Tan Graph - GCSE Maths - Steps, Examples & Worksheet

How to plot the tan graph

Remember that \tan(\theta) is a relationship between the opposite side and the adjacent side of a right angle triangle:

Let’s look at 3 triangles where we would use the tangent ratio to calculate the size of the angle \theta . For each triangle, the adjacent side is the same but the length of the opposite side and the associated angle change.

Here we can see that as \tan(\theta)=\frac{opp}{adj} , as the angle \theta increases, the length of the side opposite to the angle also increases. So for each triangle we have:

• Triangle 1: \tan(\theta)=\frac{3}{10}=0.3
• Triangle 2: \tan(\theta)=\frac{9}{10}=0.9
• Triangle 3: \tan(\theta)=\frac{15}{10}=1.5

So what would happen if the opposite side to the angle is equal to 10 ?

\tan(\theta)=\frac{10}{10}=1

So when the opposite side is equal to the adjacent side, we get \tan(\theta) = 1 .

What about when the opposite side is equal to 0 ?

\tan(\theta)=\frac{0}{10}=0

So when the opposite side is equal to 0, \; \tan(\theta) = 0.

If we plotted a graph to show the value of \tan(\theta) for each value of \theta between 0^o and 90^o , we get the following graph of the tangent function:

Let us add the values of \tan(\theta) for the three triangles from earlier into the graph to show how they would look:

We can now use the graph to find the angle \theta for triangles 1, 2, and 3 :

This graph shows that when \tan(\theta) = 0.3 , \; \theta = 17^o so we have the triangle

This graph shows that when \tan(\theta) = 0.9 , \; \theta = 42^o so we have the triangle

This graph shows that when \tan(\theta) = 1.5 , \; \theta = 56^o so we have the triangle

This shows us that we can use the graph of the tangent function to find missing angles in a triangle. More on this later as we have a large problem to resolve. How can the opposite side of a right angle triangle be equal to 0 ? What if the adjacent side is equal to 0 ?

Unfortunately there is a limit to the use of trig ratios to find angles between 0 and 90^o . For any larger or smaller angles, we need to look at the unit circle.

Explain how to plot the tan graph