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How to Throw Away Marks in Maths GCSE
One tragedy that only people who have marked external examination papers such
as GCSE will have any real idea about is the number of marks that candidates just
throw away in examinations.
Given that each mark in a GCSE examination could be worth many thousands of
pounds over your working life, it is a crazy situation to be in.
Imagine that you had to give an examiner a fifty pound note every time you threw
away a mark in your GCSE exam. I’m sure you wouldn’t be very happy about that,
but in reality you could be losing a great deal more over the course of your working
life. Marks are very often lost by:
•
•
•
•
•
•
•
•
•
•
•
•
•
Not reading the questions properly.
Not answering the question asked.
Not showing your working.
Trying to do too much in your head.
Not knowing your tables and wasting time having to work them out.
Not knowing simple formulae such as the area of a triangle.
Not knowing the names of all the common 2-D and 3-D shapes.
Not knowing the difference between a factor and a multiple.
Not being able to round answers correctly.
Not knowing the meaning of common mathematical terms such as
‘perpendicular’, ‘obtuse’, ‘corresponding angles’ and ‘quadratic’.
Not knowing the properties of shapes such as parallelograms.
Not knowing the difference between 5x and x5.
Not being able to carry out transformations properly.
The list goes on and on……
Except for the very brightest and most thorough of students (in other words, the
vast majority of everyone else) even a casual read through by an examiner will
reveal a large number of ‘throw aways’ - marks that could easily have been gained
if only the candidate had taken more care in preparing for the exam.
Please, please, please, don’t be one of those candidates. Purchase the
answers to the modules for the nominal sum of just £20 and check all
your answers thoroughly, so you don’t make these common mistakes.
You’ve been studying maths for eleven years…
Make sure you get the grade you deserve!
Home Study
Modules KS4
Higher Level
Pythagoras and
Trigonometry
Please enjoy using these free questions. If you
would like fully worked answers to the questions
in all the GCSE modules (similar to the free
Graphs module answers) you may purchase an
immediate download for just £20 by following the
instructions on the home page of our website:
http://www.gcsemathematics4u.co.uk
GCSEMathematics4U
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Home Study Modules
Higher Level: PYTHAGORAS and TRIGONOMETRY
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C
Revision Topics
What you will be practising:
Concept
Examples
Pythagoras’ Theorem given a diagram.
Given two sides in a right angled triangle, find
the third side. Includes when you need to find
the hypotenuse and when you know the
hypotenuse and need to find one of the other
sides.
4m
z
3m
Pythagoras’ Theorem given no diagram and
you need to draw one first.
1. A ship left a port and travelled 152 Km due
east and then 193 Km due north.
How far was it from its starting point?
2. Two points A and B, are drawn on a
co-ordinate grid. A is (a, b) and B is (c, d).
What is the distance between A and B?
Pythagoras’ Theorem applied to 3-D
situations.
A
7.6 cm
B
8.5 cm
C
D
7.2 cm
F
E
G
Find AG
Trigonometry – finding the values of sines,
cosines and tangents and combinations of
these functions.
Trigonometry – finding an angle given the
value of sine, cosine or tangent of the angle.
Trigonometry – solving triangles.
H
Find: a) sin 141o, b) 9 tan 37o,
c) (sin 37.5o)2 + (cos 37.5o)2
sin x = 0.7382, find the value of x.
Find x
x
8.7 cm
510
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Trigonometry – more advanced problems.
θ is one angle in a right angled triangle.
Cos θ = 12/13 and the hypotenuse is 65 cm
long. What is the length of the other two sides?
Trigonometry - given no diagram and you need Michael is flying a kite on a string 26 metres
to draw one first.
long. The angle of elevation of the kite is 42o.
Michael’s hand is 1.4 metres above the
ground.
How high is the kite above the ground?
Bearings – Solving problems using
A ship sails from a port P on a bearing of 206o
trigonometry applied to bearings.
for a distance of 56 Km to an island Q and
then changes direction to a bearing of 143o for
a distance of 98 Km to another port R.
How far is the ship now east of its starting
point and how far is it south of its starting
point?
Trigonometry and Pythagoras’ Theorem to
A point P and two flagpoles are in a straight
more complex problems.
line.
P is positioned such that P and the tops of
the flagpoles are also in a straight line. The
flagpoles are 25 metres apart, the smaller is
18 metres high and the angle of elevation from
P to the tops of the flagpoles is 27.5o.
18 m
27.5
P
o
25 m
Find the height of the taller flagpole.
Problems involving mixed trigonometry and
Pythagoras’ Theorem in 3-D
The base ABCD of a right square based
pyramid is laid horizontally on a table. The
vertex P is directly over the centre of the base.
AB = 12 cm. AP = 22 cm.
Find:
a) The height of the pyramid.
b) The angle PDB.
c) The area of one of the sloping faces.
Finding sines, cosines and tangents and their
inverses over the range of 0o to 360o.
Give the angles, x, between 0o and 360o for
each of the following:
a) sin x = 0.5693
c) cos x = –0.428
Page 3
b) sin x = –0.4792
d) cos x = 0.9283
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The sine rule and cosine rule.
Find the length x in the following triangles:
78o
56.4 m
58o
x
12 cm
54o
x
Mixed problems involving sine rule, cosine rule
and Pythagoras’ Theorem in two and three
dimensions.
Finding the area of a triangle using ½ ab Sin C
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Normally we do not give you any clues in the question section, but there is one point that is so important
we want to include it here at the beginning of this section.
It is this: When solving right angled triangles (that is, when you are trying to find angles or sides given
some information about the triangle) you only have two sets of tools available – Pythagoras’ Theorem
and Trigonometry – there is nothing else.
If you have this type of question in the examination, which you are almost certain to have, don’t sit there
scratching your head. Decide whether to use Pythagoras or Trigonometry and get on with the question.
How do you decide? Easy! If the question does not involve angles – you are not given angles and you
are not asked to find angles given the three sides of a triangle – it must be Pythagoras’ Theorem.
If you are given angles to use or are asked to find an angle, you should use Trigonometry.
Simple as that!
Q
In some more complicated examples, you may be given angles
that you don’t need to use straight away (they are ready for a
later part of the question) as well as the lengths of sides, but by
the time you get to those you will, hopefully, have had plenty of
practice and will be able to pick your way through the
questions.
1. a) State Pythagoras’ Theorem.
b) Which is the longest side in a right angled triangle?
2. Calculate the distance ‘z’ in each of the following triangles. Round the answers to three
significant figures where appropriate.
a)
4m
b)
11.8 m
c)
z
z
6 cm
12.9 m
z
5 cm
3m
3.2 cm
d)
13 m
z
e)
f)
z
8.5 mm
8.7 cm
z
12 m
6.33 mm
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3. A ladder is 15 m long and leans against a wall. The foot of the ladder is 4 m from the bottom
of the wall. How far up the wall does the ladder reach?
If the ladder is moved so that its foot is only 3 m from the base, how much further up the wall
will the ladder reach?
4. What is the length of a diagonal of a rectangle 16.2 Km by 5.7 Km ?
5. A ship left a port and travelled 152 Km due east and then 193 Km due north. How far was it
from its starting point?
6. A flagpole is 22 m high and is constrained from bending in high winds by cables each 31 m
long fixed into the ground. Each cable is stretched tight.
How far is each cable fixed into the ground from the foot of the flagpole?
7. Two points A and B, are drawn on a co-ordinate grid. A is (–2, 5) and B is (7, –9). What is
the distance between A and B?
8. The top of a chimney on one house is 9.3 metres above the ground. The top of a chimney on
another house is 6.9 metres above the ground. The chimneys are 15.6 metres apart,
measured horizontally.
A bird flies directly from the top of the higher chimney pot to the lower. How far does it fly?
9. Two points A and B, are drawn on a co-ordinate grid. A is (a, b) and B is (c, d). What is
the distance between A and B?
10. Two points A and B, are drawn on a co-ordinate grid. A is (x, y) and B is (x + a2 – 1, y + 2a).
What is the distance between A and B?
11. Show that a triangle with sides 3 367, 3 456 and 4 825 units respectively is right angled.
12. Show that a triangle with sides of length 2pq, p2 – q2 and p2 + q2 will always be a right
angled triangle, provided p and q are positive numbers and p>q.
13. What is the height of the vertical line of symmetry of this isosceles triangle?
11 cm
11 cm
9 cm
14. What is the area of each of these isosceles triangles?
a)
b)
12 cm
12 cm
8.4 m
c)
6.7 mm
4.3 mm
8.4 m
10.2 m
15 cm
Page 6
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15. A ship sails from a port P to an island Q, 100 Km south, then on to another island R,
142 Km east of Q. From R it travels south again 110 Km to another island S and
finally east a distance of 205 Km to its destination T.
a) How far is it from P to R as the crow flies?
b) How far is it from R to T as the crow flies?
c) How far is it from P to T as the crow flies?
16. Calculate the length of the diagonal from A to G in this cuboid.
A
B
7.6 cm
8.5 cm
C
D
7.2 cm
F
E
G
H
A
17.
12 cm
C
B
The diagram shows a square based
right pyramid. The length of the sides
of the base is 8 cm and the length of
the sloping edges is 12 cm.
O
M
D
8 cm
Calculate:
E
a) the distance OB
b) the vertical height of the pyramid OA
c) the distance AM, where M is the midpoint of BE
d) the total surface area of the pyramid
e) the volume of the pyramid.
18.
X
Y
Z
A line is drawn ‘diagonally’ on a can
from X to Z via Y, as shown in the
diagram. (Z is directly below X.)
The can has a diameter of 13 cm and
a height of 15 cm.
What is the length of the line XYZ ?
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19. Find the value of the following, correcting to three significant figures where appropriate:
a) sin 57o
b) sin 0o
c) sin 90o
d) sin 69o
e) sin 137o
f) cos 26o
g) cos 90o
h) cos 0o
i) cos 48o
j) cos 141o
k) tan 35o
l) tan 45o
m) tan 12o
n) tan 30o
o) tan 127o
20. Find the value of the following, correcting to three significant figures where appropriate:
a) 16.8 sin 12o
b) 9 tan 37o
c)
f) 7 cos 88o
g)
h) cos 56.4o
sin 23.5o
45
tan 36.8o
56
cos 18o
d)
e) 45.6 sin 105o
19.4
sin 69o
i) 19.5 cos 72o
j)
89.56
cos 141o
21. Find the value of sin 63o divided by cos 63o . Find the value of tan 63o.
22. Find the value of (sin 37.5o)2 + (cos 37.5o)2 .
23. Find the value of (sin 16.83o)2 + (cos 16.83o)2 .
24. Find the value of (sin 256o)2 + (cos 256o)2 .
25. Find the value of x in each of the following, giving the answers correct to one decimal place:
a) sin x = 0.7382
b) cos x = 0.5915
c) tan x = 0.3347 d)
sin x
/cos x = 0.2142
26. For each of the following triangles, find the value of sin x, cos x and tan x.
Leave your answers in fraction form.
a)
3 cm
5 cm
x
b)
26 cm
40 cm
x 10 cm
24 cm
c)
41 cm
9 cm
4 cm
sin x =
cos x =
tan x =
sin x =
cos x =
tan x =
Page 8
x
sin x =
cos x =
tan x =
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27. In each of the following triangles, find the length of the side x.
a)
b)
5 cm
x
c)
x
5 cm
430
340
510
4 cm
x
28. In each of the following triangles, find the length of the side x.
a)
b)
x
c)
x
8.7 cm
38.6
510
0
12.9 cm
7 cm
67.30
x
29. What are the values of the angles in a 3-4-5 triangle?
30. What are the values of the angles in a 5-12-13 triangle?
31. θ is one angle in a right angled triangle. If cos θ is 0.7352, what is the value of θ ?
32. θ is one angle in a right angled triangle. Sin θ = 9/41 and the opposite side to θ is 27 m
long. What is the length of the other two sides?
33. θ is one angle in a right angled triangle. Cos θ = 12/13 and the hypotenuse is 65 cm
long. What is the length of the other two sides?
34. θ is one angle in a right angled triangle. Tan θ = 48/55 and the hypotenuse is 438 cm
long. What is the length of the side adjacent to θ ?
35. Using the triangle below, give an algebraic proof that the sine of an angle divided by its
cosine is equal to the tangent of the angle.
r
p
θ
q
36. Using the same triangle as in question 35, show that the square of the sine of an angle
plus the square of the cosine of the same angle is 1.
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C
37.
b
a
B
m
n
T
A
c
Here is a neat proof of Pythagoras’ Theorem that shows the close relationship between this
theorem and trigonometry. Answer the parts of the question carefully to complete the proof.
a) In triangle ACT, cos A =
Use the small letters and write the answer as a fraction.
b) In triangle ABC, cos A =
Use the small letters and write the answer as a fraction.
c) Use your answers to parts a) and b) to complete the equation b2 =
.
d) In triangle BCT, cos B =
Use the small letters and write the answer as a fraction.
e) In triangle ABC, cos B =
Use the small letters and write the answer as a fraction.
f) Use your answers to parts d) and e) to complete the equation a2 =
g) Use your answers to parts c) and f) to complete the equation a2 + b2 =
.
.
h) Hence complete the proof of Pythagoras’ Theorem.
38. A tower is 43m high. Julia stands a certain distance from the tower and measures the angle
of elevation from the ground to the top of the tower to be 18o. How far is she from the
tower?
39. Michael is flying a kite on a string 26 metres long. The angle of elevation of the kite is 42o.
Michael’s hand is 1.4 metres above the ground. How high is the kite above the ground?
40. A rectangle ABCD has dimensions AB = 8 cm and BC = 14 cm. What is the angle ACB ?
41. A cliff is 156 metres high. A boat is 720 metres from the base of the cliff. What is the angle
of depression of the boat from the top of the cliff?
42. If Larry stands 65 m from a building and the angle of elevation to the top of the building
from the ground is 47o, how high is the building?
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43. A section of straight road 1.5 Km long rises 75 metres over its length. What is the average
angle that the road makes with the horizontal?
44. Which bearings are represented by the following compass directions:
a) East
b) West
c) South
d) North
e) North East
f) South West
45. A ship travels 135 Km on a bearing of 135o. How far is it east of its starting point?
46. An aircraft flies 340 Km on a bearing of 285o. How far is it west of its starting point and how
far is it north of its starting point?
47. A mountain P is 45 Km east and 34 Km north of another mountain Q. What is the bearing of
mountain Q from mountain P ? What is the back bearing of mountain P from mountain Q ?
48. The angle of elevation to the top of a mountain from the base camp of a mountaineering
expedition is measured very accurately to be 3.582o. The horizontal distance from the base
camp to the mountain is 56.43 Km. How high is the mountain above the base camp?
49. A ship sails from a port P on a bearing of 206o for a distance of 56 Km to an island Q and
then changes direction to a bearing of 143o for a distance of 98 Km to another port R. How
far is the ship now east of its starting point and how far is it south of its starting point?
Give each distance correct to the nearest 100 metres.
50. An aircraft flies 890 Km from an airport on a bearing of 340o and then changes to a bearing
of 203o and flies another 760 Km.
a)
b)
c)
d)
How far is the plane now west of the airport?
Is the plane now north or south of the airport and by how much?
What bearing must the plane now fly in order to return directly to the airport?
How far is the return flight assuming a direct return?
51. Michelle measures the angle of elevation to the top of a tower from the ground to be 63o.
Peter stands 22 metres behind her and measures the angle of elevation to the top of the
tower to be 35o.
a) How far is Michelle from the tower?
b) How tall is the tower?
52. A point P and two flagpoles are in a straight line. P is positioned such that P and the tops of
the flagpoles are also in a straight line. The flagpoles are 25 metres apart, the smaller is
18 metres high and the angle of elevation from P to the tops of the flagpoles is 27.5o.
18 m
Find the height of the taller flagpole.
P
27.5o
25 m
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53. Find the distance d in the diagram.
42o
23 m
39o
d
F
54.
Q
19o
R
55 m
P
98 m
S
The diagram represents a rectangular playing field PQRS. FR is a floodlight gantry in one
corner of the field.
PS = 98 metres. RS = 55 metres. The angle of elevation of the floodlight from the point Q
is 19o.
Find:
a)
b)
c)
d)
e)
The height of the floodlight gantry.
The length of the diagonal PR.
The angle of elevation of the floodlight F from point P.
The distance FS.
The angle of elevation of the floodlight from the point S.
55. The base ABCD of a right square based pyramid is laid horizontally on a table. The
vertex P is directly over the centre of the base.
AB = 12 cm. AP = 22 cm.
Find:
a) The height of the pyramid.
b) The angle PDB.
c) The area of one of the sloping faces.
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A
56.
O
B
C
In the diagram, the radius of the circle is 9.83 metres, the angle ABC is 38.5o.
Find:
a) The length of OB.
b) The area of OABC.
57. Give the angles, x, between 0o and 360o for each of the following:
a) sin x = 0.5693
b) sin x = –0.4792
c) cos x = –0.428 d) cos x = 0.9283
e) tan x = 6.723
f) tan x = –0.2735
g) sin x = 2.452
h) 1/cos x = –6.32
58. State the sine rule for a triangle ABC.
59. Find the length x in the following triangles:
a)
b)
x
78o
56.4 m
48.2o
58o
19 cm
38.6o
x
c)
d)
22o
89.4 mm
27 m
126o
146o
32.9o
x
x
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60. A triangle has vertices A, B and C.
a) Complete this form of the cosine rule:
a2 =
b) Complete this form of the cosine rule:
Cos A =
61. Find the length x in the following triangles:
a)
b)
12 cm
x
54o
14 cm
10.9 m
9.3 m
x
c)
29.5o
d)
25.8o
183 mm
157
88 cm
198 mm
o
38 cm
x
x
P
62. Find the angle marked P in the following triangles:
a)
P
18 m
b)
Q
9.365 Km
5.942 Km
12 m
19 m
Q
6.883 Km
R
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63. What happens when you try to use the cosine rule to find angle Q in this triangle?
P
4.2 cm
3.8 cm
Q
8.1 cm
R
Explain your answer.
64. Explain how, given a triangle with no right angles, you would know whether to use the sine
rule or the cosine rule, then try the questions below.
65. Peter is standing some way from the base of a small hill. On top of the hill is a vertical
tower.
Peter measures the angle of elevation to the top of the tower to be 29o and the angle to the
bottom of the tower to be 18o. The direct distance of Peter from the base of the tower is
134 metres.
Draw a diagram, labelling Peter’s position P, the base of the tower B and the top of the
tower T.
a) Find the angle TPB.
b) Find the angle PTB.
c) Find the height of the tower.
66. A plane flies on a bearing of 067o for 75 Km and then on a bearing 112o for a further 99 Km.
Find the distance, as the crow flies, of the plane from its starting position.
67. A trapezium PQRS has <QPS = 107o, PQ = 84.5 m, PS = 76.4 m and SR = 115.6 m.
a)
b)
c)
d)
e)
Find the length of the diagonal QS.
Find angle QSP.
Find angle QSR.
Find the length of QR.
Find the area of the trapezium.
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68. A surveyor is measuring a triangular piece of land TUV. TU is 56 m and TV is 65 m.
The angle UTV is 69o.
Calculate:
a) The perimeter of the triangle TUV.
b) The area of the triangle.
69. A triangle ABC has AB = 10 cm, BC = 6 cm and angle BAC = 31o.
Show that there are two possible values for angle C.
Demonstrate this difference by sketching and labelling the two possible triangles.
70. STU is a right angled triangle with the right angle at T.
P is a point somewhere on TU. Angle TPS = 37o, angle TUS = 21o and PU = 243 metres.
Find the length of ST.
71.
The diagram shows a structure made from
steel struts that is to be used as part of a
new bridge construction. It is assembled
from two rectangular frames, PSUR and
RUTQ joined at right angles along the line
RU. Additional struts PQ and ST are welded
into position.
S
20.74 m
P
7.84 m
U
The dimensions are shown on the diagram.
R
T
15.63 m
Q
Calculate the following:
a) The length of PQ to 2 d.p.
d) <PQR
b) The distance PT to 2 d.p.
e) <UQS
c) The length QU to 2 d.p.
72. Three points X, Y and Z lie on a circle. The angle YXZ is 42o, the angle XYZ is 57o and the
distance YZ is 300 mm.
a) Calculate the length of XY.
b) Hence, or otherwise, find the area of triangle XYZ.
c) Find the radius of the circle.
73. A triangle XYZ has XY = 54 cm, YZ = 71 cm and <XYZ = 4α
a) Find the area of the triangle in terms of sin 4α .
b) What value of α gives the maximum area of the triangle?
c) What is the maximum area of the triangle?
d) There are two possible values of α that give an area of 1200 cm2. What are these two
possible values?
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