One to one maths interventions built for KS4 success

Weekly online one to one GCSE maths revision lessons now available

Learn more

Tan Graph

Here we will learn about tangent graphs, including how to recognise the graph of the tangent function, sketch the tangent curve and label important values, and interpret the tangent graph.

There are also tangent graph worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is the tan graph?

The tan graph is a visual representation of the tangent function for a given range of angles.

The horizontal axis of a trigonometric graph represents the angle, usually written as \theta , and the y -axis is the tangent function of that angle.

Tan Graph image 1 1

  • repeats every 360^o
  • Not a continuous curve
  • Vertical asymptotes at 90^o \pm 180^o

We will only look at the graph of the tangent function in this lesson for angles in degrees although this can also be represented in radians.

What is the tan graph?

What is the tan graph?

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:

Tan Graph image 2 1

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.

Tan Graph image 3 1

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:

Tan Graph image 4 1

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

Tan Graph image 5 1

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

Tan Graph image 6 1

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

Tan Graph image 7 1

Tan Graph image 8 1

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

Tan Graph image 9 1

Tan Graph image 10 1

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

Tan Graph image 11 1

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

Explain how to plot the tan graph

Unit circle

The unit circle is a circle of radius 1 with the centre at the origin. We can label the values where the circle intersects the axes because we know that the radius of the unit circle is 1 unit.

Tan Graph image 12 1

We can still construct a triangle within the unit circle with the angle starting from the positive x axis.

Tan Graph image 13 1

Looking at the trigonometric ratios of sine and cosine, we can say that:

  • \sin(\theta)=\frac{a}{1} \; \text{so} \; a=\sin(\theta)
  • \cos(\theta)=\frac{b}{1} \; \text{so} \; b=\cos(\theta)
  • \tan(\theta)=\frac{a}{b} \; \text{so} \; \tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}

This means that the width of the triangle is equal to \cos(\theta) and the height of the triangle is equal to \sin(\theta) . The point on the unit circle is therefore the coordinate ( \cos(\theta) , \; \sin(\theta) ):

Tan Graph image 14 1

When do we get positive and negative values for tan(θ)

As \tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)} , we can now calculate when the tangent function is positive and negative using division of negative numbers.

Tan Graph image 15 1

\tan(\theta) is as periodic function as we could continue to turn around the origin from the positive x axis beyond 360^o .

We can also turn anticlockwise which would give us the negative values for the value of \tan(\theta) .

The period of the function is 180^o

What is an asymptote?

An asymptote is a line that a curve gets closer and closer to but never touches as the curve goes towards infinity.

Tan Graph image 16 1

We know that \tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)} .

As \tan(\theta) can be expressed as a fraction, we have an issue when \cos(\theta) = 0 as you cannot divide by 0 . This means that the value for \tan(\theta) is undefined every time \cos(\theta) = 0 , which occurs every at 90^o \pm 180^o when y=0

Given the angles in the table below, the gradient of the graph increases continuously as the value of \theta tends towards 90^o .

Tan Graph image 17 1

We already know that the graph of the tangent function is periodic and so a vertical asymptote occurs every 180^o before and after the asymptote shown above. See the graph below:

Tan Graph image 18 1

The graph of the tangent function

Tan Graph image 19 1

Key features include:

  • Periodic (repeats every 360^o )
  • Not a continuous curve
  • Vertical asymptotes at 90^o \pm 180^o
  • The curve always has a positive gradient

Unlike the graphs for the sine and cosine functions, the tangent function is not a wave. This is because the graph is not continuous

There is therefore no amplitude to the wave as there is no maximum or minimum value.

The tangent function can change when we transform the function. For more information, see the lesson on transformations of trigonometric functions.

You should be able to interpret the unit circle to determine the value of \theta . Here are a couple of examples to help you.

Tan Graph image 20 1
Tan Graph image 21 1

Now that we have generated the graph of \tan(\theta) , we need to be able to interpret it to find values of \theta and \tan(\theta) for any value between the range -360 \leq \theta \leq 360^o (values for between -360^o and 360^o .

The reciprocal to \tan(\theta)

The inverse function to \tan(\theta) is \tan^{-1}(\theta) . The reciprocal to \tan(\theta) is called the cotangent or \cot(\theta) where \cot(\theta)=\frac{1}{\tan(\theta)}=\frac{\cos(\theta)}{\sin(\theta)} .

How to interpret tangent graphs

In order to interpret tangent graphs:

  1. Draw a straight line from the axis of the known value to the tangent curve.
  2. Draw a straight, perpendicular line at the intersection point to the other axis.
  3. Read the value where the perpendicular line meets the other axis.

Tangent graph worksheet

Get your free tan graph worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOON
x

Tangent graph worksheet

Get your free tan graph worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOON

Tangent graph examples

Example 1: 0° ≤ θ ≤ 90°

Use the graph of \tan(\theta) to estimate the value for \theta when \tan(\theta)=1.5 for 0^o \leq \theta \leq 90^o

Tan Graph example 1 1

  1. Draw a straight line from the axis of the known value to the tangent curve. 

Tan Graph example 1 step 1 1

2Draw a straight, perpendicular line at the intersection point to the other axis.

Tan Graph example 1 step 1 2

3Read the value where the perpendicular line meets the other axis.

When \tan(\theta)=1.5, \; \theta=56^o

Example 2: 90° ≤ θ ≤ 270°

Use the graph of \tan(\theta) to estimate the value for \theta when \tan(\theta)=0.5 for 90^o \leq \theta \leq 270^o

Tan Graph example 2 1

Tan Graph example 2 step 1 1

Tan Graph example 2 step 2 1

When \tan(\theta)=0.5, \; \theta=207^o

Example 3: 270 ≤ θ ≤ 360°

Use the graph of \tan(\theta) to estimate the value for \tan(\theta) when \theta=300^o .

Tan Graph example 3 1

Tan Graph example 3 step1 1

Tan Graph example 3 step 2 1

When \theta=300^o, \; \tan(\theta)=-1.75 .

Example 4: −180 ≤ θ ≤ 0°

Use the graph of \tan(\theta) to estimate the value for \theta when \tan(\theta)=-0.5 for -180 \leq \theta \leq 0^o

Tan Graph example 4 1

Tan Graph example 4 step 1 1

Tan Graph example 4 step 2 1

When \tan(\theta)=-0.5, \; \theta=-27^o

Example 5: −360° ≤ θ ≤ 0°

Use the graph of \tan(\theta) to estimate the value for \tan(\theta) when \theta=-255^o .

Tan Graph example 5 1

Tan Graph example 5 step 1 1

Tan Graph example 5 step 2 1

When \theta=-225^o, \; \tan(\theta)=-1 .

Example 6: −360° ≤ θ ≤ 360°

Use the graph of \tan(\theta) to estimate the value for \theta when \tan(\theta)=0.25 for -360^o \leq \theta \leq 360^o

Tan Graph example 6 1

Tan Graph example 6 step 1 1

Tan Graph example 6 step 2 1

When \tan(\theta)=0.25, \; \theta=-346^o, \; \theta=-166^o, \; \theta=14^o \; \text{and} \; \theta=194^o

Common misconceptions

  • Drawing the horizontal line only to one intersection point 

When you are finding the value of \theta using a trigonometric graph, only one value is calculated when there are more points of intersection.

For example, only the value of 65^o is read off the graph. 

Tan Graph Common Misconceptions 1 1

  • Sine and cosine graphs switched

The sine and cosine graphs are very similar and can easily be confused with one another. A tip to remember is that you “sine up” from 0 for the sine graph so the line is increasing whereas you “cosine down” from 1 so the line is decreasing for the cosine graph.

Tan Graph Common Misconceptions 2 1

  • Asymptotes are drawn incorrectly for the graph of the tangent function

The tangent function has an asymptote at 90^o because this value is undefined. As the curve repeats every 180^o , the next asymptote is at 270^o and so on.

  • The graphs are sketched using a ruler

Each trigonometric graph is a curve and therefore the only time you are required to use a ruler is to draw a set of axes. Practice sketching each curve freehand and label important values on each axis.

  • Value given out of range

When finding a value of \theta using a trigonometric graph, you must make sure that the value of \theta is within the range specified in the question. 

For example, the range of values for \theta is given as 0^o \leq \theta \leq 360^o and only the value of \theta=240^o is written for the solution, whereas the solution \theta=300^o is also correct.

Tan Graph Common Misconceptions 3 1

Practice tangent graphs questions

1. Use the graph of \tan(\theta) to calculate the value of \theta when \tan(\theta)=3 for 0^o \leq \theta \leq 180^o

 

Tan Graph Practice Question 1 1

\theta=252^o
GCSE Quiz False

\theta=-108^o
GCSE Quiz False

\theta=-288^o
GCSE Quiz False

\theta=72^o
GCSE Quiz True

Tan Graph Practice Question 1 explanation 1

2. Use the graph of \tan(\theta) to calculate the value of \theta when \tan(\theta)=-1.2 for -180^o \leq \theta \leq 90^o

 

Tan Graph Practice Question 2 1

\theta=130^o
GCSE Quiz False

\theta=-50^o
GCSE Quiz True

\theta=-230^o
GCSE Quiz False

\theta=310^o
GCSE Quiz False

Tan Graph Practice Question 2 explanation 1

3. Use the graph of \tan(\theta) to calculate the value of \theta when \tan(\theta)=0 for -180^o \leq \theta \leq 180^o

 

Tan Graph Practice Question 3 1

\theta=0^o
GCSE Quiz True

\theta=-180^o
GCSE Quiz False

\theta=180^o
GCSE Quiz False

\theta=180^o, \; 0^o, \; 180^o
GCSE Quiz False

Tan Graph Practice Question 3 explanation 1

4. Use the graph of \tan(\theta) to calculate the value of \tan(\theta) when \theta=150^o

 

Tan Graph Practice Question 4 1

\tan(150)=-0.8
GCSE Quiz False

\tan(150)=0.6
GCSE Quiz False

\tan(150)=-0.6
GCSE Quiz True

\tan(150)=-0.2
GCSE Quiz False

Tan Graph Practice Question 4 explanation 1

5. Use the graph of \tan(\theta) to calculate the value of \tan(\theta) when \theta=-320^o

 

Tan Graph Practice Question 5 1

\tan(-320)=0.8
GCSE Quiz True

\tan(-320)=-0.5
GCSE Quiz False

\tan(-320)=-0.8
GCSE Quiz False

\tan(-320)=0.5
GCSE Quiz False

Tan Graph Practice Question 5 explanation 1

6. What value for \theta would return the same value for 240^o for -90^o \leq \theta \leq 270^o

 

Tan Graph Practice Question 6 1

\tan(240)= \tan(-120)
GCSE Quiz False

\tan(240) = \tan(-240)
GCSE Quiz False

\tan(240) = \tan(120)
GCSE Quiz False

\tan(240) = \tan(60)
GCSE Quiz True

Tan Graph Practice Question 6 explanation 1

Tangent graphs GCSE questions

1. Below is a sketch of the graph of y=\tan(\theta) for 0 \leq \theta \leq 360^o .

 

Tan Graph GCSE questions 1 1

 

If \tan(x)=1.4 , what other value for would return the same value of \tan(\theta) ?

 

(1 mark)

Show answer
x+180

(1)

2. (a) Write an equation in terms of \theta for the size of the angle ABC .

 

Tan Graph GCSE questions 2a 1

 

(b) Use the graph of y=\tan(\theta) to estimate the value of \theta in triangle ABC .

Tan Graph GCSE questions2b 1

(4 marks)

Show answer

(a)

 

\tan(\theta)=\frac{25}{15}=1.6…

(1)

\theta =tan^{-1}(1.67)

(1)

 

(b)

 

Tan Graph GCSE questions 2b2 1

(1)

tan^{-1}(1.67) = [55-65^o]

(1)

3. Write down the solutions to the equation \tan(\theta)=2 for 360 \leq \theta \leq 720^o .

 

Tan Graph GCSE questions 3a 1

 

(2 marks)

Show answer

[60-70^o] and [240-250^o] highlighted on graph

(1)

[420-430^o] and [600-610^o]

(1)

4. Below are the graphs of y=\sin(\theta) and y=\tan(\theta). Use the graph to find the solutions for \sin(\theta) = \tan(\theta) for -360 \leq \theta \leq 360^o .

 

Tan Graph Practice Question 2 1-1

 

(3 marks)

Show answer
\theta =-360^o, \; -180^o

(1)

\theta = 0

(1)

\theta=180^o, \; -360^o

(1)

Learning checklist

You have now learned how to:

  •  recognise, sketch and interpret graphs of trigonometric functions (with arguments in degrees) y = \tan x for angles of any size

The next lessons are

  • Plotting graphs of nonlinear equations
  • Transformations of linear graphs
  • Transformations of non-linear graphs
  • Transforming trigonometric graphs

Still stuck?

Prepare your KS4 students for maths GCSEs success with Third Space Learning. Weekly online one to one GCSE maths revision lessons delivered by expert maths tutors.

Find out more about our GCSE maths revision programme.