Study Advice Service
This is one of a series of worksheets designed to help you increase your confidence in
handling Mathematics. This worksheet contains both theory and exercises which cover:

1. Pythagoras’ Theorem
2. Introduction to trigonometry
3. Using trigonometry to find an unknown side
4. Using trigonometry to find an unknown angle
5. Trigonom
etric diagrams and identities
There are often different ways of doing things in Mathematics and the methods suggested in
the worksheets may not be the ones you were taught. If you are successful and happy with
the methods you use it may not be necessary
for you to change them. If you have problems
or need help in any part of the work then there are a number of ways you can get help.
For students at the University of Hull
Ask your lecturers
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dy Advice Servi
ce
i
n the Brynmor
Jones Library
if you are in Hull;
you can also contact
us by email
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In session organised for your department
Look at one of the many textbooks in the library.
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Help Service
Use any other facilities that may be available.
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Mathematics
Worksheet
Trigonometry &
Pythagoras’ Theorem
1. Pythagoras’ Theorem
Pythagoras’ Theorem is used to find the lengths of unknown sides in triangles. It can be used in
the following conditions:
1. The triangle is a right

angle
d triangle, i.e. contains an angle of 90 degrees.
2. Two of the three sides are already known.
Two main uses of Pythagoras’ Theorem are converting a vector to magnitude and direction form
and finding the resultant force given a force in the horizontal and
vertical directions.
The rule is:
In words:
The square of the hypotenuse is equal to the sum of the
squares of the other two sides.
Note that this is sometimes expressed as
.
Here
and
are the two shorter sides of the triangle

the ones which are
attached to the right

angle.
or
is t
he hypotenuse, the longest side;
the
side tha
t lies opposite the right

angle.
E
xamples
1 Finding the hypotenuse
I
f
find
.
Substitute the known values into
P
ythagoras’
T
heorem:
Evaluate the Left Hand Side
(LHS)
.
Take the
positive
square root of both sides:
This can now be left as
or
a
calculator
may be used
to find
to 3 d.p.
2 Finding a shorte
r side
If
find
.
Substitute the known values into Pythagoras’ Theorem:
This becomes
.
Subtract 25 from both sides to
get
on
its own:
.
Take the positive square root of both sides:
.
So
.
Pythagoras’ Theorem is often used to find the length of vectors. The theorem can be extended to
3
dimensions by squaring all 3 components and adding, then square rooting. For more information
on this, please refer to Vectors 1, available from
www.hull.ac.uk/studyadvice
.
a
b
c
3
7
c
5
b
13
Exercise 1
1 For each of
the following triangles find the length of the hypotenuse:
a)
b)
c)
2 For each of the following triangles find the length of the unknown side:
a)
b)
c)
2. Introduction to trigonometry
Basic trigonometry uses the rules sine, cosin
e and tangent. These functions are actually infinite
series and would prove very difficult and time

consuming to calculate to a reasonable degree of
accuracy by hand. Fortunately scientific calculators are able to deal with these functions.
The most commo
n use of sine, cosine and tangent is with right

angled triangles. They are used to
find unknown sides and angles.
These functions are reliant on either knowing an angle and a side or the lengths of two sides.
The formulae for sine, cosine and tangent are:
Where
is used to denote the angle of interest, and
, for example, is the value of the sine
function acting o
n
.
These rules are often remembered by writing down
SOH
CAH
TOA
You may find it useful to include a ‘/’ between the second and third letters in each row to remind
you of the division.
Hypotenuse, adjacent and opposite refe
r to the lengths of the sides of the triangle. Note that whilst
the position of the hypotenuse is fixed (it is always the side opposite the right angle), the positions
of the opposite and adjacent sides are dependant on the location of the angle that is be
ing used.
The position of the angle of interest determines the labels on the sides.
c
6
8
7
12
c
c
4
9
13
b
7
20
c
b
6
12
4
a
Hypotenuse
Opposite
Adjacent
Angle
Hypotenuse
Opposite
Adjacent
Angle
3. Using trigonometry to find an unknown side
Trigonometry can be used to find an unknown side of a triangle when you know only one angle an
d
the length of one side.
Given a right

angled triangle such as:
How can
the length of side c
be found
?
A
n angle and a side
are known
. Look for a trig
onometric
formula which includes both the known
side and the unknown side.
includes
all of the necessary information
as the side opposite the angle is
known and the hypotenuse is the side that
is to be determined
.
Substituting the values in gives:
.
Rearrange:
(for h
elp with rearranging equations see Algebra 3)
All that remains is to substitute in the value of
(found via your calculator)
and work out the
value of the fraction…
.
Hence the length of side c is 10.
Another example:
Here we use cosine, as the side we need is
a
djacent to
the known angle, and the
hypotenuse
is
known.
So
, rearranging
,
Notes:
Remember to c
heck that
your
calculator is in degrees if using degrees or in radians if using
radians. This can normally be altered via the mode button.
5
c
1
2
b
Depending on
the make and age of the
calculator
being used
it may be necessary to type in
either
30
or
30
to get the value of
.
Always work with the numbers as they are shown on the calculator screen until the final result is
produced. Then this figure can be rounded. Rounding figures
part

way through the calculation will
result in a less accurate answer.
Exercise 2
1 Find the lengths of the missing sides in the following triangles:
a)
b)
c)
4. Using trigonometry to find an unknown angle
Trigonometry can
be used to find an unknown angle of a right

angled triangle when you know only
the length of two sides.
Given a right

angled triangle such as:
How do we find the size of angle
?
We can use the same formulae as w
e have been using for finding unknown sides.
In the above triangle t
he sides that are known are, in relation to
, the opposite side and the
hypotenuse.
So, use a formula which uses both the opposite side and the hypotenuse.
can be used here.
Substituting in the values of the known sides gives:
.
To get from
to a value for
you need the inverse sine operation.
This is on
most calculators as
and is usually accessed using a 2
nd
function or shift key, then
the sin key.
to 2 d.p. So
Note that I used
in the calculatio
n rather than 0.42. This is because 0.42 is rounded and so is
less accurate than
.
12
5
11
a
b
d
c
8
10
e
f
Another example:
Here the known sides are the
a
djacent and
the hypotenuse, so we use
cosine.
.
So,
to 2 d.p.
Notes:
Remember to check that your calculator is in degrees if you are using degrees or in radians if you
are using radians. This can normally be altered via the mode button.
Depending on your calculator you may need to ty
pe in either
2
nd
/shift
sin
30
or
30
2
nd
/shift
sin
to
get the value of
.
Exercise 3
Find the size of the marked angles
a)
b)
c)
5. Trigonometric diagrams and identities
Diagrams
There are 2 diagra
ms that can be memorised in order to recall certain values of sin, cos and tan
quickly.
The first is a right

angled isosceles triangle with two sides of length 1.
Because this is an Iso
sceles triangle, it has 2 an
gles the same.
The size of these
angles is
Hence this triangle will provide us with the values for
and
Using
Pythagoras’ Theorem, we find that the length of the hypotenuse is
equal to
.
Using the formulae for sin, cos and tan on either angle, we can now find that:
You may wish to confirm these answers yourself.
6
16
1
1
8
3
6
4
9
11
The second diagram is used to find sin, cos and tan for angles of
and
It is half of an equilateral triangle with sides of length 2.
This gives us angles of
and
as the angles of an equilateral triangle are all
. Here we
know the length of the hypotenuse, but one of the shorter sides is unknown. Using Pythagoras’
Theorem we get
, hence t
he length of the missing side is
.
Using the formulae for sin, cos and tan, we can now find that:
and
Again, you may wish to check these.
Notes:
If you have 2 angles that add up to
, then the sine of the first will equal the cosine of the
second, for example see
and
.
It is best to leave in the ‘surds’ or square

root signs. Results such as
are exact answers,
rounding them off will only make them less accurate.
Identities
There are a number of identitie
s in trigonometry. Identities are facts that will always be true, no
matter whether numbers (or in this case angles) are changed. These can be used to simplify long
algebraic arguments.
The one you are most likely to encounter is:
This can be seen using a diagram:
If you are given a right

angled triangle with
hypotenuse length 1, then the
adjacent side can be written as
by rearranging the formula for
cosine and the opposite side as
by rearranging the sine rule.
Since the hypotenuse in this case is 1, these simplify to
and
.
Using Pythagoras’ Theorem, the squares of these sides must sum to give
the square of th
e hypotenuse, hence
Other identities which may be of use are:
(Note the
sign)
2
2
2
1
2
1
For fu
rther identities, please refer to ‘a useful guide to facts and formulae’ produced by
Loughborough University, copies of which are available (free) from the Study Advice Desk.
Answers
Exercise 1
1. a) 10
b) 13.89 to 2 d.p.
c) 9.85 to 2 d.p.
2
. a) 11.31 to 2 d.p. b) 19.08 to 2 d.p. c) 10.95 to 2 d.p.
Exercise 2
1.
to 2 d.p.,
to 2 d.p.,
to 2 d.p.
to 2
d.p.,
to 2.d.p.
Exercise 3
a)
b)
c)
(all to 2 d.p.)
The information in this leaflet can be made
available in an
alterna
tive format on request
. T
elephone 01482 466199
©
2009
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