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Mr. Konstantinou IB Math Studies 12 1
Right‐Angled Trigonometry Review Sheet 28/11/2012 Content
Right-angled trigonometry.
Use of the ratios of sine, cosine and tangent.
Amplifications/exclusions
In examinations: problems incorporating Pythagoras’ theorem will be set.
Use of the inverse trigonometric functions on a GDC is expected, but a
detailed understanding of the functions themselves is not expected.
2
Content
The sine rule:
a
b
c
=
=
.
sin A sin B
sin C
The cosine rule a2 = b2 + c2 – 2bc cos A;
cos A =
b2 + c2 – a2
.
2bc
Area of a triangle:
1
ab sin C
2
Construction of labelled diagrams from verbal statements.
Amplifications/exclusions
Not required: radian measure.
In examinations: students will not be asked to derive the sine and cosine rules.
The ambiguous case could be taught, but will not be examined.
3
Content
Geometry of three-dimensional shapes: cuboid; prism; pyramid; cylinder;
sphere; hemisphere; cone.
Lengths of lines joining vertices with vertices, vertices with midpoints and
midpoints with midpoints; sizes of angles between two lines and between lines
and planes.
Amplifications/exclusions
Included: surface area and volume of these shapes.
Included: only right prisms and square-based right pyramids.
1.
The height of a vertical cliff is 450 m. The angle of elevation from a ship to the top
of the cliff is 23°. The ship is x metres from the bottom of the cliff.
(a)
Draw a diagram to show this information.
Diagram:
(b)
Calculate the value of x.
(Total 4 marks)
2.
The diagram shows a water tower standing on horizontal ground. The height of the
tower is 26.5 m.
A
xm
From a point A on the ground the angle of elevation to the top of the tower is 28°.
(a)
On the diagram, show and label the angle of elevation, 28°.
(b)
Calculate, correct to the nearest metre, the distance x m.
(Total 4 marks)
3.
The following diagram shows a carton in the shape of a cube 8 cm long on each side:
B
C
D
A
F
E
G
H
(a)
The longest rod that will fit on the bottom of the carton would go from E to G.
Find the length l of this rod.
(b)
Find the length L of the longest rod that would fit inside the carton.
(Total 4 marks)
4.
A rectangular block of wood with face ABCD leans against a vertical wall, as shown
in the diagram below. AB = 8 cm, BC = 5 cm and angle BÂE = 28°.
C
F
Wall
D
B
28º
E
Ground
Find the vertical height of C above the ground.
(Total 4 marks)
5.
ABCD is a trapezium with AB = CD and [BC] parallel to [AD]. AD = 22 cm,
BC = 12 cm, AB = 13 cm.
Diagram not to scale
B
A
(a)
C
E
D
Show that AE = 5 cm.
(2)
(b)
Calculate the height BE of the trapezium.
(2)
(c)
Calculate
(i)
BÂE;
(ii)
BĈ D.
(3)
(d)
Calculate the length of the diagonal [CA].
(3)
(Total 10 marks)
6.
The diagram shows a cuboid 22.5 cm by 40 cm by 30 cm.
H
G
E
F
40 cm
D
C
30 cm
A
22.5 cm
(a)
Calculate the length of [AC].
(b)
Calculate the size of GÂC .
B
(Total 4 marks)
7.
In the diagram below, PQRS is the square base of a solid right pyramid with vertex
V. The sides of the square are 8 cm, and the height VG is 12 cm. M is the midpoint
of [QR].
Diagram not to scale
V
VG = 12 cm
P
Q
8 cm
G
S
(a)
8 cm
M
R
(i)
Write down the length of [GM].
(ii)
Calculate the length of [VM].
(2)
(b)
Find
(i)
the total surface area of the pyramid;
(ii)
the angle between the face VQR and the base of the pyramid.
(4)
(Total 6 marks)
8.
Andrew is at point A in a park. A deer is 3 km directly north of Andrew, at point D.
Brian is 1.8 km due west of Andrew, at point B.
(a)
Draw a diagram to represent this information.
(b)
Calculate the distance between Brian and the deer.
(c)
Brian looks at Andrew, and then turns through an angle θ to look at the deer.
Calculate the value of θ.
(Total 8 marks)
9.
The following diagram shows the rectangular prism ABCDEFGH. The length is 5
cm, the width is 1 cm, and the height is 4 cm.
C
H
B
G
Diagram not to
D
E
A
F
scale
(a)
Find the length of [DF].
(b)
Find the length of [CF].
(Total 8 marks)
10.
The figure below shows a hexagon with sides all of length 4 cm and with centre at
O. The interior angles of the hexagon are all equal.
F
A
O
E
B
4 cm
D
C
The interior angles of a polygon with n equal sides and n equal angles (regular
polygon) add up to (n – 2) × 180°.
(a)
Calculate the size of angle A B̂ C.
(b)
Given that OB = OC, find the area of the triangle OBC.
(c)
Find the area of the whole hexagon.
(Total 8 marks)
In the diagram below ABEF, ABCD and CDFE are all rectangles. AD = 12 cm, DC
= 20 cm and DF = 5 cm.
M is the midpoint of EF and N is the midpoint of CD.
E
B
C
M
cm
N
F
20
11.
5 cm
A
(a)
Calculate
12 cm
(i)
D
the length of AF;
(ii) the length of AM.
(3)
(b)
Calculate the angle between AM and the face ABCD.
(3)
(Total 6 marks)
12.
The following diagram shows a sloping roof. The surface ABCD is a rectangle. The
angle ADE is 55°. The vertical height, AF, of the roof is 3 m and the length DC is 7
m.
B
C
A
7m
3m
55°
E
F
D
(a)
Calculate AD.
(b)
Calculate the length of the diagonal DB.
(Total 8 marks)
13.
OABCD is a square based pyramid of side 4 cm as shown in the diagram.
The vertex D is 3 cm directly above X, the centre of square OABC.
M is the midpoint of AB.
(a)
Find the length of XM.
(b)
Calculate the length of DM.
(c)
Calculate the angle between the face ABD and the base OABC.
Diagram
not to scale
D
B
C
X
O
M
A
(Total 8 marks)
A cross-country running course is given in the diagram below. Runners start and finish
at point O.
O
Not to
scale
150
0
m
C
500 m
14.
110°
B
(a)
800 m
A
Show that the distance CA is 943 m correct to 3 s.f.
(2)
(b)
Show that angle BCA is 58.0° correct to 3 s.f.
(2)
(c)
(i)
Calculate the angle CAO.
(ii)
Calculate the distance CO.
(5)
(d)
Calculate the area enclosed by the course OABC.
(4)
(e)
Gonzales runs at a speed of 4 m s–1. Calculate the time, in minutes, taken for
him to complete the course.
(3)
(Total 16 marks)
15.
The diagram shows a point P, 12.3 m from the base of a building of height h m. The
angle measured to the top of the building from point P is 63°.
hm
P
(a)
63°
12.3
Calculate the height h of the building.
Consider the formula h = 4.9t2, where h is the height of the building and t is the time
in seconds to fall to the ground from the top of the building.
(b)
Calculate how long it would take for a stone to fall from the top of the
building to the ground.
(Total 6 marks)
16.
A child’s toy is made by combining a hemisphere of radius 3 cm and a right circular
cone of slant height l as shown on the diagram below.
diagram not to scale
l
3 cm
(a)
Show that the volume of the hemisphere is 18π cm3.
(2)
The volume of the cone is two-thirds that of the hemisphere.
(b)
Show that the vertical height of the cone is 4 cm.
(4)
(c)
Calculate the slant height of the cone.
(2)
(d)
Calculate the angle between the slanting side of the cone and the flat surface
of the hemisphere.
(3)
(e)
The toy is made of wood of density 0.6 g per cm3. Calculate the weight of the
toy.
(3)
(f)
Calculate the total surface area of the toy.
(5)
(Total 19 marks)
17.
Find the volume of the following prism.
Diagram not to scale
8 cm
42°
5.7 cm
(Total 4 marks)
18.
D
A
4.5 cm
3 cm
B
25°
3 cm
C
In the diagram, AB = BC = 3 cm, DC = 4.5 cm, angle AB̂C = 90° and angle AĈD
= 25°.
(a)
Calculate the length of AC.
(b)
Calculate the area of triangle ACD.
(c)
Calculate the area of quadrilateral ABCD.
(Total 8 marks)
19.
Points P(0,–4), Q (0, 16) are shown on the diagram.
y
Q
8
0
2
4
6
8
10
12
14
16
18
x
P
(a)
Plot the point R (11,16).
(b)
Calculate angle QP̂R.
M is a point on the line PR. M is 9 units from P.
(c)
Calculate the area of triangle PQM.
(Total 6 marks)
20.
The figure below shows a rectangular prism with some side lengths and diagonal
lengths marked. AC = 10 cm, CH = 10 cm, EH = 8cm, AE 8 cm.
E
8 cm
H
8 cm
D
A
10 cm
F
10 cm
B
(a)
(not to scale)
G
C
Calculate the length of AH.
(2)
(b)
Find the size of angle AĈH.
(3)
(c)
Show that the total surface area of the rectangular prism is 320 cm2.
(3)
(d)
A triangular prism is enclosed within the planes ABCD, CGHD and ABGH.
Calculate the volume of this prism.
(3)
(Total 11 marks)
21.
The diagram below shows a child’s toy which is made up of a circular hoop, centre
O, radius 7 cm. The hoop is suspended in a horizontal plane by three equal strings
XA, XB, and XC. Each string is of length 25 cm. The points A, B and C are equally
spaced round the circumference of the hoop and X is vertically above the point O.
X
25 cm
B
C
7 cm
diagram not to scale
A
(a)
Calculate the length of XO.
(2)
(b)
Find the angle, in degrees, between any string and the horizontal plane.
(2)
(c)
Write down the size of angle AÔB.
(1)
(d)
Calculate the length of AB.
(3)
(e)
Find the angle between strings XA and XB.
(3)
(Total 11 marks)