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Maths · Further Trigonometry

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Trigonometric Graphs and Exact Values

Sketching and using the sine, cosine and tangent graphs, and solving equations between 0 and 360 degrees using exact values.

  • Higher
  • 9 key terms
  • All boards

Learning Objectives

  1. 1Sketch and recognise the graphs of \(y = \sin x\), \(y = \cos x\) and \(y = \tan x\).
  2. 2State the key values of each graph, including where each crosses the axis and its maximum and minimum.
  3. 3Use the symmetry of the graphs to find more than one angle with the same sine or cosine.
  4. 4Solve equations such as \(\sin x = \dfrac{1}{2}\) between \(0^\circ\) and \(360^\circ\), using exact values.

Trigonometry beyond right-angled triangles

In a right-angled triangle the angle is always below \(90^\circ\), but the sine, cosine and tangent functions are defined for any angle. Their graphs repeat, and have a shape that you should know well. The graphs show why an equation such as \(\sin x = \dfrac{1}{2}\) has more than one solution between \(0^\circ\) and \(360^\circ\). On a non-calculator paper, the values come from the exact values table that you met in right-angled trigonometry, and the symmetry of the graph gives the other solutions.

The sine and cosine graphs

Both graphs are smooth waves that repeat every \(360^\circ\) and stay between \(-1\) and \(1\).

  • \(y = \sin x\)

    Starts at 0 when \(x = 0^\circ\), reaches a maximum of 1 at \(90^\circ\), crosses zero at \(180^\circ\), has a minimum of \(-1\) at \(270^\circ\) and returns to 0 at \(360^\circ\).

  • \(y = \cos x\)

    Starts at 1 when \(x = 0^\circ\), crosses zero at \(90^\circ\), has a minimum of \(-1\) at \(180^\circ\), crosses zero at \(270^\circ\) and returns to 1 at \(360^\circ\).

  • The shape

    The cosine graph is the sine graph moved \(90^\circ\) to the left.

  • Range

    The values of sine and cosine are never greater than 1 or less than \(-1\).

The tangent graph

The tangent graph is different. It has no maximum or minimum, and it has gaps where the function is undefined.

  • \(y = \tan x\)

    Starts at 0 when \(x = 0^\circ\), rises towards infinity as \(x\) approaches \(90^\circ\), then comes up from minus infinity after \(90^\circ\).

  • Asymptotes

    The lines \(x = 90^\circ\) and \(x = 270^\circ\) are asymptotes, which the graph approaches but never touches.

  • Zeros

    The graph crosses the axis at \(0^\circ\), \(180^\circ\) and \(360^\circ\).

  • Period

    The tangent graph repeats every \(180^\circ\), and its values can be any number.

Reading a value from the graph

Use the graph of \(y = \cos x\) to write down the value of \(\cos 180^\circ\) and the values of \(x\) between \(0^\circ\) and \(360^\circ\) where \(\cos x = 0\).

Show the solutionHide the solution
  1. 1 Find 180 The cosine graph reaches its minimum at \(x = 180^\circ\), so \(\cos 180^\circ = -1\).
  2. 2 Find the zeros The curve crosses the \(x\)-axis where \(y = 0\).
  3. 3 Read the values It crosses at \(90^\circ\) and \(270^\circ\).

Answer\(\cos 180^\circ = -1\), and \(\cos x = 0\) when \(x = 90^\circ\) and \(x = 270^\circ\)

Solving \(\sin x = k\) and \(\cos x = k\)

An equation has more than one solution because the graph reaches the same height more than once.

  • Step 1

    Find the first angle from the exact values table, for example \(\sin 30^\circ = \dfrac{1}{2}\).

  • Sine

    The graph is symmetrical about \(90^\circ\), so the second solution is \(180^\circ - x\).

  • Cosine

    The graph is symmetrical about \(180^\circ\), so the second solution is \(360^\circ - x\).

  • Check

    Look at the graph to see how many solutions to expect between \(0^\circ\) and \(360^\circ\).

Solving a cosine equation

Solve \(\cos x = -\dfrac{1}{2}\) for \(0^\circ \le x \le 360^\circ\).

Show the solutionHide the solution
  1. 1 Use the positive value \(\cos 60^\circ = \dfrac{1}{2}\), so the related acute angle is \(60^\circ\).
  2. 2 Negative cosine Cosine is negative between \(90^\circ\) and \(270^\circ\).
  3. 3 First solution \(180^\circ - 60^\circ = 120^\circ\).
  4. 4 Second solution \(180^\circ + 60^\circ = 240^\circ\).

Answer\(x = 120^\circ\) and \(x = 240^\circ\)

Test yourself

  1. 1

    What are the maximum and minimum values of \(\sin x\)?

    Show answerHide answer

    1 and \(-1\).

  2. 2

    Where does the cosine graph start?

    Show answerHide answer

    At 1, when \(x = 0^\circ\).

  3. 3

    Where are the asymptotes of the tangent graph?

    Show answerHide answer

    At \(90^\circ\) and \(270^\circ\).

  4. 4

    What is the period of the tangent graph?

    Show answerHide answer

    \(180^\circ\).

  5. 5

    If \(\sin 30^\circ = \dfrac{1}{2}\), what is the other solution of \(\sin x = \dfrac{1}{2}\)?

    Show answerHide answer

    \(150^\circ\).

Exam technique: trigonometric graphs

Most marks come from knowing the shapes and the values.

  • Sketching

    Label the axes with \(90\), \(180\), \(270\) and \(360\), and mark the maximum and minimum.

  • Find all the solutions

    Use the graph to find out how many solutions there are, then use symmetry to find them.

  • Exact values

    Without a calculator, the numbers will be exact values, so learn the table.

  • The range

    If a question gives \(\sin x = 2\), there is no solution, because sine is never above 1.

Summary and exam focus

  • Sine and cosine are waves between \(-1\) and \(1\) that repeat every \(360^\circ\).
  • Tangent has asymptotes at \(90^\circ\) and \(270^\circ\), and repeats every \(180^\circ\).
  • For \(\sin x = k\), the second solution is \(180^\circ - x\). For \(\cos x = k\), it is \(360^\circ - x\).
  • Use the exact values table to solve equations without a calculator.

Exam focus

Solve \(\sin x = \dfrac{\sqrt{3}}{2}\) for \(0^\circ \le x \le 360^\circ\). (3 marks) (3 marks)

\(\sin 60^\circ = \dfrac{\sqrt{3}}{2}\), so \(x = 60^\circ\) is one solution. The second is \(180^\circ - 60^\circ = 120^\circ\). Check that you have found both.

Key terms

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

Sine
The ratio of the opposite side to the hypotenuse.
Cosine
The ratio of the adjacent side to the hypotenuse.
Tangent
The ratio of the opposite side to the adjacent side.
Period
The distance after which a graph repeats.
Asymptote
A line that a graph approaches but never touches.
Amplitude
Half the distance between the maximum and minimum values.
Exact value
A value written with fractions and surds, not as a decimal.
Symmetry
A balance in a graph, so that values are repeated.
Periodic
Repeating at regular intervals.

Questions and answers

15 questions set on this lesson, with the mark schemes and model answers open.

1. Exam question Solve 2 marks Core

The diagram shows the graph of \(y = \cos x\) for \(0^\circ \le x \le 360^\circ\), and the line \(y = -0.5\). Solve \(\cos x = -0.5\) for \(0^\circ \le x \le 360^\circ\). [2 marks]

The graph of cosine from 0 to 360 degrees with the line y equals minus 0.5 crossing the curve twice.

Mark scheme — 2 marks available

  • \(120\) — M1
  • \(240\) — A1

Model answer

\(\cos 60^\circ = 0.5\), and cosine is negative between \(90^\circ\) and \(270^\circ\), so \(x = 180 - 60 = 120^\circ\) or \(x = 180 + 60 = 240^\circ\).

2. Exam question Write down 3 marks Core

The diagram shows the graphs of two curves, \(P\) and \(Q\), for \(0^\circ \le x \le 360^\circ\). One is \(y = \sin x\) and the other is \(y = \cos x\). (a) State which curve is \(y = \sin x\). [1 mark] (b) Solve \(\sin x = \cos x\) for \(0^\circ \le x \le 360^\circ\). [2 marks]

Two wave graphs, one starting at the origin and one starting at 1, crossing at two points.

Mark scheme — 3 marks available

  • (a) Curve P, because it starts at the origin — B1
  • (b) \(45\) — B1
  • (b) \(225\) — B1

Model answer

(a) \(P\), because it passes through the origin. (b) The curves cross at \(x = 45^\circ\), where \(\sin 45^\circ = \cos 45^\circ = \dfrac{\sqrt{2}}{2}\), and at \(x = 225^\circ\).

3. Exam question Write down 3 marks Core

Write down the exact value of (a) \(\sin 135^\circ\) [1 mark] (b) \(\cos 135^\circ\) [1 mark] (c) \(\tan 150^\circ\) [1 mark]

Mark scheme — 3 marks available

  • (a) \(\dfrac{\sqrt{2}}{2}\) — B1
  • (b) \(-\dfrac{\sqrt{2}}{2}\) — B1
  • (c) \(-\dfrac{1}{\sqrt{3}}\) or \(-\dfrac{\sqrt{3}}{3}\) — B1

Model answer

(a) \(\dfrac{\sqrt{2}}{2}\). (b) \(-\dfrac{\sqrt{2}}{2}\). (c) \(-\dfrac{\sqrt{3}}{3}\), which is the same as \(-\dfrac{1}{\sqrt{3}}\).

4. Exam question Solve 3 marks Core

Solve \(2\cos x + 1 = 0\) for \(0^\circ \le x \le 360^\circ\). [3 marks]

Mark scheme — 3 marks available

  • \(\cos x = -\dfrac{1}{2}\) — M1
  • \(120\) — A1
  • \(240\) — A1

Model answer

\(\cos x = -\dfrac{1}{2}\), so \(x = 120^\circ\) or \(x = 240^\circ\).

5. Exam question Solve 3 marks Stretch

Solve \(\tan x = -\sqrt{3}\) for \(0^\circ \le x \le 360^\circ\). [3 marks]

Mark scheme — 3 marks available

  • Uses \(60^\circ\) as the related angle — M1
  • \(120\) — A1
  • \(300\) — A1

Model answer

\(\tan 60^\circ = \sqrt{3}\), and tangent is negative between \(90^\circ\) and \(180^\circ\), and between \(270^\circ\) and \(360^\circ\). So \(x = 180 - 60 = 120^\circ\) or \(x = 360 - 60 = 300^\circ\).

6. Exam question Write down 2 marks Stretch

(a) Write down the period of the graph of \(y = \tan x\). [1 mark] (b) How many solutions does \(\cos x = 0.3\) have for \(0^\circ \le x \le 720^\circ\)? [1 mark]

Mark scheme — 2 marks available

  • (a) \(180^\circ\) — B1
  • (b) 4 — B1

Model answer

(a) \(180^\circ\). (b) Four solutions, two in each \(360^\circ\).

7. Multiple choice 1 mark Easier

What is the maximum value of \(y = \sin x\)?

  1. A 0
  2. B 1 Correct
  3. C 90
  4. D \(\infty\)

Why: The sine graph is a wave between \(-1\) and \(1\).

8. Multiple choice 1 mark Easier

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

  1. A 1 Correct
  2. B 0
  3. C \(-1\)
  4. D \(\dfrac{1}{2}\)

Why: The cosine graph starts at its maximum, 1.

9. Multiple choice 1 mark Easier

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

  1. A \(x = 0^\circ\) and \(x = 180^\circ\)
  2. B \(x = 45^\circ\) and \(x = 135^\circ\)
  3. C \(x = 180^\circ\) and \(x = 360^\circ\)
  4. D \(x = 90^\circ\) and \(x = 270^\circ\) Correct

Why: The tangent is undefined at \(90^\circ\) and \(270^\circ\).

10. Multiple choice 1 mark Core

How many solutions does \(\sin x = \dfrac{1}{2}\) have for \(0^\circ \le x \le 360^\circ\)?

  1. A 1
  2. B 3
  3. C 2 Correct
  4. D 4

Why: The line \(y = \dfrac{1}{2}\) crosses the sine curve twice, at \(30^\circ\) and \(150^\circ\).

11. Multiple choice 1 mark Core

If \(\sin 30^\circ = \dfrac{1}{2}\), what is the other solution of \(\sin x = \dfrac{1}{2}\) between \(0^\circ\) and \(360^\circ\)?

  1. A \(120^\circ\)
  2. B \(150^\circ\) Correct
  3. C \(210^\circ\)
  4. D \(330^\circ\)

Why: The second solution is \(180^\circ - 30^\circ = 150^\circ\).

12. Multiple choice 1 mark Core

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

  1. A \(180^\circ\) Correct
  2. B \(360^\circ\)
  3. C \(90^\circ\)
  4. D \(45^\circ\)

Why: The tangent graph repeats every \(180^\circ\).

13. Multiple choice 1 mark Core

Solve \(\cos x = \dfrac{1}{2}\) for \(0^\circ \le x \le 360^\circ\).

  1. A \(60^\circ\) and \(120^\circ\)
  2. B \(30^\circ\) and \(330^\circ\)
  3. C \(60^\circ\) and \(240^\circ\)
  4. D \(60^\circ\) and \(300^\circ\) Correct

Why: \(\cos 60^\circ = \dfrac{1}{2}\), and the second solution is \(360^\circ - 60^\circ = 300^\circ\).

14. Multiple choice 1 mark Stretch

Solve \(\sin x = -\dfrac{1}{2}\) for \(0^\circ \le x \le 360^\circ\).

  1. A \(30^\circ\) and \(150^\circ\)
  2. B \(120^\circ\) and \(240^\circ\)
  3. C \(210^\circ\) and \(330^\circ\) Correct
  4. D \(150^\circ\) and \(210^\circ\)

Why: Sine is negative between \(180^\circ\) and \(360^\circ\), so the solutions are \(180 + 30 = 210\) and \(360 - 30 = 330\).

15. Multiple choice 1 mark Stretch

Which equation has no solutions?

  1. A \(\cos x = -1\)
  2. B \(\sin x = 1.5\) Correct
  3. C \(\tan x = 5\)
  4. D \(\sin x = 0\)

Why: Sine and cosine are never greater than 1, so \(\sin x = 1.5\) has no solution.