# 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 = 36.9^{o} and x = 270^{o}.

**(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)**