A LevelMathsEdexcelPureExam QuestionsTrigonometryCompound & Double Angle Formulae

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

Exam code: 9MA0

5 hours49 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?

5a
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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?

5b
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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?

6
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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?

7
2 marks

Show that

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

How did you do?

8a
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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?

8b
1 mark

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

How did you do?

9
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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?

10
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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?

11a
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?

11b
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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?

12
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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
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?

1b
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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?

2
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4 marks

Solve, for 0 less or equal than x less than pi over 2, the equation

4 sin space x equals sec space x

How did you do?

3a
4 marks

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

Solutions relying entirely on calculator technology are not acceptable.

Show that

cos 3 A identical to 4 cos cubed A minus 3 cos A

How did you do?

3b
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4 marks

Hence solve, for negative 90 degree less or equal than x less or equal than 180 degree, the equation

1 minus cos space 3 x equals sin squared x

How did you do?

4a
4 marks

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

Solutions relying entirely on calculator technology are not acceptable.

Given that 1 plus cos 2 theta plus sin 2 theta not equal to 0 prove that

fraction numerator 1 minus cos 2 theta plus sin 2 theta over denominator 1 plus cos 2 theta plus sin 2 theta end fraction identical to tan theta

How did you do?

4b
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4 marks

Hence solve, for 0 less than x less than 180 degree

fraction numerator 1 minus cos 4 x plus sin 4 x over denominator 1 plus cos 4 x plus sin 4 x end fraction equals 3 sin 2 x

giving your answers to one decimal place where appropriate.

How did you do?

5a
4 marks

Prove

fraction numerator cos 3 theta over denominator sin theta end fraction plus fraction numerator sin 3 theta over denominator cos theta end fraction identical to 2 cot 2 theta space space space space space theta not equal to open parentheses 90 n close parentheses degree comma space n element of straight integer numbers

How did you do?

5b
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3 marks

Hence solve, for 90 degree less than theta less than 180 degree, the equation

fraction numerator cos 3 theta over denominator sin theta end fraction plus fraction numerator sin 3 theta over denominator cos theta end fraction equals 4

giving any solutions to one decimal place.

How did you do?

6a
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3 marks

Express 2 cos theta plus 8 sin theta in the form R cos open parentheses theta minus alpha close parentheses, where R and alpha are constants, 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?

6b
3 marks

The first three terms of an arithmetic sequence are

cos space x space space space space space space space space cos space x plus sin space x space space space space space space space space cos space x plus 2 sin space x space space space space space space space space space space space space space x not equal to n pi

Given that S subscript 9 represents the sum of the first 9 terms of this sequence as x varies,

(i) find the exact maximum value of S subscript 9

(ii) deduce the smallest positive value of x at which this maximum value of S subscript 9 occurs.

How did you do?

7a
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?

7b
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2 marks

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

How did you do?

8a
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?

8b
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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?

9a
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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?

9b
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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?

10
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?

11a
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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?

11b
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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?

12a
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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?

12b
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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?

13
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?

14a
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?

14b
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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?

15
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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?

16
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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?

17
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?

18
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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?

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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6 marks

Solve, for negative 180 degree less or equal than theta less or equal than 180 degree, the equation

5 sin 2 theta equals 9 tan theta

giving your answers, where necessary, to one decimal place.

[Solutions based entirely on graphical or numerical methods are not acceptable.]

How did you do?

2b
2 marks

Deduce the smallest positive solution to the equation

5 sin open parentheses 2 x minus 50 degree close parentheses equals 9 tan open parentheses x minus 25 degree close parentheses

How did you do?

3a
4 marks

Given that

2 sin left parenthesis x minus 60 degree right parenthesis equals cos left parenthesis x minus 30 degree right parenthesis

show that

tan space x equals 3 square root of 3

How did you do?

3b
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4 marks

Hence or otherwise solve, for 0 less or equal than theta less than 180 degree

2 sin space 2 theta equals cos left parenthesis 2 theta plus 30 degree right parenthesis

giving your answers to one decimal place.

How did you do?

4a
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3 marks

Express 2 cos theta minus sin theta in the form R cos open parentheses theta plus 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 the value of alpha in radians to 3 decimal places.

How did you do?

4b
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3 marks
Diagram of a circular wheel with centre C. Radial lines come out from the centre of the wheel and the water level is indicated with a horizontal line. Point 'P' and it's height above the water level, 'H metres', are labelled.
Figure 6

Figure 6 shows the cross-section of a water wheel.

The wheel is free to rotate about a fixed axis through the point C.

The point P is at the end of one of the paddles of the wheel, as shown in Figure 6.

The water level is assumed to be horizontal and of constant height.

The vertical height, H metres, of P above the water level is modelled by the equation

H equals 3 plus 4 cos open parentheses 0.5 t close parentheses minus 2 sin open parentheses 0.5 t close parentheses

where t is the time in seconds after the wheel starts rotating.

Using the model, find

(i) the maximum height of P above the water level,

(ii) the value of t when this maximum height first occurs, giving your answer to one decimal place.

How did you do?

4c
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4 marks

In a single revolution of the wheel, P is below the water level for a total of T seconds.

According to the model, find the value of T giving your answer to 3 significant figures.

(Solutions based entirely on calculator technology are not acceptable.)

How did you do?

5a
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3 marks

Express sin x plus 2 cos x in the form R sin open parentheses x plus alpha close parentheses where R and alpha are constants, 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?

5b
1 mark

The temperature, theta °C, inside a room on a given day is modelled by the equation

theta equals 5 plus sin open parentheses fraction numerator pi t over denominator 12 end fraction minus 3 close parentheses plus 2 cos open parentheses fraction numerator pi t over denominator 12 end fraction minus 3 close parentheses space space space space space space space space 0 less or equal than t less than 24

where t is the number of hours after midnight.

Using the equation of the model and your answer to part (a), deduce the maximum temperature of the room during this day.

How did you do?

5c
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3 marks

Find the time of day when the maximum temperature occurs, giving your answer to the nearest minute.

How did you do?

6
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5 marks

Solve, for 0 less or equal than theta less than 360 degree, the equation

5 sin theta minus 5 cos theta equals 2

giving your answers to one decimal place.

(Solutions based entirely on graphical or numerical methods are not acceptable.)

How did you do?

7a
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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?

7b
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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?

8
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?

9a
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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?

9b
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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?

10a
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?

10b
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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?

11
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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?

12
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
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?