Name ______ Topic 3 Problem Set: Trigonometry

Name ______Topic 3 Problem Set: Trigonometry

Part I – No Calculator (Questions 1-9)

1. Given that sin x = , where x is an acute angle, find the exact value of

(a) cos x;

(b) cos 2x.

(Total 6 Marks)

2. Write the expression 3 sin2 x + 4 cos x in the form a cos2 x + b cos x + c. (2 Marks)

3. If sinx=-13 and πx3π2, find tanx. (4 Marks)

4. If A is an obtuse angle in a triangle and sin A = 45, calculate the exact value of sin 2A.

(4 Marks)

5. Given the values of sin and cos , determine the quadrant in which lies.

, (2 Marks)

8. Evaluate and express your answer in simplified form.

sin5π6+cosπ+tan5π4 (4 Marks)

9. If 32,-12 is a coordinate on the unit circle find the value of the angle θ. (2 Marks)

Part II – Calculator Permitted

10. The diagram below shows a circle of radius r and centre O. The angle= θ.

The length of the arc AB is 24 cm. The area of the sector OAB is 180 cm2.

Find the value of r and of θ. (6 Marks)

11. The following diagram shows a circle of centre O, and radius 15 cm. The arc ACB subtends an angle of 2 radians at the centre O.

Find

(a) the length of the arc ACB;

(b) the area of the shaded region.

(6 Marks)

12. Find the acute angle between the line given and the x-axis.

y=4x+1

13. Consider the equation 3 cos 2x + sin x = 1

(a) Write this equation in the form f (x) = 0 , where f (x) = p sin2 x + q sin x + r , and p , q , r є .

(b) Factorize f (x).

14. The following diagram shows a triangle ABC, where BC = 5 cm, = 60°, = 40°.

(a) Calculate AB.

(b) Find the area of the triangle.

15. [Show your work for this problem on a separate sheet of paper and attach.]

The following diagram shows two semi-circles. The larger one has centre O and radius 4 cm. The smaller one has centre P, radius 3 cm, and passes through O. The line (OP) meets the larger semi-circle at S. The semi-circles intersect at Q.

(a) (i) Explain why OPQ is an isosceles triangle.

(ii) Use the cosine rule to show that cos = .

(iii) Hence show that sin = .

(iv) Find the area of the triangle OPQ.

(7)

(b) Consider the smaller semi-circle, with centre P.

(i) Write down the size of

(ii) Calculate the area of the sector OPQ.

(3)

(d) Hence calculate the area of the shaded region.

(4)

(Total 17 marks)

16. [Show your work for this problem on a separate sheet of paper and attach.]

A farmer owns a triangular field ABC. One side of the triangle, [AC], is 104 m, a second side, [AB], is 65 m and the angle between these two sides is 60°.

(a) Use the cosine rule to calculate the length of the third side of the field.

(3)

(b) Given that sin 60° = find the area of the field in the form where p is an integer.

(3)

Let D be a point on [BC] such that [AD] bisects the 60° angle. The farmer divides the field into two parts A1 and A2 by constructing a straight fence [AD] of length x metres, as shown on the diagram below.

(ii) Find a similar expression for the area of A2.

(iii) Hence, find the value of x in the form , where q is an integer.

(7)

(Total 13 marks)

17. The following graph shows the depth of water, y metres, at a point P, during one day.

The time t is given in hours, from midnight to noon.

(a) Use the graph to write down an estimate of the value of t when

(i) the depth of water is minimum;

(ii) the depth of water is maximum;

(iii) the depth of the water is increasing most rapidly.

(3)

(b) The depth of water can be modelled by the function y = A cos (B (t – 1)) + C.

(i) Show that A = 8.

(ii) Write down the value of C.

(iii) Find the value of B.

(6)

(c) A sailor knows that he cannot sail past P when the depth of the water is less than 12 m. Calculate the values of t between which he cannot sail past P.

(2)

(Total 11 marks)

18. The diagram below shows the graph of f (x) = 1 + tan for −360° £ x £ 360°.

(a) On the same diagram, draw the asymptotes.

(2)

(b) Write down

(i) the period of the function;

(ii) the value of f (90°).

(2)

(Total 6 marks)

19. Consider g (x) = 3 sin 2x.

(a) Write down the period of g.

......

(1)

(b) On the diagram below, sketch the curve of g, for 0 £ x £ 2p.

(3)

......

(2)

(Total 6 marks)