C3 Trigonometry

340 min

283 marks

1.(a)Prove that for all values of x,

sin x + sin (60° - x) º sin (60° + x).

(4)

(b)Given that sin 84° - sin 36° = sin a°, deduce the exact value of the acute angle a. (2)

(c)Solve the equation

4 sin 2x + sin (60° - 2x) = sin (60° + 2x) – 1

for values of x in the interval 0 £ x < 360°, giving your answers to one decimal place. (5)

(Total 11 marks)

2.On separate diagrams, sketch the curves with equations

(a)y = arcsin x, -1 £ x £ 1,

(b)y = sec x, - £ x £ , stating the coordinates of the end points of your curves in each case.

(4)

Use the trapezium rule with five equally spaced ordinates to estimate the area of the region bounded by the curve with equation y = sec x, the x-axis and the lines x = and x = -, giving your answer to two decimal places. (4)

(Total 8 marks)

3.Find, giving your answers to two decimal places, the values of w, x, y and z for which (a)e-w = 4,

(2)

(b)arctan x = 1,

(2)

(c)ln (y + 1) – ln y = 0.85

(4)

(d)cos z + sin z = , -p < z < p.

(5)

(Total 13 marks)

4.(a)Using the formulae

sin (A ± B) = sin A cos B ± cos A sin B,

cos (A ± B) = cos A cos B sin A sin B,

show that

(i)sin (A + B) – sin (A – B) = 2 cos A sin B,

(2)

(ii)cos (A – B) – cos (A + B) = 2 sin A sin B.

(2)

(b)Use the above results to show that

= cot A.

(3)

Using the result of part (b) and the exact values of sin 60° and cos 60°,

(c)find an exact value for cot 75° in its simplest form.

(4)

(Total 11 marks)

5.In a particular circuit the current, I amperes, is given by I = 4 sin q – 3 cos q, q > 0,

where q is an angle related to the voltage.

Given that I = R sin (q - a), where R > 0 and 0 £ a < 360°,

(a)find the value of R, and the value of a to 1 decimal place. (4)

(b)Hence solve the equation 4 sin q – 3 cos q = 3 to find the values of q between 0 and 360°.

(5)

(c)Write down the greatest value for I.

(1)

(d)Find the value of q between 0 and 360° at which the greatest value of I occurs. (2)

(Total 12 marks)

6.(a)Express sin x + Ö3 cos x in the form R sin (x + a), where R > 0 and 0 < a < 90°. (4)

(b)Show that the equation sec x + Ö3 cosec x = 4 can be written in the form sin x + Ö3 cos x = 2 sin 2x.

(3)

(c)Deduce from parts (a) and (b) that sec x + Ö3 cosec x = 4 can be written in the form sin 2x – sin (x + 60°) = 0.

(1)

(d)Hence, using the identity sin X – sin Y = 2 cos , or otherwise, find the values of x in the interval 0 £ x £ 180°, for which sec x + Ö3 cosec x = 4. (5)

(Total 13 marks)

7.(i)Given that cos(x + 30)° = 3 cos(x – 30)°, prove that tan x° = -. (5)

(ii)(a)Prove that º tan q .

(3)

(b)Verify that q = 180° is a solution of the equation sin 2q = 2 – 2 cos 2q. (1)

(c)Using the result in part (a), or otherwise, find the other two solutions, 0 < q < 360°, of the equation sin 2q = 2 – 2 cos 2q. (4)

(Total 13 marks)

8.(a)Prove that

(4)

(b)Hence, or otherwise, prove

tan2 = 3 – 2Ö2.

(5)

(Total 9 marks)

9.(i)(a)Express (12 cos q – 5 sin q) in the form R cos (q + a), where R > 0 and 0 < a < 90°.

(4)

(b)Hence solve the equation

12 cos q – 5 sin q = 4,

for 0 < q < 90°, giving your answer to 1 decimal place.

(3)

(ii)Solve

8 cot q – 3 tan q = 2,

for 0 < q < 90°, giving your answer to 1 decimal place.

(5)

(Total 12 marks)

10.(i)Given that sin x = , use an appropriate double angle formula to find the exact value of sec 2x. (4)

(ii)Prove that

cot 2x + cosec 2x º cot x, (x ¹ , n Î ).

(4)

(Total 8 marks)

11.

This diagram shows an isosceles triangle ABC with AB = AC = 4 cm and Ð BAC = 2q .

The mid-points of AB and AC are D and E respectively. Rectangle DEFG is drawn, with F and G on BC. The perimeter of rectangle DEFG is P cm. (a)Show that DE = 4 sin q.

(2)

(b)Show that P = 8 sinq + 4 cosq.

(2)

(c)Express P in the...