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Trigonometry (sine & cosine rules)
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49 questions at your level
Difficulty
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Level 3
Level 4
Level 5
The sine rule
16 questions
Lesson
Not started
Mark as done
In triangle
A
B
C
ABC
A
B
C
,
b
=
9.1
b = 9.1
b
=
9.1
cm,
c
=
12.4
c = 12.4
c
=
12.4
cm and angle
B
=
39
°
B = 39°
B
=
39°
.
Angle
C
C
C
is
obtuse
.
Work out the size of angle
C
C
C
. Give your answer correct to 1 decimal place.
●●●●●
Level 5
4 marks
Start
→
Mark as done
Two people,
A
A
A
and
B
B
B
, stand on level ground on the same side of a vertical tower, in line with its base.
A
A
A
and
B
B
B
are
50
50
50
m apart, and
B
B
B
is nearer the tower.
From
A
A
A
the angle of elevation of the top of the tower,
T
T
T
, is
32
°
32°
32°
. From
B
B
B
the angle of elevation of
T
T
T
is
55
°
55°
55°
.
(a)
Work out the length
B
T
BT
B
T
. Give your answer correct to 3 significant figures.
(b)
Hence work out the height of the tower. Give your answer correct to 3 significant figures.
●●●●●
Level 5
4 marks
Start
→
Mark as done
A
B
C
D
ABCD
A
B
C
D
is a quadrilateral.
9.4 cm
112°
41°
63°
A
B
C
D
A
B
=
9.4
AB = 9.4
A
B
=
9.4
cm \qquad angle
A
B
C
=
112
°
ABC = 112°
A
B
C
=
112°
\qquad angle
C
A
D
=
41
°
CAD = 41°
C
A
D
=
41°
\qquad angle
A
C
D
=
63
°
ACD = 63°
A
C
D
=
63°
The area of triangle
A
B
C
ABC
A
B
C
is
38
38
38
cm²
Calculate the area of triangle
A
C
D
ACD
A
C
D
.
Give your answer correct to 3 significant figures.
(5 marks)
●●●●●
Level 5
5 marks
Start
→
Mark as done
A
B
C
D
ABCD
A
B
C
D
is a quadrilateral.
8 cm
9.5 cm
11 cm
72°
58°
A
B
C
D
A
B
=
8
AB = 8
A
B
=
8
cm \qquad
A
D
=
11
AD = 11
A
D
=
11
cm \qquad
C
D
=
9.5
CD = 9.5
C
D
=
9.5
cm
angle
B
A
D
=
72
°
BAD = 72°
B
A
D
=
72°
\qquad angle
B
C
D
=
58
°
BCD = 58°
B
C
D
=
58°
Find the size of angle
B
D
C
BDC
B
D
C
.
Give your answer correct to the nearest degree.
(5 marks)
●●●●●
Level 5
5 marks
Start
→
The cosine rule
22 questions
Lesson
Not started
Mark as done
A ship leaves a harbour
H
H
H
and sails
11
11
11
km on a bearing of
062
°
062°
062°
to a point
A
A
A
.
It then sails
8
8
8
km on a bearing of
145
°
145°
145°
to a point
B
B
B
.
11 km
8 km
H
A
B
N
N
Diagram NOT accurately drawn
(a)
Show that angle
H
A
B
=
97
°
HAB = 97°
H
A
B
=
97°
.
(b)
Work out the distance
H
B
HB
H
B
. Give your answer correct to 1 decimal place.
(c)
Work out the bearing of
B
B
B
from
H
H
H
. Give your answer correct to the nearest degree.
●●●●●
Level 5
8 marks
Start
→
Mark as done
In triangle
A
B
C
ABC
A
B
C
,
b
=
7.5
b = 7.5
b
=
7.5
cm,
c
=
10.2
c = 10.2
c
=
10.2
cm and angle
A
=
118
°
A = 118°
A
=
118°
.
10.2 cm
a
7.5 cm
118°
Diagram NOT accurately drawn
(a)
Work out the length of
a
a
a
. Give your answer correct to 1 decimal place.
(b)
Explain why
a
a
a
must be longer than
7.5
2
+
10.2
2
\sqrt{7.5^2 + 10.2^2}
7.
5
2
+
10.
2
2
.
●●●●●
Level 5
5 marks
Start
→
Mark as done
Two ships leave a port at the same time. Ship
A
A
A
sails
12
12
12
km on a bearing of
040
°
040°
040°
. Ship
B
B
B
sails
18
18
18
km on a bearing of
115
°
115°
115°
.
Work out the distance between the two ships. Give your answer correct to 3 significant figures.
●●●●●
Level 5
4 marks
Start
→
Mark as done
A
B
C
D
ABCD
A
B
C
D
is a quadrilateral.
9.4 cm
112°
41°
63°
A
B
C
D
A
B
=
9.4
AB = 9.4
A
B
=
9.4
cm \qquad angle
A
B
C
=
112
°
ABC = 112°
A
B
C
=
112°
\qquad angle
C
A
D
=
41
°
CAD = 41°
C
A
D
=
41°
\qquad angle
A
C
D
=
63
°
ACD = 63°
A
C
D
=
63°
The area of triangle
A
B
C
ABC
A
B
C
is
38
38
38
cm²
Calculate the area of triangle
A
C
D
ACD
A
C
D
.
Give your answer correct to 3 significant figures.
(5 marks)
●●●●●
Level 5
5 marks
Start
→
Mark as done
A
B
C
D
ABCD
A
B
C
D
is a quadrilateral.
8 cm
9.5 cm
11 cm
72°
58°
A
B
C
D
A
B
=
8
AB = 8
A
B
=
8
cm \qquad
A
D
=
11
AD = 11
A
D
=
11
cm \qquad
C
D
=
9.5
CD = 9.5
C
D
=
9.5
cm
angle
B
A
D
=
72
°
BAD = 72°
B
A
D
=
72°
\qquad angle
B
C
D
=
58
°
BCD = 58°
B
C
D
=
58°
Find the size of angle
B
D
C
BDC
B
D
C
.
Give your answer correct to the nearest degree.
(5 marks)
●●●●●
Level 5
5 marks
Start
→
Mark as done
The diagram shows a circle, centre
O
O
O
, radius
r
r
r
cm, and two regular octagons.
r cm
O
The sides of the larger octagon are tangents to the circle.
The vertices of the smaller octagon lie on the circle.
By considering perimeters, show that
4
2
−
2
<
π
<
8
(
2
−
1
)
4\sqrt{2 - \sqrt{2}} < \pi < 8(\sqrt{2} - 1)
4
2
−
2
<
π
<
8
(
2
−
1
)
(4 marks)
●●●●●
Level 5
4 marks
Start
→
Area of a triangle & 3D trigonometry
21 questions
Lesson
Not started
Mark as done
The diagram shows a circle of radius
10
10
10
cm. A chord divides the circle, and the angle at the centre is
80
°
80°
80°
.
O
10 cm
80°
Diagram NOT accurately drawn
Work out the area of the shaded segment. Give your answer correct to 1 decimal place.
●●●●●
Level 5
4 marks
Start
→
Mark as done
The diagram shows a cuboid measuring
12
12
12
cm by
5
5
5
cm by
9
9
9
cm.
12 cm
9 cm
5 cm
Diagram NOT accurately drawn
Work out the angle between the long diagonal of the cuboid and its base. Give your answer correct to 1 decimal place.
●●●●●
Level 5
3 marks
Start
→
Mark as done
The diagram shows a pyramid with a square base of side
10
10
10
cm. Each sloping edge has length
13
13
13
cm.
10 cm
13 cm
Diagram NOT accurately drawn
(a)
Work out the vertical height of the pyramid. Give your answer correct to 1 decimal place.
(b)
Work out the angle between a sloping edge and the base. Give your answer correct to 1 decimal place.
●●●●●
Level 5
5 marks
Start
→
Mark as done
A pyramid has a square base of side
12
12
12
cm and a vertical height of
8
8
8
cm. The apex is directly above the centre of the base.
Work out the angle between a sloping edge and the base. Give your answer correct to 1 decimal place.
●●●●●
Level 5
3 marks
Start
→
Mark as done
A pyramid has a square base of side
8
8
8
cm. The apex
V
V
V
is directly above the centre of the base, at a height of
6
6
6
cm.
(a)
Work out the length of a sloping edge of the pyramid. Give your answer correct to 3 significant figures. [3]
(b)
Work out the angle between a sloping edge and the base. Give your answer correct to 1 decimal place. [2]
●●●●●
Level 5
5 marks
Start
→
Mark as done
A
B
C
D
ABCD
A
B
C
D
is a quadrilateral.
9.4 cm
112°
41°
63°
A
B
C
D
A
B
=
9.4
AB = 9.4
A
B
=
9.4
cm \qquad angle
A
B
C
=
112
°
ABC = 112°
A
B
C
=
112°
\qquad angle
C
A
D
=
41
°
CAD = 41°
C
A
D
=
41°
\qquad angle
A
C
D
=
63
°
ACD = 63°
A
C
D
=
63°
The area of triangle
A
B
C
ABC
A
B
C
is
38
38
38
cm²
Calculate the area of triangle
A
C
D
ACD
A
C
D
.
Give your answer correct to 3 significant figures.
(5 marks)
●●●●●
Level 5
5 marks
Start
→
Mark as done
A
B
C
D
E
F
G
H
ABCDEFGH
A
B
C
D
E
F
G
H
is a cuboid.
A
B
C
D
E
F
G
H
A
C
=
7.5
AC = 7.5
A
C
=
7.5
cm \qquad
A
G
=
15
AG = 15
A
G
=
15
cm
Work out the size of the angle between
A
G
AG
A
G
and the plane
A
B
C
D
ABCD
A
B
C
D
.
(2 marks)
●●●●●
Level 5
2 marks
Start
→
Mark as done
A
B
C
D
E
F
G
H
ABCDEFGH
A
B
C
D
E
F
G
H
is a cuboid.
12 cm
34°
49°
A
B
C
D
E
F
G
H
A
B
=
12
AB = 12
A
B
=
12
cm \qquad angle
B
A
C
=
34
°
BAC = 34°
B
A
C
=
34°
\qquad angle
G
B
C
=
49
°
GBC = 49°
GB
C
=
49°
Work out the size of the angle between
A
G
AG
A
G
and the plane
A
B
C
D
ABCD
A
B
C
D
.
Give your answer correct to the nearest degree.
(4 marks)
●●●●●
Level 5
4 marks
Start
→