Download Sides and angles

Coordinate geometry and
trigonometry
Introduction
3
Angles
4
Classifying angles
4
Relationship between angles
5
Parallel lines
8
Alternate angles
8
Corresponding angles
9
Co-interior angles
9
Triangles
Types of triangles
12
12
Similar triangles
14
Quadrilaterals
17
Special quadrilaterals
Pythagoras’ theorem
Using Pythagoras’s theorem
Sides and angles
Adjacent and opposite sides
The trigonometric ratios—sin, cos and tan
20
20
24
24
30
Labelling angles
35
Similar triangles—a reminder
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1
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Introduction
In this module we will mainly be learning about trigonometry and
coordinate geometry.
However, both of these topics will be referring back to various aspects of
geometry and Pythagoras’ theorem.
You have already done some work in both of these topics so we will just be
going over those parts, which are relevant to this module.
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Angles
Classifying angles

An angle that is less than 90º is called an acute angle.

A 90º angle is a right angle.

An angle between 90º and 180º is called an obtuse angle.

A 180º angle is a straight angle.

An angle greater than 180º is called a reflex angle.

A 360º angle is a revolution.
Naming an angle
When naming an angle make sure the middle letter is the vertex.
This angle may be named as PQR or RQP.
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Activity 1
Classify and name each angle.
Answers to Activity 1
1
CKQ or QKC is obtuse.
2
PTZ or ZTP is a straight angle.
3
GHL or LHG is an acute angle.
4
ACA is a revolution. (Note: Usually you are not asked to name
revolutions.)
5
TNF or FNT is a right angle.
6
WSK or KSW is a reflex angle. (To distinguish a reflex angle from
the smaller angle on the same diagram we usually write ref WSK.)
Relationship between angles
Complementary angles
These are two angles that add to 90º.
Supplementary angles
These are two angles that add to 180º.
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Angles at a point
The sum of all the angles at a point is one revolution or 360º.
xº + yº + zº = 360º
Vertically opposite angles
When two lines intersect, two pairs of equal vertically opposite angles are
formed.
Activity 2
1
What is the complement of 36º?
2
What is the supplement of 58º?
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3
Find the value of the pronumeral.
Answers to Activity 2
1
54º
2
122º
3
(a)
62º
(g)
54º
(b)
127º
(h)
52º
(c)
324º
(i)
317º
(d)
25º
(j)
235º
(e)
21º
(k)
328º
(f)
96º
(l)
37º
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Parallel lines
Alternate angles
The angles marked with a bar are called alternate angles.
Alternate angles formed on parallel lines are equal.
There are two pairs of alternate angles.
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Corresponding angles
The angles marked with a bar are called corresponding angles.
Corresponding angles formed on parallel lines are equal.
There are four pairs of corresponding angles.
Co-interior angles
The angles marked a and b are called co-interior angles.
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Co-interior angles on parallel lines add to 180º. They are supplementary.
There are two pairs of co-interior angles.
Activity 3
1
10
Name the types of angles shown in each diagram.
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Find the value of the pronumeral.
Answers to Activity 3
1
2
(a)
corresponding angles
(f)
alternate angles
(b)
alternate angles
(g) corresponding angles
(c)
co-interior angles
(h) co-interior angles
(d)
corresponding angles
(i)
(e)
corresponding angles
alternate angles
(a) 106º
(b) 74º
(c)
(d)
(e)
(f)
106º
112º
20º
180 – 50 = 130º
360 – 130 = 230º
x = 230º
(g) 320º
(h) 30º
(i) 140º
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Triangles
Angles in a triangle add to 180º.
Types of triangles
Scalene triangle

All sides different length

All angles different size
Isosceles triangle

Two sides equal

Two angles equal
Equilateral triangle
12

All sides equal

All angles equal 60º
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Right-angled triangle

Contains one right angle
Activity 4
Find the size of the missing angle in 1–4 and then name the type of triangle. Name the type
of triangle in 5–7.
Answers to Activity 4
1
50º; isosceles triangle
2
90º; right-angled triangle
3
60º; equilateral triangle
4
65º; scalene triangle
5
Isosceles triangle
6
Equilateral triangle
7
Scalene triangle
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Similar triangles
Similar triangles are identical in shape but different in size.
In similar triangles:

the corresponding angles are equal

the corresponding sides are in the same ratio.
Note: When we are referring to similar triangles, ‘corresponding’ just means
matching.
Example
Find x.
The triangles are similar
because all angles of one
equal the corresponding
angles of the other.
The fractions formed by matching sides are equal. (Matching sides are
opposite equal angles.)
Solve this equation by multiplying both sides by 21.
Use your calculator.
x=7
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Activity 5
Find the value of the pronumeral.
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Answers to Activity 5
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Quadrilaterals
A quadrilateral is a four-sided shape. The angles in a quadrilateral add
to 360º.
Special quadrilaterals
Trapezium

One pair of parallel sides
Kite

Two pairs of adjacent sides equal

One pair of opposite angles equal
Parallelogram

Two pairs of parallel sides

Two pairs of equal sides

Opposite angle equal

Diagonals bisect each other
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Rhombus

Has the same properties as a parallelogram
Extra properties:

All sides equal

Diagonals perpendicular to each other
Rectangle

Same properties as a parallelogram
Extra properties:

All angles 90º

Diagonals equal to each other
Square

Same properties as a rectangle
Extra properties:
18

All sides equal

Diagonals perpendicular to each other
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Activity 6
Name the quadrilaterals.
Answers to Activity 6
1
Square
2
Parallelogram
3
Trapezium
4
Kite
5
Rectangle
6
Rhombus
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Pythagoras’ theorem
Earlier, you studied Pythagoras’ theorem. Since it will be needed when we
do trigonometry, we will briefly revise it.
In this right-angled triangle (ABC), a is the hypotenuse and b and c are the
other two sides.
For this triangle, Pythagoras’ theorem states: a2 = b2 + c2.
Remember:
1
The side opposite the right angle is the hypotenuse.
2
The hypotenuse can be named in two ways, using the two capital letters
at the ends of the line—BC—or using the one small letter ‘a’.
Using Pythagoras’s theorem
Pythagoras’ theorem is used when we know two sides of a right-angled
triangle and want to find the third side.
Example
First it’s always a good idea to state the theorem:
a2 = b2 + c2
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Now substitute the numbers and letters you know—make sure the
hypotenuse is by itself.
Example
This is an equation. To get x2 by itself, subtract 92 from both sides.
Notice how in the first example the numbers were next to each other so we
added.
In the second example the numbers are separated by the equals sign so when
we moved them together we subtracted.
Some students get confused when to add and when to subtract; if you set
your work out as shown above you should have no problems.
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Activity 7
1
Find the length of the hypotenuse (correct to one decimal place).
2
Find the length of the short side (correct to two decimal places).
3
Find the length of the missing side (correct to one decimal place).
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Answers to Activity 7
Answers to one or two decimal places.
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Sides and angles
Trigonometry has been used by people since the time of Alexander the Great
(about 300 BC). In those early days, trigonometry had two main uses. The
first of these involved practical applications in navigation and land
surveying. The second was the rapid development of astronomy and the
measurement of the earth, moon and sun.
Nowadays, trigonometry is still used for investigating the universe and in
navigation and surveying. It also has many other practical uses.
Trigonometry is about the relationships between the sides and the angles of
triangles. In this part of the section we are going to look at some of these
relationships.
Adjacent and opposite sides
In the notes on Pythagoras’ theorem you found out that the side opposite the
right angle is called the hypotenuse. In trigonometry the other two sides are
also named. They are called the adjacent and opposite with reference to the
two other angles.
Here is the triangle ABC in which B = 90.
On the diagram, write in the names of the sides: a, b and c.
Does your triangle look like this?
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
b is the hypotenuse because it is opposite the right angle at B.
Now look at the angle C and the other two sides of the triangle:

The side c is opposite C.

The remaining side a is adjacent to C (side between the marked angle
and the right angle).
So, when we are considering C:

the side c is opposite C

the side a is adjacent to C.
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Now look at ABC again, but this time consider A:

Again b is the hypotenuse.

The side a is opposite A.

The remaining side c is adjacent to A.
The hypotenuse is always opposite the right angle and is never called an
adjacent side.
Notice that:

b is the hypotenuse

a is opposite to A and is adjacent to C

c is opposite to C and is adjacent to A.
Look up the word ‘adjacent’ in your dictionary.
Can you see why it’s used for the side lying alongside the angle?
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Activity 8
1
Here is the right-angled triangle PQR.
Which of these statements are correct?
(a) p is the hypotenuse
(b) p is adjacent to R
(c) r is opposite to R
(d) r is adjacent to Q
(e) p is opposite to Q
(f) q is adjacent to R.
Now fill in the words missing from these sentences:
r is ______________________ to R and is ____________________ to Q.
q is ______________________ to Q and is ____________________ to R.
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2
You are given enough information in this table to enable you to fill in the missing
sides and angles. The first one has been done for you.
Triangle
Angle
Hypotenuse
Opposite side
Adjacent side
Y
x
y
z
B
N
s
p
Answers to Activity 8
1
The correct statements are:
(a) p is the hypotenuse
(c) r is opposite to R
(d) r is adjacent to Q
(f) q is adjacent to R.
Did you fill in these missing words correctly?
r is opposite to R and is adjacent to Q.
q is opposite to Q and is adjacent to R.
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Triangle
Angle
Hypotenuse
Opposite
side
Adjacent
side
B
c
b
a
N
l
n
m
S
t
s
r
R
q
r
p
Did you do these correctly? Well done!
If you did not, go over the notes on sides and angles again.
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The trigonometric ratios –
sin, cos and tan
Remember that the sides of a triangle are named with reference to a given
angle, as shown here with reference to angle x.
In trigonometry:
We abbreviate sine to sin (still pronounced sine), cosine to cos and tangent
to tan.
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So in the triangle drawn below:
sin x =
=
cos x =
=
tan x =
=
opposite
hypotenuse
O
H
adjacent
hypotenuse
A
H
opposite
adjacent
O
A
This rhyme may help you remember these ratios:
SOH CAH TOA
This reminds us that:
sin x =
O
H
(S is O over H)
cos x =
A
H
(C is A over H)
tan x =
O
A
(T is O over A)
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Example
In triangle DEF we will use angle F to determine the trig ratios. As shown
below, in all trig questions you should label the sides as O, A and H before
you start.
In this triangle:
sin F =
O
H
cos F =
A
H
tan F =
O
A
=
f
e
=
d
e
=
f
d
For the triangle below, complete the missing parts in the statements that
follow:
sin A =
=
tan C =
=
32
O
H
b
O
A
cos A =
A
H
tan A =
O
A
=
c
=
a
cos C =
A
H
sin C =
O
H
=
=
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Does your work look like this?
O
H
sin A =
cos A =
A
H
tan A =
O
A
=
a
b
=
c
b
=
a
c
tan C =
O
A
cos C =
A
H
sin C =
O
H
=
c
a
=
a
b
=
c
b
Well done!
Now try these questions:
1
sin X =
H
=
=
cos Z =
sin Z =
H
=
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cos X =
=
A
tan X =
A
=
O
tan Z =
O
=
33
2
sin P =
cos Q =
tan Q =
=
=
=
Do your answers look like these?
1
O
H
4
=
5
cos X =
A
H
4
=
5
sin Z =
O
H
6
=
10
3
=
5
cos Q =
sin X =
cos Z =
2
sin P =
A
H
3
=
5
tan X =
O
A
4
=
3
O
H
3
=
5
tan Z =
A
H
6
=
10
3
=
5
tan Q =
O
A
3
=
4
O
A
8
=
6
4
=
3
If you had all of these answers, well done! If not, you should go back over
the work on trigonometric ratios before going on.
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Labelling angles
When we are using right-angled triangles in trigonometry, we often don’t
label all of the vertices of the triangle.
Pronumerals (such as x, y and z) or Greek letters are often used to indicate
the angle we are considering. Commonly used Greek letters are  (theta)
and  (phi).
For example, in this triangle:
tan  =
c
b
sin  =
c
a
cos  =
b
a
Similar triangles—a reminder
Similar triangles are triangles which have the same shape but are different
sizes. Similar triangles are equiangular and their sides are in proportion.
Equiangular means that the three angles of one triangle are equal to the three
angles of the other triangle (or triangles).
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The diagram below shows three similar right-angled triangles.
For Triangle 1, the side opposite to A is 12 units and the side adjacent to
A is 10 units.
O
A
12
=
10
So the value of tan A =
= 1.2
For Triangle 2, the side opposite to A is 6 units and the side adjacent to
A is 5 units.
O
A
6
=
5
So the value of tan A =
= 1.2
For Triangle 3, what is the value of tan A?
Did you find that the value is also 1.2?
So we have discovered that tan A = 1.2 in all three triangles, irrespective of
the size of the triangle in which it is found.
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Check that the values of both sin A and cos A are the same in all three
triangles.
You will have to use Pythagoras’ theorem to find the length of the
hypotenuse in each case and then use your calculator to get a decimal value
so that you can compare your answers.
Did you get these results?
sin A = 0.77 (to two decimal places)
cos A = 0.64 (to two decimal places).
Well done!
We have now discovered that sin A, cos A and tan A have the same values
in similar triangles, irrespective of the size of the triangle.
Activity 9
1
Here is the right-angled triangle JKL.
Select the correct statements from this list:
(a) sin L =
l
k
(b) tan K =
k
l
(c) cos L =
j
k
(d) cos K =
l
j
(e) tan L =
l
k
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2
For this right-angled triangle, write down the tangent, sine and cosine ratios of .
3
Complete these statements for triangle LMN above:
tan N =….
sin M =….
cos N =….
38
cos… =
n
l
tan… =
m
n
sin… =
n
l
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Answers to Activity 9
1
The correct statements are:
k
l
l
(d) cos K =
j
(b) tan K =
(e) tan L =
2
3
tan  =
l
k
y
z
y
sin  =
x
z
cos  =
x
n
tan N =
m
m
sin M =
l
m
cos N =
l
n
cos M =
l
m
tan M =
n
n
sin N =
l
Did you notice that sin M = cos N for this triangle?
Congratulations! You have completed Section 1 of this module. Have a break
before starting the Check your progress exercises. These will revise all of the
work covered in this section.
If you have any difficulties, go back and revise the appropriate examples.
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Check your progress
1
What is the complement of 72º?
2
What is the supplement of 58º?
3
Find the value of x.
4
Name the types of triangles.
5
Find the value of x.
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6
Name the quadrilaterals.
7
(a) Find x correct to two decimal places.
(b) In ABC below, C = 90º, c = 40 cm and a = 24 cm.
Find the length of AC.
(c) In the triangle below, find the value of h correct to one decimal place.
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8
Complete the following table referring to UVW.
Angle
Hypotenuse
Adjacent side
Opposite side
V
W
9
Here is a right-angled triangle RST.
Complete the following statements:
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Answers to Check your progress
1
90º – 72º = 18º
2
180º – 58º = 122º
3
(a) x = 90º – 21º = 69º
(b)
(c)
(d)
(e)
x = 180º – 53º = 127º
x = 360º – 158º = 202º
x = 27º (vertically opposite angles)
x = 180º – 118º = 62º (co-interior angles)
(f) x = 118º (corresponding)
(g) x = 118º (alternate).
4
(a) equilateral triangle
(b) right-angled triangle
(c) isosceles triangle
(d) scalene triangle.
6
(a) square
(b)
(c)
(d)
(e)
kite
rectangle
trapezium
parallelogram
(f) rhombus.
7
(a)
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(b)
(c)
8
Angle
Hypotenuse
Adjacent side
Opposite side
V
u
w
v
W
u
v
w
How did you go?
Hope you did well!
You have now completed the first section in the module Coordinate
Geometry and Trigonometry.
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