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Box of Formulae
Page 1 of 12
C2 Chapter 10
1
Special angles
1. i. Show that each of the following can be expressed in the form
an integer.
a. sin 90
b. sin 60
c. sin 45
a
, where a is
2
d. sin 30
e. sin 0
(5)
2. The diagram shows a right-angled triangle ABC where    BAC and    ABC
B

c
A
a

b
C
i. Write down an expression for sin  and show that cos   sin  .
(2)
ii. Hence show that cos  90     sin  for all angles 0    90 . You should
separate consideration to any special cases.
(3)
iii. Use the triangle to show tan 90    tan   1 for all angles 0    90 .
Explain why this relationship is not valid in the cases when   0 and   90
(5)
Page 2 of 12
give
3. Find the value of each of the following. You must show your working.
i. sin30  4cos60
ii. 2sin 2 60  cos 60
iii. 5tan 60 cos30
v.
 sin 60  3cos 30 
2
iv.
 sin 45  cos 45 
vi.
3
 2 sin 45
tan 30
4. i. Given that a tan30  tan 45 , find the exact value of a.
2
(18)
(2)
ii. Verify that a 2 tan 30  tan 60
(3)
iii. Find the value of a 3 tan 30
(2)
Page 3 of 12
C2 Chapter 10
2
Solution of trigonometric equations
40
1. Solve each of the following equations for 0  x  360
i. sin x  0
ii. cos x  0
iii. 2sin x  1
iv. 3tan x  3
v. 4cos x  12
vi. 2 3 sin x  3
(12)
2. Solve each of the following equations for 180  x  180 . Give all answers correct
to one decimal place.
i. tan x  4
ii. 4sin x  3
iii. 5cos x 1  3
iv.
1
tan x  1  0
2
(8)
Page 4 of 12
3. It is given that cos   
4
where 90    180 .
5
i. Use the fact that sin 2   cos 2   1 to show that sin  
3
.
5
(4)
ii. Use an appropriate identity to find the value of tan  .
(2)
4. Solve each of the following equations for 180    180
i.
sin 2   0.25
ii. 3 tan 2   1
iii.
1
2
cos 2 
(12)
5. Solve the equation 2 cos 2   cos   1  0 for 0    360
(4)
Page 5 of 12
6. For each of the following equations, use an appropriate identity to find their solution
0    360 . Give answers correct to one decimal place where appropriate.
i. 2 cos 2   3sin   3  0
(6)
ii. 5sin 2   8cos   1  0
(6)
iii. 2sin   3cos
(3)
Page 6 of 12
iv. 2 tan  
1
0
2 cos 
(5)
v. 2 tan  
3
0
sin 
(8)
C2 Chapter 10
3
Triangles without right angles
1. For each of the following triangles (not drawn to scale), use the sine rule to find the
length of the side labelled x. Give each answer correct to 1 decimal place.
i.
x
12cm
25 
130 
(3)
ii.
78 
x
37 
(4)
19 mm
Page 7 of 12
2. i. Given that sin   sin 180    , write down the exact value of sin120 .
(1)
The diagram shows triangle ABC, together with a line pointing due North. Point C
lies due East of A.
B
N
not to scale
16 cm
120
A
5 cm
C
ii. Show that the area of triangle ABC is 20 3 cm2
(2)
iii. Use the cosine rule to
a. show that the length of AB = 19 cm,
(3)
b. find angle ABC , correct to the nearest whole degree.
iv. Find the bearing of;
a. B from C
(3)
(2)
b. A from B (to the nearest whole degree).
Page 8 of 12
(2)
C2 Chapter 10
4
Circular measure
1. Convert each of the following angles into degrees. Give answers correct to one
decimal place where appropriate.
c
c
2
1
5 c

i.
ii.
iii. 2.3c
iv.
(4)
3
2
8
2. Convert each of the following angles into radians. Express each answer in terms of 
i. 40
ii. 135
iii. 220
iv. 12.5
(4)
3. The diagram shows the sector of a circle with radius 8 cm and centre C. Points A and
B lie on the circle and subtend an angle of 0.8 radians at the centre.
A
Not to scale
8cm
C
0.8
B
i. Show that the length of the minor arc AB shown in the diagram is 6.4 cm.
(1)
ii. Find the perimeter of the sector. Give your answer correct to 1 decimal place.
(2)
ˆ giving your answer in terms of  .
iii. Calculate the size of angle BAC
(2)
Page 9 of 12
4. In the diagram, A and B are points on the circumference of a circle which has centre C
and radius 9 cm. Angle ACB   radians and sector ACB has area 56.7 cm2.
A
Not to scale
9cm
C

B
i. Show that   1.4 and hence show that triangle ACB occupies approximately
sector ACB.
(4)
c
ii. Calculate the length of the chord AB, correct to 1 decimal place.
Page 10 of 12
70% of
(3)
C2 Chapter 10
5
More trigonometrical graphs
1. The diagram shows the graph of y  sin x for 0  x  360
y
1
O
360 
x
-1
i. On the same pair of axes, sketch the graph of y  sin 3 x for 0  x  360 . (2)
ii. Hence, or otherwise, solve the equation
a. sin3x  0 for 0  x  360
b. sin 3x  1 for 0  x  360
(2)
(2)
2. i. Describe the transformation which maps the graph of y  cos x onto the graph of
y  cos  0.5x  .
(2)
ii. On the same pair of axes, sketch the graph of y  cos x and y  cos  0.5x  for
0  x  360 , labelling each graph clearly.
(2)
iii. Verify that the graphs y  cos x and y  cos  0.5x  intersect at the point  240,  0.5 .
Hence, with reference to your sketch, write down the solution to the
inequality
for 0  x  360 .
cos x  cos  0.5x 
(3)
Page 11 of 12
3. i. Use the identity tan  
sin 
to solve the equation 2sin   tan  for 0  x  180
cos 
(5)
ii. On the same pair of axes, sketch for 0  x  180 , the graphs of y  tan  and
y  2sin  . Clearly label the points where the two graphs intersect with their
co-ordinates.
(3)
4. i. Sketch the graph of y  sin 4 x for 0  x  180
(2)
ii. With reference to your sketch, state the values of k for which the solution to the
sin 4x  k for 0  x  180 consists of exactly two values of x.
equation
(2)
Page 12 of 12