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 outline and marking scheme for question: 1
Give yourself marks for mentioning any of the points below:
a) Using the Rcos formula give:
and
Therefore
(2 marks)
b) (i) input values into the Rcos formula
solving the above equation gives x = 36.9o and x = 270o.
(2 marks)
(ii) solving the given in-equality gives:
(2 marks)
(Marks available: 6)
2.
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 = 
c) Find sinA in the form
where p and q are positive integers to be determined.
(Marks available: 7)
Answer outline and marking scheme for question: 2
Give yourself marks for mentioning any of the points below:
a) Applying the cosine rule gives:
(2 marks)
b) Rearranging gives:
(2 marks)
c) Appling the sine rule gives:
(3 marks)
Total 7 marks
3. a) Express 2 sin θ cos 6θ as a difference of two sines.
b) Hence prove the identity
c) Deduce that
(Marks available: 7)
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:
(3 marks)
c) Substitute θ = 2π/7.
Putting this into the equation in (b) gives:
.
(3 marks)
(Marks available: 7)