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Maths · More trigonometry

Graph of the tangent function

Sketch and read the graph of \(y = \tan x\), know its asymptotes and period of \(180^\circ\), and solve \(\tan x = k\) between \(0^\circ\) and \(360^\circ\).

  • Higher
  • 6 key terms
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Warm-up

Answer each one, then check.

  1. 1

    What does TOA mean?

    Show answerHide answer

    \(\tan = \text{opposite} \div \text{adjacent}\)

  2. 2

    What is \(\tan 45^\circ\)?

    Show answerHide answer

    \(1\)

  3. 3

    What is \(\tan 0^\circ\)?

    Show answerHide answer

    \(0\)

  4. 4

    What is the period of sine and cosine?

    Show answerHide answer

    \(360^\circ\)

  5. 5

    Can you divide by zero?

    Show answerHide answer

    No

Learning Objectives

  1. 1Sketch the graph of \(y = \tan x\) for \(0^\circ \le x \le 360^\circ\).
  2. 2Recall key values and the position of the asymptotes.
  3. 3Use the period of \(180^\circ\) to find a second solution.
  4. 4Read solutions from the graph.

TANGENT GRAPH

The graph of \(y = \tan x\) repeats every \(180^\circ\) and has vertical asymptotes at \(x = 90^\circ\) and \(x = 270^\circ\).

Unlike sine and cosine, tangent has no maximum or minimum: it goes up to infinity and down to minus infinity.

Key Values of Tangent

Learn these, and note the gaps.

  • \(\tan x\)

    \(0^\circ\): \(0\). \(45^\circ\): \(1\). \(90^\circ\): undefined. \(135^\circ\): \(-1\) | \(0\) | \(1\) | undefined | \(-1\) | \(0\)

Features of the Graph

  • Period

    Repeats every \(180^\circ\).

  • Asymptotes

    Vertical lines at \(x = 90^\circ\) and \(270^\circ\); the graph never touches them.

  • Roots

    \(x = 0^\circ,\ 180^\circ,\ 360^\circ\).

  • Range

    Every value of \(y\); no maximum or minimum.

  • Shape

    Rises steeply through each root.

Finding Two Solutions

The graph repeats every \(180^\circ\).

  1. 1 Calculator

    Find \(x = \tan^{-1}(k)\); this is the first solution
  2. 2 Add the period

    Second solution \(= x + 180^\circ\)
  3. 3 Negative values

    If \(\tan^{-1}(k)\) is negative, add \(180^\circ\) and \(360^\circ\)
  4. 4 Check the range

    Only keep answers between \(0^\circ\) and \(360^\circ\)

Solving tan x = 2

Solve \(\tan x = 2\) for \(0^\circ \le x \le 360^\circ\), correct to 1 decimal place.

Show the solutionHide the solution
  1. 1 Calculator \(x = \tan^{-1}(2) = 63.4^\circ\)
  2. 2 Add \(180^\circ\) \(63.4^\circ + 180^\circ = 243.4^\circ\)

Answer\(x = 63.4^\circ\) and \(x = 243.4^\circ\)

Solving a Negative Value

Solve \(\tan x = -1\) for \(0^\circ \le x \le 360^\circ\).

Show the solutionHide the solution
  1. 1 Calculator gives \(\tan^{-1}(-1) = -45^\circ\)
  2. 2 Add \(180^\circ\) \(-45^\circ + 180^\circ = 135^\circ\)
  3. 3 Add \(360^\circ\) \(-45^\circ + 360^\circ = 315^\circ\)

Answer\(x = 135^\circ\) and \(x = 315^\circ\)

Comparing the Three Graphs

sin x and cos x

  • Period \(360^\circ\).
  • Smooth waves between \(-1\) and \(1\).
  • No asymptotes.
  • Solutions: \(x\) and \(180^\circ - x\) (sin), \(x\) and \(360^\circ - x\) (cos).

tan x

  • Period \(180^\circ\).
  • Takes every value.
  • Asymptotes at \(90^\circ\) and \(270^\circ\).
  • Solutions: \(x\) and \(x + 180^\circ\).

Tangent Table

Use a calculator to fill in \(\tan x\) for \(x = 60^\circ, 80^\circ, 85^\circ, 89^\circ\) (1 d.p.). Then describe what happens as \(x\) gets close to \(90^\circ\).

1. Type each value into the calculator.

2. Describe the pattern.

A good answer shows: \(1.7\), \(5.7\), \(11.4\), \(57.3\). The values get bigger and bigger, which is why there is an asymptote at \(90^\circ\).

Can I...?

  1. 1Sketch the tangent graph.
  2. 2State the period.
  3. 3Mark the asymptotes.
  4. 4Recall tan 45.
  5. 5Use tan inverse.
  6. 6Find the second solution.
  7. 7Solve for negative values.
  8. 8Read solutions from a graph.

Summary & Exam Focus

  • Tangent repeats every \(180^\circ\).
  • Vertical asymptotes at \(90^\circ\) and \(270^\circ\).
  • Solutions are \(x\) and \(x + 180^\circ\).
  • Tangent takes every value, so \(\tan x = k\) always has solutions.

Exam focus

Solve \(\tan x = 0.5\) for \(0^\circ \le x \le 360^\circ\). Give answers correct to 1 decimal place. (3 marks) (3 marks)

The period is \(180^\circ\), so the second solution is \(x + 180^\circ\).

Key terms

The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.

Asymptote
A line that a graph gets closer to but never touches.
Period
The length after which the graph repeats.
Undefined
Has no value, e.g. \(\tan 90^\circ\).
Inverse tangent
\(\tan^{-1}\), the calculator function that finds an angle.
Root
A value of \(x\) where \(y = 0\).
Range
The set of values \(y\) can take.

Practice questions

Have a go at each one before you open its answer.

  1. Question 1 Solve 3 marks

    Solve \(\tan x = 0.5\) for \(0^\circ \le x \le 360^\circ\). Give your answers correct to 1 decimal place.

    Show answerHide answer

    Model answer

    \(x = 26.6^\circ\) and \(x = 206.6^\circ\)

    Mark scheme

    • \(\tan^{-1}(0.5) = 26.6^\circ\) — B1
    • Adds \(180\) — M1
    • 206.6 — A1
  2. Question 2 Write down 2 marks

    Write down the value of (a) \(\tan 45^\circ\) (b) \(\tan 0^\circ\)

    Show answerHide answer

    Model answer

    (a) 1 (b) 0

    Mark scheme

    • 1 — B1
    • 0 — B1
  3. Question 3 Use the graph 3 marks

    The graph shows \(y = \tan x\) for \(0^\circ \le x \le 180^\circ\). Use the graph to solve \(\tan x = 1\), and write down the equation of the asymptote.

    The graph of tangent x from 0 to 180 degrees on a grid with an asymptote at 90 degrees and a dotted line at y equals 1.
    Show answerHide answer

    Model answer

    \(x = 45^\circ\); the asymptote is \(x = 90^\circ\).

    Mark scheme

    • 45 — B1
    • \(x = 90\) — B1
    • Equation written in the form \(x = 90\) — B1
  4. Question 4 Solve 3 marks

    Solve \(\tan x = -2\) for \(0^\circ \le x \le 360^\circ\). Give your answers correct to 1 decimal place.

    Show answerHide answer

    Model answer

    \(x = 116.6^\circ\) and \(x = 296.6^\circ\)

    Mark scheme

    • \(\tan^{-1}(-2) = -63.4^\circ\) — B1
    • \(-63.4 + 180\) or \(-63.4 + 360\) — M1
    • Both answers — A1
  5. Question 5 Explain 2 marks

    Explain why \(\tan 90^\circ\) has no value.

    Show answerHide answer

    Model answer

    The tangent graph has an asymptote at \(90^\circ\): the values get larger and larger and never reach a value.

    Mark scheme

    • Refers to the asymptote or the graph — M1
    • Values grow without limit — C1
  6. Question 6 Write down 2 marks

    \(\tan 30^\circ = 0.58\) (to 2 decimal places). Write down another angle between \(0^\circ\) and \(360^\circ\) with tangent 0.58.

    Show answerHide answer

    Model answer

    \(210^\circ\)

    Mark scheme

    • \(30 + 180\) — M1
    • 210 — A1

Quick check

  1. What is the period of \(y = \tan x\)?

    1. A\(90^\circ\)
    2. B\(180^\circ\)
    3. C\(270^\circ\)
    4. D\(360^\circ\)
    Show answerHide answer

    B: \(180^\circ\)

    The graph repeats every 180 degrees.

  2. Where are the asymptotes of \(y = \tan x\) between \(0^\circ\) and \(360^\circ\)?

    1. A0 and 180
    2. B180 and 360
    3. C90 and 270
    4. D45 and 225
    Show answerHide answer

    C: 90 and 270

    At 90 and 270 degrees.

  3. \(\tan 45^\circ = 1\). Another solution between 0 and 360 is...

    1. A135
    2. B315
    3. C180
    4. D225
    Show answerHide answer

    D: 225

    \(45 + 180 = 225\).

  4. What is the maximum value of \(\tan x\)?

    1. AIt has no maximum
    2. B1
    3. C90
    4. D360
    Show answerHide answer

    A: It has no maximum

    There is none: the graph goes up without limit.

  5. \(\tan^{-1}(-1)\) on a calculator gives...

    1. A\(45^\circ\)
    2. B\(-45^\circ\)
    3. C\(135^\circ\)
    4. DError
    Show answerHide answer

    B: \(-45^\circ\)

    The calculator gives \(-45^\circ\); add 180 to get 135 degrees.

  6. The roots of \(\tan x\) between \(0^\circ\) and \(360^\circ\) are...

    1. A90 and 270
    2. B45 and 225
    3. C0, 180 and 360
    4. D0 and 360 only
    Show answerHide answer

    C: 0, 180 and 360

    Where the curve crosses the x-axis: 0, 180 and 360 degrees.

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