A LevelMathsAQAPureExam QuestionsTrigonometryCompound & Double Angle Formulae

Compound & Double Angle Formulae (AQA A Level Maths: Pure): Exam Questions

Exam code: 7357

4 hours40 questions
Easy
Medium
Hard
Very Hard
1
5 marks

(i) State the exact value of \text{cos} \; 60^{\circ}.

(ii) State the exact value of \text{cos} \; 45^{\circ}.

(iii) Write down the exact value of \text{cos} \; 105^{\circ}.

(iv) Hence show that \text{cos} \; 60^{\circ} + \text{cos} \; 45^{\circ} \neq \text{cos} \; 105^{\circ}.

How did you do?

2a
1 mark

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

By writing 15^{\circ} as (45^{\circ} - 30^{\circ}), express \text{sin} \; 15^{\circ} in terms of the sine and cosine of 45^{\circ} and 30^{\circ}.

How did you do?

2b
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3 marks

Hence show that

\text{sin} \; 15^{\circ} = \frac{\sqrt{6} - \sqrt{2}}{4}

How did you do?

3a
2 marks

By substituting B = A into the identity for \text{sin} \left(A + B\right), show that

\text{sin} \; 2A \equiv 2 \; \text{sin} \; A \; \text{cos} \; A

How did you do?

3b
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2 marks

Hence show that the exact value of \text{sin} \; 120^{\circ} is \frac{\sqrt{3}}{2}.

How did you do?

4a
1 mark

Write down the expansion of \text{sin} \left(\theta + \alpha\right) in terms of \text{sin} \; \theta, \text{cos} \; \theta, \text{sin} \; \alpha and \text{cos} \; \alpha.

How did you do?

4b
1 mark

Hence show that

R \; \text{sin} \left(\theta + \alpha\right) \equiv R \; \text{cos} \; \alpha \; \text{sin} \; \theta + R \; \text{sin} \; \alpha \; \text{cos} \; \theta

How did you do?

5
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3 marks

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Show that

5 \; \text{cos} \left(\theta - \frac{\pi}{6}\right) \equiv \frac{5\sqrt{3}}{2} \; \text{cos} \; \theta + \frac{5}{2} \; \text{sin} \; \theta

How did you do?

6
2 marks

Show that

\text{cos}^{2} \; x + \text{cos} \; 2x \equiv 3 \; \text{cos}^{2} \; x - 1

How did you do?

7a
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4 marks

(i) Show that

R \; \text{sin} \left(\theta + \alpha\right) \equiv R \; \text{cos} \; \alpha \; \text{sin} \; \theta + R \; \text{sin} \; \alpha \; \text{cos} \; \theta

where R and \alpha are constants with R > 0 and 0 < \alpha < \frac{\pi}{2}.

(ii) Hence show that

\sqrt{3} \; \text{sin} \; \theta + \text{cos} \; \theta \equiv 2 \; \text{sin} \left(\theta + \frac{\pi}{6}\right)

How did you do?

7b
1 mark

Write down the maximum value of \sqrt{3} \; \text{sin} \; \theta + \text{cos} \; \theta.

How did you do?

8
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3 marks

Sketch the graph of y = \text{tan} \; 2\theta for 0 \leq \theta \leq 2\pi.

Show on your sketch the coordinates of the points where the graph crosses the coordinate axes.

How did you do?

9
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2 marks

"If A and B are any two angles, then \text{sin} \left(A + B\right) \equiv \text{sin} \; A + \text{sin} \; B."

Disprove this statement by means of a counter example.

How did you do?

10a
2 marks

By substituting B = A into the identity for \text{cos} \left(A + B\right), show that

\text{cos} \; 2A \equiv \text{cos}^{2} \; A - \text{sin}^{2} \; A

How did you do?

10b
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2 marks

Hence, or otherwise, show that

\text{cos} \; 2A \equiv 1 - 2 \; \text{sin}^{2} \; A

How did you do?

11
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2 marks

A student observes that when A = B, the following relationship holds:

\text{sin} \left(A - B\right) = \text{sin} \left(0\right) = 0 = \text{sin} \; A - \text{sin} \; A = \text{sin} \; A - \text{sin} \; B

The student concludes that \text{sin} \left(A - B\right) \equiv \text{sin} \; A - \text{sin} \; B is true in general.

Disprove this statement by means of a counter example.

How did you do?

1a
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3 marks

Express 3 cos space theta plus 4 sin space theta in the form R cos space open parentheses theta minus alpha close parentheses, where R greater than 0 and 0 less than alpha less than pi over 2.

Give the exact value of R and give the value of alpha in radians to 3 decimal places.

How did you do?

1b
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3 marks

Hence solve the equation 3 cos space theta plus 4 sin space theta equals 2.5 for 0 less or equal than theta less than 2 pi.

Write your answers to 3 significant figures.

How did you do?

2a
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4 marks

Solve, for -\pi \leq \theta \leq \pi, the equation

\text{cos}^{2} \; \theta - \text{sin}^{2} \; \theta = \frac{1}{2}

How did you do?

2b
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5 marks

Solve, for 0 \leq x \leq \pi, the equation

4 \; \text{sin} \; x \; \text{cos} \; x = -\sqrt{3}

How did you do?

3
3 marks

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Show that

\frac{5 \; \text{sin} \; 2x}{\text{tan} \; x} \equiv 10 \; \text{cos}^{2} \; x \qquad x \neq \frac{k\pi}{2}

How did you do?

4a
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3 marks

(i) Show that

R \; \text{cos} \left(x + \alpha\right) \equiv R \; \text{cos} \; \alpha \; \text{cos} \; x - R \; \text{sin} \; \alpha \; \text{sin} \; x

where R and \alpha are constants.

(ii) Hence show that

\text{cos} \; x - \sqrt{3} \; \text{sin} \; x \equiv 2 \; \text{cos} \left(x + \frac{\pi}{3}\right)

How did you do?

4b
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3 marks

Hence solve, for 0 \leq x \leq 2\pi, the equation

\text{cos} \; x - \sqrt{3} \; \text{sin} \; x = 1

How did you do?

5a
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4 marks

Express 5 \; \text{sin} \; \theta + 12 \; \text{cos} \; \theta in the form R \; \text{sin} \left(\theta + \alpha^{\circ}\right), where R > 0 and 0^{\circ} < \alpha < 90^{\circ}.

How did you do?

5b
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3 marks

Sketch the graph of y = 5 \; \text{sin} \; x + 12 \; \text{cos} \; x for 0^{\circ} \leq x \leq 360^{\circ}.

Show on your sketch the coordinates of the points where the graph crosses the coordinate axes.

How did you do?

6
3 marks

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Show that

2 \; \text{cosec} \; 2A \equiv \text{cosec} \; A \; \text{sec} \; A \qquad A \neq \frac{k\pi}{2}

How did you do?

7a
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4 marks

Solve, for - \pi \leq \theta \leq \pi, the equation

\text{sin} \; 2\theta = \frac{1}{2}

How did you do?

7b
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4 marks

Solve, for 0 \leq \theta \leq 2\pi, the equation

\text{cos} \; 2\theta = \frac{\sqrt{3}}{2}

How did you do?

8a
2 marks

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

By writing 285^{\circ} as (315^{\circ} - 30^{\circ}), express \text{cos} \; 285^{\circ} in terms of the sine and cosine of 315^{\circ} and 30^{\circ}.

How did you do?

8b
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3 marks

Hence show that

\text{cos} \left(285^{\circ}\right) = \frac{\sqrt{6} - \sqrt{2}}{4}

How did you do?

9a
3 marks

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Use the difference of two squares to show that

\text{cos}^{4} \; x - \text{sin}^{4} \; x \equiv \text{cos} \; 2x

How did you do?

9b
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3 marks

Hence solve, for -\frac{\pi}{2} \leq x \leq \frac{\pi}{2}, the equation

\text{cos}^{4} \; x - \text{sin}^{4} \; x = \frac{\sqrt{2}}{2}

How did you do?

10a
2 marks

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

By writing 210^{\circ} as (180^{\circ} + 30^{\circ}), express \text{tan} \; 210^{\circ} in terms of \text{tan} \; 180^{\circ} and \text{tan} \; 30^{\circ}.

How did you do?

10b
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2 marks

Hence show that \text{tan} \left(210^{\circ}\right) = \frac{\sqrt{3}}{3}.

How did you do?

11
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5 marks

Express 2 \; \text{cos} \; \theta - 5 \; \text{sin} \; \theta in the form R \; \text{cos} \left(\theta + \alpha\right), where R > 0 and 0 < \alpha < \frac{\pi}{2}.

Give the exact value of R, and give the value of \alpha in radians correct to 3 significant figures.

How did you do?

12a
1 mark

Show that

R \; \text{sin} \left(\theta + \alpha\right) \equiv R \; \text{cos} \; \alpha \; \text{sin} \; \theta + R \; \text{sin} \; \alpha \; \text{cos} \; \theta

where R and \alpha are constants.

How did you do?

12b
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3 marks

Hence show that

3 \; \text{sin} \; \theta + 2 \; \text{cos} \; \theta \equiv \sqrt{13} \; \text{sin} \left(\theta + 0.588\right)

where 0.588 is measured in radians to 3 decimal places.

How did you do?

13
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3 marks

(i) Disprove the following statement by means of a counter example:

\text{sin} \left(A - B\right) \equiv \text{sin} \; A + \text{sin} \; B

(ii) Find a value for A and a value for B, where A \neq 0 and B \neq 0, such that

\text{sin} \left(A - B\right) = \text{sin} \; A + \text{sin} \; B

How did you do?

14
3 marks

By writing 2A as (A + A) show that

\text{tan} \; 2A \equiv \frac{2 \; \text{tan} \; A}{1 - \text{tan}^{2} \; A}

You must clearly state any trigonometric identities you use in your proof.

How did you do?

15
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6 marks

Given that a and b are positive constants, and that

a \; \text{sin} \; \theta + b \; \text{cos} \; \theta \equiv R \; \text{sin} \left(\theta + \alpha\right)

where R > 0 and 0 < \alpha < \frac{\pi}{2},

(i) find an expression for \alpha in terms of a and b,

(ii) find an expression for R in terms of a and b.

How did you do?

16
4 marks

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Show that

\frac{\text{sin} \left(A + B\right) + \text{sin} \left(A - B\right)}{\text{cos} \left(A + B\right) + \text{cos} \left(A - B\right)} \equiv \text{tan} \; A \qquad A, B \neq \left(k + \frac{1}{2}\right)\pi

How did you do?

17
3 marks

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Show that

2 - 2 \; \text{cot} \; 2A \; \text{tan} \; A \equiv \text{sec}^{2} \; A \qquad A \neq k\pi

How did you do?

1a
5 marks

By using the identity for \text{sin} \left(A + B\right) and the substitution \text{cos} \; 2A \equiv 1 - 2 \; \text{sin}^{2} \; A, show that

\text{sin} \; 3A \equiv 3 \; \text{sin} \; A - 4 \; \text{sin}^{3} \; A

How did you do?

1b
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4 marks

Hence solve, for -\pi \leq \theta \leq \pi, the equation

3 \; \text{sin} \; \theta - 4 \; \text{sin}^{3} \; \theta = \frac{1}{2}

How did you do?

2a
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5 marks

Solve, for -\pi \leq \theta \leq \pi, the equation

\text{sin} \; 2\theta = \text{sin} \; \theta

How did you do?

2b
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4 marks

Solve, for 0 \leq x \leq 2\pi, the equation

\text{cos} \; 2x + \text{sin}^{2} \; x = 0

How did you do?

3a
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4 marks

Express 2 \; \text{sin} \; \theta + 4 \; \text{cos} \; \theta in the form R \; \text{cos} \left(\theta - \alpha\right), where R > 0 and 0 < \alpha < \frac{\pi}{2}.

Give the exact value of R, and give the value of \alpha in radians to 3 significant figures.

How did you do?

3b
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3 marks

Hence solve, for -\pi \leq \theta \leq \pi, the equation

2 \; \text{sin} \; \theta + 4 \; \text{cos} \; \theta = 3

giving your answers to 3 significant figures.

How did you do?

4a
5 marks

By writing 3A as (2A + A) and using the identity for \text{tan} \left(A + B\right), show that

\text{tan} \; 3A \equiv \frac{3 \; \text{tan} \; A - \text{tan}^{3} \; A}{1 - 3 \; \text{tan}^{2} \; A}

How did you do?

4b
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3 marks

Hence solve, for 0 \leq x \leq \pi, the equation

\frac{6 \; \text{tan} \; x - 2 \; \text{tan}^{3} \; x}{1 - 3 \; \text{tan}^{2} \; x} = 2

How did you do?

5
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7 marks

(i) Express 2 \; \text{sin} \; x - 2 \; \text{cos} \; x in the form R \; \text{sin} \left(x - \alpha^{\circ}\right), where R > 0 and 0^{\circ} < \alpha < 90^{\circ}.

Give the exact value of R and the value of \alpha.

(ii) Hence sketch the curve with equation

y = 2 \left(\text{sin} \; x - \text{cos} \; x\right) \qquad 0^{\circ} \leq x \leq 360^{\circ}

Show on your sketch the coordinates of the points where the curve crosses the coordinate axes, and state the exact coordinates of the maximum and minimum turning points.

How did you do?

1a
3 marks

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

By writing \left(X + Y - Z\right) as \left(\left(X + Y\right) - Z\right) and using the identities for \text{sin} \left(A \pm B\right) and \text{cos} \left(A \pm B\right), show that

\text{sin} \left(X + Y - Z\right) \equiv \text{sin} \; X \; \text{cos} \; Y \; \text{cos} \; Z + \text{cos} \; X \; \text{sin} \; Y \; \text{cos} \; Z - \text{cos} \; X \; \text{cos} \; Y \; \text{sin} \; Z + \text{sin} \; X \; \text{sin} \; Y \; \text{sin} \; Z

How did you do?

1b
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4 marks

Hence show that

\text{sin} \; 165^{\circ} = \frac{\sqrt{6} - \sqrt{2}}{4}

How did you do?

2a
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5 marks

Solve, for 0 \leq \theta < 2\pi, the equation

\text{cos} \; 2\theta = \text{cos} \; \theta

How did you do?

2b
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6 marks

Solve, for -\pi \leq x \leq \pi, the equation

\text{tan} \; 2x = 3 \; \text{tan} \; x

How did you do?

3
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5 marks

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Show that

\text{tan} \; 2\theta \; \text{tan} \; \theta \equiv \text{sec} \; 2\theta - 1

How did you do?

4a
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4 marks

Show that 5 \; \text{sin} \; \theta - 3 \; \text{cos} \; \theta can be expressed in the form R \; \text{sin} \left(\theta - \alpha\right), where R = \sqrt{34} and \alpha = 0.540 radians to 3 significant figures.

How did you do?

4b
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5 marks

Hence or otherwise, solve for 0 \leq x \leq 2\pi, the equation

3 \; \text{cos} \; 2x + 5 \; \text{sin} \; 2x = 0.4

How did you do?

5a
4 marks

By using the double angle identity for \text{cos} \; 2A, show that \text{cos} \; 4A can be expressed in the form

a \; \text{cos}^{4} \; A + b \; \text{cos}^{2} \; A + c

where a, b and c are constants to be found.

How did you do?

5b
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5 marks

Hence solve, for 0 \leq x \leq \pi, the equation

2 \; \text{cos} \; 4x = 7 \; \text{sin}^{2} \; x - 2

How did you do?

6
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7 marks

Two right-angled triangles are shown in the diagram below. Angles A and B have been labelled.

q9-5-6-compund-and-double-angle-formulae-a-level-only-edexcel-a-level-pure-maths-veryhard

Given that \alpha = A + B, find the exact values of \text{sin} \; \alpha, \text{cos} \; \alpha and \text{tan} \; \alpha.

How did you do?

7
4 marks

(i) Explain briefly why \theta = 0 is not a solution to the equation

3\theta \; \text{cot} \; 2\theta = 0

(ii) Given that \theta is small and measured in radians, use the small angle approximations to find the value of

3\theta \; \text{cot} \; 2\theta

How did you do?