Exam-style Questions: Trigonometry

1. a) Express 2cos x - sin x in the form Rcos (x + a), where R is a positive constant and α is an angle between 0° and 360°.

b) Given that 0 ≤ x

(i) solve 2cos x - sin x = 1

(ii) deduce the solution set of the inequality 2cos x- sin x ≥ 1.

(Marks available: 6)

Answer

Answer outline and marking scheme for question: 1

Give yourself marks for mentioning any of the points below:

a) Using the Rcos formula give:

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and

Copyright S-cool

Therefore

Copyright S-cool

(2 marks)

b) (i) input values into the Rcos formula

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solving the above equation gives x = 36.9o and x = 270o.

(2 marks)

(ii) solving the given in-equality gives:

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(2 marks)

(Marks available: 6)

2.

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The diagram shows the triangle ABC in which AB = 7 cm, BC = 9cm and CA = 8cm.

a) Use the cosine rule to find cos C, giving your answer as a fraction in its lowest
terms.

b) Hence show that sin C = Copyright S-cool

c) Find sinA in the formCopyright S-cool where p and q are positive integers to be determined.

(Marks available: 7)

Answer

Answer outline and marking scheme for question: 2

Give yourself marks for mentioning any of the points below:

a) Applying the cosine rule gives:

Copyright S-cool

(2 marks)

b) Rearranging gives:

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Copyright S-cool

(2 marks)

c) Appling the sine rule gives:

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(3 marks)

Total 7 marks

3. a) Express 2 sin θ cos 6θ as a difference of two sines.

b) Hence prove the identity

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c) Deduce that

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(Marks available: 7)

Answer

Answer outline and marking scheme for question: 3

Give yourself marks for mentioning any of the points below:

a) Using the sine rule:

2sin θ cost 6θ = sin 7θ - sin 5θ

(1 mark)

b) Applying the sine rule again:

2sin θ cost 4θ = sin 5θ - sin 3θ

2sin θ cos 2θ = sin 3θ - sin θ

Adding the three expressions above, gives:

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(3 marks)

c) Substitute θ = 2π/7.

Putting this into the equation in (b) gives:

Copyright S-cool.

(3 marks)

(Marks available: 7)