c2 trigonometry exam questions

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C2 Trigonometry Exam Questions 1. [Jan 05 Q4] (a) Show that the equation

5 cos2 x = 3(1 + sin x)

can be written as

5 sin2 x + 3 sin x – 2 = 0.

(2)

(b) Hence solve, for 0 x < 360, the equation

5 cos2 x = 3(1 + sin x),

giving your answers to 1 decimal place where appropriate. (5)

2. [June 05 Q5] Solve, for 0 x 180, the equation

(a) sin (x + 10) = 2

3, (4)

(b) cos 2x = –0.9, giving your answers to 1 decimal place. (4)

3. [Jan 06 Q8] (a) Find all the values of , to 1 decimal place, in the interval 0 < 360

for which

5 sin ( + 30) = 3.

(4)

(b) Find all the values of , to 1 decimal place, in the interval 0 < 360 for which

tan2 = 4.

(5)

4. [June 06 Q6] (a) Given that sin = 5 cos , find the value of tan . (1)

(b) Hence, or otherwise, find the values of in the interval 0 < 360 for which

sin = 5 cos ,

giving your answers to 1 decimal place. (3)

5. [Jan 07 Q6] Find all the solutions, in the interval 0 ≤ x < 2, of the equation

2 cos2 x + 1 = 5 sin x,

giving each solution in terms of . (6)

6. [June 07 Q9] (a) Sketch, for 0 ≤ x ≤ 2, the graph of y = sin

6

x . (2)

(b) Write down the exact coordinates of the points where the graph meets the coordinate axes.

(3)

(c) Solve, for 0 ≤ x ≤ 2, the equation


sin

6

x = 0.65,

giving your answers in radians to 2 decimal places. (5)

7. [Jan 08 Q4] (a) Show that the equation

3 sin2 – 2 cos2 = 1

can be written as

5 sin2 = 3.

(2)

(b) Hence solve, for 0 < 360, the equation

3 sin2 – 2 cos2 = 1,

giving your answer to 1 decimal place. (7)

8. [June 08 Q9] Solve, for 0 x < 360°,

(a) sin(x – 20) = 2

1

, (4)

(b) cos 3x = –2

1. (6)


4 sin2 x + 9 cos x – 6 = 0

can be written as

4 cos2 x – 9 cos x + 2 = 0.

(2)

(b) Hence solve, for 0 x < 720°,

4 sin2 x + 9 cos x – 6 = 0,


10. [June 09 Q7] (i) Solve, for –180° θ < 180°,

(1 + tan θ )(5 sin θ − 2) = 0.

(4)

(ii) Solve, for 0 x < 360°,

4 sin x = 3 tan x.

(6)


5 sin x = 1 + 2 cos2 x

can be written in the form

2 sin2 x + 5 sin x – 3 = 0.

(2)


(b) Solve, for 0 x < 360,

2 sin2 x + 5 sin x – 3 = 0.

(4)

12. [June 10 Q5] (a) Given that 5 sin θ = 2 cos θ, find the value of tan θ . (1)

(b) Solve, for 0 x<360°,

5 sin 2x = 2 cos 2x ,



3 sin2 x + 7 sin x = cos2 x − 4


4 sin2 x + 7 sin x + 3 = 0.

(2)

(b) Hence solve, for 0 x < 360°,

3 sin2 x + 7 sin x = cos2 x − 4

giving your answers to 1 decimal place where appropriate. (5)

14. [June 11 Q7] (a) Solve for 0 x < 360°, giving your answers in degrees to 1 decimal place,

3 sin (x + 45°) = 2.

(4)

(b) Find, for 0 x < 2, all the solutions of

2 sin2 x + 2 = 7cos x,

giving your answers in radians.

You must show clearly how you obtained your answers. (6)

15. [Jan 12 Q9] (i) Find the solutions of the equation sin (3x – 15) = 2

1, for which 0 x

180. (6)

(ii)

Figure 4 shows part of the curve with equation

y = sin (ax − b), where a > 0, 0 < b < .

The curve cuts the x-axis at the points P, Q and R as shown.


Given that the coordinates of P, Q and R are

0,

10

,

0,

5

3 and

0,

10

11 respectively, find

the values of a and b. (4)

16. [June 12 Q6] (a) Show that the equation

tan 2x = 5 sin 2x


(1 – 5 cos 2x) sin 2x = 0.

(2)

(b) Hence solve, for 0 x 180°,

tan 2x = 5 sin 2x,

giving your answers to 1 decimal place where appropriate.

You must show clearly how you obtained your answers.

(5)

17. [Jan 13 Q4] Solve, for 0 x < 180°,

cos (3x − 10°) = −0.4,

giving your answers to 1 decimal place. You should show each step in your working. (7)

18. [June 13 Q8] (i) Solve, for –180° x < 180°,

tan(x – 40) = 1.5,


(ii) (a) Show that the equation

sin tan = 3 co s + 2


4 cos2 + 2 cos – 1 = 0.

(3)

(b) Hence solve, for 0 < 360°,

sin tan = 3 cos + 2,

showing each stage of your working. (5)

19. [June 13(R) Q9] (i) Solve, for 0 ≤ θ < 180°

sin (2θ – 30°) + 1 = 0.4


(ii) Find all the values of x, in the interval 0 ≤ θ < 360° , for which

9 cos2 x – 11 cos x + 3 sin2 x = 0


You must show clearly how you obtained your answers.


Solutions

Question 1


Question 2

Question 3


Question 4

Question 5

Question 6


Question 7

Question 8

Question 9


Question 10

Question 11

Question 12


Question 13

Question 14

Question 15


Question 16

Question 17


Question 18

Question 19

c2 trigonometry exam questions

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