Advanced Trigonometry Sample Test Questions
1.
A triangle has sides of length 4, 5, 7 units. Find, to the nearest tenth of a degree, the size of the
largest angle.
Working:
Answer:
......................................................................
(Total 4 marks)
2.
The diagram below shows a quadrilateral ABCD. AB = 4, AD = 8, CD =12, B Ĉ D = 25,
BÂD =.
(a)
Use the cosine rule to show that BD = 4 5  4 cos  .
(2)
Let  = 40.
(b)
(i)
Find the value of sin CB̂D .
(ii)
Find the two possible values for the size of CB̂D .
(iii)
Given that CB̂D is an acute angle, find the perimeter of ABCD.
(12)
(c)
Find the area of triangle ABD.
(2)
(Total 16 marks)
1
3.
The following diagram shows a sector of a circle of radius r cm, and angle  at the centre. The
perimeter of the sector is 20 cm.
20  2r
.
r
(a)
Show that  =
(b)
The area of the sector is 25 cm2. Find the value of r.
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
(Total 6 marks)
2
4.
The following diagram shows a semicircle centre O, diameter [AB], with radius 2.
Let P be a point on the circumference, with PÔB =  radians.
(a)
Find the area of the triangle OPB, in terms of .
(2)
(b)
Explain why the area of triangle OPA is the same as the area triangle OPB.
(3)
Let S be the total area of the two segments shaded in the diagram below.
(c)
Show that S = 2( − 2 sin  ).
(3)
(d)
Find the value of  when S is a local minimum, justifying that it is a minimum.
(8)
(e)
Find a value of  for which S has its greatest value.
(2)
(Total 18 marks)
3
5.
The following diagram shows a pentagon ABCDE, with AB = 9.2 cm, BC = 3.2 cm, BD = 7.1
cm, AÊD =110, AD̂E = 52 and AB̂D = 60.
(a)
Find AD.
(4)
(b)
Find DE.
(4)
(c)
The area of triangle BCD is 5.68 cm2. Find DB̂C .
(4)
(d)
Find AC.
(4)
(e)
Find the area of quadrilateral ABCD.
(5)
(Total 21 marks)
4
6.
The diagram below shows two circles which have the same centre O and radii 16 cm and 10 cm
respectively. The two arcs AB and CD have the same sector angle  = 1.5 radians.
A
B
D
C
O
Find the area of the shaded region.
Working:
Answer:
…………………………………………..
(Total 6 marks)
7.
In triangle PQR, PQ is 10 cm, QR is 8 cm and angle PQR is acute. The area of the triangle is 20
cm2. Find the size of angle PQ̂R.
Working:
Answers:
........................................................
(Total 6 marks)
5
8.
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 OP̂Q =
(iii)
Hence show that sin OP̂Q =
(iv)
Find the area of the triangle OPQ.
1
.
9
80
.
9
(7)
(b)
Consider the smaller semi-circle, with centre P.
(i)
Write down the size of OP̂Q.
(ii)
Calculate the area of the sector OPQ.
(3)
(c)
Consider the larger semi-circle, with centre O. Calculate the area of the sector QOS.
(3)
(d)
Hence calculate the area of the shaded region.
(4)
(Total 17 marks)
6
9.
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.
C
104 m
A2
A
30°
D
x
30°
A1
65 m
B
(c)
65x
.
4
(i)
Show that the area of Al is given by
(ii)
Find a similar expression for the area of A2.
(iii)
Hence, find the value of x in the form q 3 , where q is an integer.
(7)
(d)
(i)
Explain why sin AD̂C  sin AD̂B .
(ii)
Use the result of part (i) and the sine rule to show that
BD 5
 .
DC 8
(5)
(Total 18 marks)
7
10.
The diagram below shows a circle, centre O, with a radius 12 cm. The chord AB subtends at an
angle of 75° at the centre. The tangents to the circle at A and at B meet at P.
A
12 cm
P diagram not to
scale
O 75º
B
(a)
Using the cosine rule, show that the length of AB is 12 21 – cos 75 .
(2)
(b)
Find the length of BP.
(3)
(c)
Hence find
(i)
the area of triangle OBP;
(ii)
the area of triangle ABP.
(4)
(d)
Find the area of sector OAB.
(2)
(e)
Find the area of the shaded region.
(2)
(Total 13 marks)
8
11.
2
, BC = 6, AB̂C = 45°.
2
The diagram shows a triangle ABC in which AC = 7
A
2
7 2
Diagram
not to scale
B
(a)
Use the fact that sin 45° =
45°
C
6
6
2
to show that sin BÂC = .
2
7
(2)
The point D is on (AB), between A and B, such that sin BD̂C =
(b)
(i)
Write down the value of BD̂C + BÂC .
(ii)
Calculate the angle BCD.
(iii)
Find the length of [BD].
6
.
7
(6)
(c)
Show that
Area of ΔBDC
BD
=
.
Area of ΔBAC BA
(2)
(Total 10 marks)
9
12.
In triangle ABC, AC = 5, BC = 7, Â = 48°, as shown in the diagram.
C
5
A
48°
7
diagram not to scale
B
Find B̂, giving your answer correct to the nearest degree.
Working:
Answer:
......................................................................
(Total 6 marks)
13.
Two boats A and B start moving from the same point P. Boat A moves in a straight line at
20 km h–1 and boat B moves in a straight line at 32 km h–1. The angle between their paths is
70°.
Find the distance between the boats after 2.5 hours.
Working:
Answer:
......................................................................
(Total 6 marks)
10
14.
The diagrams below show two triangles both satisfying the conditions
AB = 20 cm, AC = 17 cm, AB̂C = 50°.
Diagrams not
to scale
Triangle 1
Triangle 2
A
A
B
C
(a)
Calculate the size of AĈB in Triangle 2.
(b)
Calculate the area of Triangle 1.
B
C
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 4 marks)
11
15.
O is the centre of the circle which has a radius of 5.4 cm.
O
A
B
The area of the shaded sector OAB is 21.6 cm2. Find the length of the minor arc AB.
Working:
Answer:
......................................................................
(Total 4 marks)
12
16.
Town A is 48 km from town B and 32 km from town C as shown in the diagram.
C
32km
A
48km
B
Given that town B is 56 km from town C, find the size of angle CÂB to the nearest degree.
Working:
Answer:
....................................................................
(Total 4 marks)
13
17.
The following diagram shows a triangle with sides 5 cm, 7 cm, 8 cm.
5
7
Diagram not to scale
8
Find
(a)
the size of the smallest angle, in degrees;
(b)
the area of the triangle.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 4 marks)
14