary

Trigonometric graphs

In a nutshell

Trigonometric graphs are a class of "wave" graphs. They repeat themselves after a point, which makes them periodic. There are three main trigonometric graphs: sine, cosine and tangent and there are key points on each to learn.

Sine graph

The graph of $y=\sin(x)$ is pictured above. The $x$ -axis displays degrees, the unit of angle. You can see that the graph passes through the origin and has peaks where $y=1$ and throughs where $y=-1$ . The $x$ -coordinates are evenly spaced out such that the peaks are a horizontal distance of $180$ from each other and the troughs also have the same gap between them. The curve goes on forever in the positive and negative $x$ -directions but never goes higher than $y=1$ nor lower than $y=-1$ .

The key points you need to know are the coordinates of the peaks, troughs and the $x$ -intercepts. It repeats itself every $360$ on the $x$ -axis.

Cosine graph

Maths; Graphs; KS4 Year 10; Trigonometric graphs - Higher

The graph of $y=\cos(x)$ is very similar to the sine graph above, but is shifted horizontally by $90$ units (when the $x$ -axis is in degrees). Hence the cosine graph (often referred to as the "cos" graph) intersects the $y$ -axis at $(0,1)$ . Like the sine graph, the cosine graph repeats itself every $360$ degrees on the $x$ -axis.

Example 1

Give the coordinates of two points where $y=\cos(x)$ intersects the $x$ -axis.

Reading off the graph, you can see that two examples of $x$ -intercepts are $(90,0)$ and $(-90,0)$ .

Note: The cosine graph is mirrored in the $y$ -axis.

Tangent graph

The graph of $y=\tan(x)$ is different to the other two trigonometric graphs shown here in that it has a different shape and it repeats itself every $180$ degrees rather than just every $360$ degrees. You still need to know key points on it: it passes through the $x$ -axis at the origin and again every $180$ degrees in the positive and negative $x$ -directions.

At $x=90$ (and also at every jump of $180$ ), there is what is called an asymptote. This is a point on a curve where it gets closer and closer to a point, but never reaches it. So the graph of $y=\tan(x)$ has no value when $x=90$ , but very close to $x=90$ , it has a $y$ -value that is very big in the positive direction (if $x<90$ ) or very big in the negative direction (if $x>90$ ).

Example 2

Using a calculator, find $\tan(89.9)$ and also $\tan(90.1)$ .What happens if you type $\tan(90)$ ?

Typing these into a scientific calculator gives

$\tan(89.9)=\underline{572.96}$

$\tan(90.1)=\underline{-572.96}$ 

each to two decimal places. Typing $\tan(90)$ gives a "math error" since there is no value for this - the graph never reaches $x=90$ . Another way of thinking of this is that coming from the left, $y=\tan(x)$ gets bigger as it approaches $x=90$ . It shoots off infinitely. But coming from the right, it shoots off to negative infinity. Since $y=\tan(x)$ cannot be infinity, nor negative infinity (and especially not both), it does not have a value.

Learn with Basics

Length:

Constructing triangles

Trigonometric ratios: SOH CAH TOA

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