Trigonometry 1 – Core 2 Revision
1.
The diagram shows a triangle ABC.
B
4.8 cm
30°
A
5 cm
C
The lengths of AC and BC are 5 cm and 4.8 cm respectively.
The size of the angle BCA is 30°.
(a)
Calculate the area of the triangle ABC.
(2)
(b)
Calculate the length of AB, giving your answer to three significant figures.
(3)
(Total 5 marks)
2.
The diagram shows a triangle ABC.
B
12 cm
100°
A
4.8 cm
C
The lengths of AC and BC are 4.8 cm and 12 cm respectively.
The size of the angle BAC is 100°.
(a)
Show that angle ABC = 23.2°, correct to the nearest 0.1°.
(3)
(b)
Calculate the area of triangle ABC, giving your answer in cm2 to three significant figures.
(3)
(Total 5 marks)
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3.
The diagram shows a sector OAB of a circle with centre O and radius 20 cm. The angle between
the radii OA and OB is q radians.
A
20 cm
O
28 cm
20 cm
B
The length of the arc AB is 28 cm.
(a)
Show that q = 1.4.
(2)
(b)
Find the area of the sector OAB.
(2)
(c)
The point D lies on OA. The region bounded by the line BD, the line DA and the arc AB is
shaded.
A
D
15 cm
O
1.4
20 cm
B
The length of OD is 15 cm.
(i)
Find the area of the shaded region, giving your answer to three significant figures.
(3)
(ii)
Use the cosine rule to calculate the length of BD, giving your answer to three
significant figures.
(3)
(Total 10 marks)
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4.
The triangle ABC, shown in the diagram, is such that BC = 6 cm, AC = 5 cm and AB = 4 cm.
The angle BAC is q.
A
4 cm
5 cm
B
(a)
C
6 cm
1
Use the cosine rule to show that cos q = .
8
(3)
(b)
(c)
Hence use a trigonometrical identity to show that sin q =
3 7
.
8
(3)
Hence find the area of the triangle ABC.
(2)
(Total 8 marks)
5.
The triangle ABC, shown in the diagram, is such that AC = 8 cm, CB = 12 cm and angle ACB =
θ radians.
A
8 cm
C
12 cm
B
The area of triangle ABC = 20 cm2.
(a)
Show that θ = 0.430 correct to three significant figures.
(3)
(b)
Use the cosine rule to calculate the length of AB, giving your answer to two significant
figures.
(3)
(c)
The point D lies on CB such that AD is an arc of a circle centre C and radius 8 cm. The
region bounded by the arc AD and the straight lines DB and AB is shaded in the diagram.
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A
8 cm
8 cm
C
D
B
12 cm
Calculate, to two significant figures:
the length of the arc AD;
(i)
(2)
(ii)
the area of the shaded region.
(3)
(Total 11 marks)
6.
The diagram shows a sector of a circle of radius 5 cm and angle θ radians.
5 cm
5 cm
The area of the sector is 8.1 cm2.
(a)
Show that q = 0.648.
(2)
(b)
Find the perimeter of the sector.
(3)
(Total 5 marks)
7.
The diagram shows a sector OAB of a circle with centre O.
A
B
6 cm
6 cm
c
1.2
O
The radius of the circle is 6 cm and the angle AOB is 1.2 radians.
(a)
Find the area of the sector OAB.
(2)
(b)
Find the perimeter of the sector OAB.
(3)
(Total 5 marks)
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8.
The diagram shows a shaded segment of a circle of radius r cm. The segment is formed by
drawing an arc on one side of an equilateral triangle of side r cm with the centre at the opposite
vertex.
r
r
r
(a)
Show that the ratio of the length of the arc to the side of the triangle is p:3.
(2)
(b)
Show that the area of the triangle is
3 2 2
r cm .
4
(2)
(c)
Given that the area of the shaded segment is 10 cm2, find, to 3 significant figures, the
value of r.
(3)
(Total 7 marks)
9.
The diagram shows a triangle ABC and the arc AB of a circle whose centre is C and whose
radius is 24 cm.
A
24 cm
C
32 cm
24 cm
B
The length of the side AB of the triangle is 32 cm. The size of the angle ACB is q radians.
(a)
Show that q = 1.46 correct to three significant figures.
(3)
(b)
Calculate the length of the arc AB to the nearest cm.
(2)
(c)
(i)
Calculate the area of the sector ABC to the nearest cm2.
(2)
(ii)
Hence calculate the area of the shaded segment to the nearest cm2.
(3)
(Total 10 marks)
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10.
The diagram shows a sector OAB of a circle with centre O and radius r cm.
A
r cm
O
1.5
c
r cm
B
The angle AOB is 1.5 radians. The perimeter of the sector is 56 cm.
(a)
Show that r = 16.
(3)
(b)
Find the area of the sector.
(2)
(Total 5 marks)
11.
The area of a sector of a circle with radius 6 cm is 9 cm2.
Calculate
(a)
the size of the angle of the sector in radians;
(2)
(b)
the perimeter of the sector.
(2)
(Total 4 marks)
12.
The diagram shows a sector of a circle, centre O and radius 10 cm. The angle of the
sector is q radians and the arc length is 4 cm.
10 cm
O
4 cm
10 cm
(a)
Find the value of q.
(2)
(b)
Find the area of the sector.
(2)
(Total 4 marks)
13.
The area of a sector of a circle of radius 10 cm is 75 cm2. Find the arc length of this sector.
(Total 3 marks)
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14.
The diagram shows a triangle ABC with AB = 3 cm, AC = 4 cm and angle BAC = q radians.
A
3 cm
3 cm
B
4 cm
D
C
The point D lies on AC such that AD = 3 cm, and ABD is a sector of a circle with centre A and
radius 3 cm.
(a)
Write down, in terms of q:
(i)
the area of the sector ABD;
(2)
(ii)
the area of triangle ABC.
(2)
(b)
Show that, for small values of q, the area of the shaded region, bounded by the lines BC
and DC and the arc BD, is approximately 1.5q square centimetres.
(2)
(Total 6 marks)
15.
The diagrams show a square of side 6 cm and a sector of a circle of radius 6 cm and angle q
radians.
6 cm
6 cm
6 cm
q
6 cm
The area of the square is three times the area of the sector.
(a)
Show that q =
2
3
.
(2)
(b)
Show that the perimeter of the square is 1 12 times the perimeter of the sector.
(3)
(Total 5 marks)
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16.
A
B
25 cm
25 cm
O
The diagram shows the sector OAB of a circle with centre O. The radius of the circle is exactly
25 cm. The perimeter of the sector OAB is 83 cm correct to the nearest centimetre.
(a)
Write down the least possible perimeter.
(1)
(b)
Show that angle AOB cannot be less than 1.3 radians.
(2)
(c)
Hence calculate the smallest possible area of sector OAB.
(2)
(Total 5 marks)
A
4 cm
17.
2 cm
B
q°
3 cm
C
The diagram shows a triangle ABC with AB = 4cm, AC = 2cm, BC = 3cm and angle ABC = q°.
(a)
Calculate the exact value of cos q º.
(3)
(b)
Hence find the exact value of cot q º.
(2)
(Total 5 marks)
18.
The diagram shows a shape ABCDE. The shape consists of a square ABCD, with sides of length
5 cm, and a sector ADE of a circle with centre A and radius 5 cm. The angle of the sector is q
radians.
A
B
5 cm
5 cm
E
D
(a)
C
Find the area of the sector ADE in terms of q.
(2)
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(b)
The area of the sector ADE is a quarter of the area of the square ABCD.
(i)
Find the value of q.
(2)
(ii)
Find the perimeter of the shape ABCDE.
(2)
(Total 6 marks)
19.
The diagram shows a shape ABCD consisting of two sectors of circles joined along the side AC.
The centres of the circles are A and C. Each sector has radius 5 cm and angle 0.6 radians.
A
B
5 cm
0.6
0.6
D
5 cm
C
Calculate:
(a)
the area of the shape ABCD;
(2)
(b)
the perimeter of the shape ABCD.
(3)
(Total 5 marks)
20.
The diagram shows a sector of a circle of radius 8 cm. The sector has angle q radians. The
perimeter of the sector is P cm and its area is A cm2.
8 cm
q
8 cm
(a)
Show that P = 8(q + 2)
(2)
(b)
Find A in terms of q.
(2)
(c)
Given that A = P, find the value of q.
(3)
(Total 7 marks)
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21.
The following diagram shows a sector OAB of a circle with centre O and radius 3 cm.
The angle AOB is 1 radian.
B
3 cm
1c
O
(a)
3 cm
A
Find:
(i)
the perimeter of the sector OAB;
(2)
(ii)
the area of the sector OAB.
(2)
(b)
The sector OCD has radius r cm and angle q radians, as shown below.
D
r cm
c
O
(i)
r cm
C
The area of the sector OCD is equal to the area of the sector OAB. Show that
q = 92 .
r
(2)
(ii)
The perimeter of the sector OCD is twice the perimeter of the sector OAB. Show
that
2r + 9 = 18.
r
(2)
(iii) Deduce that
2r2 – 18r + 9 = 0.
(2)
(iv)
Given that r > 3, find the value of r and the corresponding value of q.
Give each answer to three significant figures.
(4)
(Total 14 marks)
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22.
(a)
The diagram shows the sector OPQ of a circle of centre O and radius x cm.
The angle POQ is 60°.
Q
P
x cm
x cm
60°
O
(i)
Write 60° in radians in terms of p.
(1)
(ii)
Find the length of the arc PQ in terms of x.
(1)
(b)
A thin metal component consists of a rectangle PQRS of height h cm and width x cm
and the shaded segment shown in the diagram above.
Q
P
h cm
R
x cm
S
The perimeter of the component is 100 cm. Show that
h = 50 –
(c)
x px
–
2 6
(2)
The surface area of the front face of the component is A cm2.
(i)
Write down the exact value of sin 60°.
(1)
(ii)
(iii)
Show that A = 50x –
Find
1 2
3 2
x –
x .
2
4
(5)
dA
and hence find the value of x for which A has a stationary value.
dx
(3)
(iv)
d2 A
and hence determine whether the stationary value is a maximum or a
dx 2
minimum.
Find
(2)
(Total 15 marks)
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B
A
r
23.
r
q
O
The diagram shows a circular path with centre O and radius r, together with two other paths
along the radii AO and OB. The size of the angle AOB is q radians, where q < p. The widths of
the paths may be neglected in the calculations. Peter runs along the radii AO and OB, then along
the minor arc BA. Mary runs along the major arc AB.
(a)
Given that Peter and Mary run the same distance, show that
q=p–1
(3)
(b)
Given that they each run 410 metres, find the radius of the circular path correct to the
nearest metre.
(3)
(Total 6 marks)
24.
The diagram shows a circle with centre P and radius 6 cm. The angle RPS is
p
radians.
3
R
6 cm
p
3
P
6 cm
S
(a)
Find, in terms of p, the length of the arc RS.
(2)
(b)
Find the area of:
(i)
the sector PRS;
(2)
(ii)
the triangle PRS;
(2)
(iii)
the shaded segment.
(1)
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(c)
The diagram below shows a logo consisting of two overlapping circles with centres
P and Q and both with radius 6 cm. The circles intersect at R and S, and the angle
RPS is p radians.
3
R
6cm
6cm
p
–
3
p
–
3
P
6cm
Q
6cm
S
Show that the area of the front face of the logo is approximately 220 cm2.
(3)
(Total 10 marks)
25.
(a)
C
A
a
B
The diagram shows an equilateral triangle ABC with sides of length 6 cm and an arc BC
of a circle with centre A.
(i)
Write down, in radians, the value of the angle a.
(1)
(ii)
Find the length of the arc BC.
(2)
(iii)
2
Show that the area of the triangle ABC is 9 3 cm .
(3)
(iv)
Show that the area of the sector ABC is 6p cm2.
(3)
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(b)
C
A
B
The diagram shows an ornament made from a flat sheet of metal. Its boundary consists
of three arcs of circles. The straight lines AB, AC and BC are each of length 6 cm.
The arcs BC, AC and AB have centres A, B and C respectively.
(i)
The boundary of the ornament is decorated with gilt edging. Find the total length of
the boundary, giving your answer to the nearest centimetre.
(2)
(ii)
Find the area of one side of the ornament, giving your answer to the nearest square
centimetre.
(3)
(Total 14 marks)
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