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Trigonometry
Copyright  2011 Pearson Canada Inc.
T-1
§1
Angles and Radian
Measure
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Angles
A ray is a part of a line that has only one endpoint
and extends forever in the opposite direction. A
rotating ray is often a useful means of thinking
about angles.
An angle is formed
by two rays that have
a common endpoint.
One ray is called the
initial side and the
other the terminal side.
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Common Angles
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Measuring Angles Using Radians
A central angle is angle whose vertex is at the
centre of a circle.
Central angle
Intercepted arc
An intercepted arc is the distance along the
circumference of the circle between the initial and
terminal side of a central angle.
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One-Radian Angle
If the length of the intercepted arc is equal to the
circle’s radius, then we say the central angle
measures one radian.
For 1-radian angle,
the intercepted arc
and the radius are
equal.
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Radian Measure
Angles measured in radians.
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Radian Measure
Let θ be a central angle in a circle of radius r and
let s be the length of its intercepted arc.
The measure of θ is:
s
  radians.
r
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Radian Measure
Example:
A central angle, θ , in a circle of radius 5 centimetres
intercepts an arc of length 20 centimetres. What is
the radian measure of θ?
s 20 cm
 
4
r 5cm
20 cm
The radian measure of θ is 4.
5 cm
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Converting Between Degrees and
Radians
The measure of one complete rotation in radians is:
s 2 r
 
 2 radians
r
r
The measure of one complete
rotation is also 360˚, so
360˚ = 2π radians.
Dividing both sides by 2 gives: 180˚ = π radians
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Converting Between Degrees and
Radians
Conversion Between Degrees and Radians
Using the basic relationship π radians = 180˚,
1. To convert degrees to radians, multiply degrees by
 radians
180
2.
To convert radians to degrees, multiply radians by
180
 radians
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Converting Between Degrees and
Radians
Example:
Convert each angle in degrees to radians.
135˚
120˚
135  135 
120  120 
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 radians
180
 radians
180
135
3

radians 
radians
180
4
120
2

radians 
radians
180
3
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Converting Between Degrees and
Radians
Example:
Convert each angle in radians to degrees.
5

6
3
5
5 radians
180
5 180
radians 


 150
6
6
 radians
6

3
radians 
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 radians
3
180
180


 60
 radians
3
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§2
Angles and the Cartesian
Plane
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Drawing Angles in Standard Position
An angle is in standard position on the xy-plane if
 its vertex is at the origin and
 its initial side lies along the positive x-axis.
y
x
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Drawing Angles in Standard Position
A positive angle is generated by a counterclockwise
rotation form the initial side to the terminal side.
A negative angle is generated by a clockwise rotation
form the initial side to the terminal side.
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Drawing Angles in Standard Position
y
The xy-plane is
divided into four
quadrants.
Quadrant II
Quadrant I
x
Quadrant III
Quadrant IV
If the terminal side of the angle lies on the x-axis or
y-axis the angle is called a quadrantal angle.
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Angles Formed by Revolution of
Terminal Sides
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Drawing Angles in Standard Position
Example:
Draw and label each angle in standard position.

y

Terminal side
3
Vertex
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 

3
Initial side
x
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Drawing Angles in Standard Position
Example:
Draw and label each angle in standard position.

y
 
2
Vertex
Initial side
x
Terminal side
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 

2
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Degree and Radian Measures of
Common Angles
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Coterminal Angles
Two angles with the same initial and terminal side
but possibly different rotations are called coterminal
angles.
Coterminal Angles Measured in Degrees
An angle of θ˚ (an angle measured in degrees) is coterminal
with angles of θ˚ + 360˚k, where k is an integer.
Two coterminal angles for an angle of θ˚ can be
found by adding 360˚ to θ˚ and subtracting 360˚ from
θ˚.
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Coterminal Angles
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Coterminal Angles
Example:
Assume the following angle is in standard position.
Find a positive angle less than 360˚ that is
coterminal with it.
460˚
460˚ – 360˚ = 100˚
Angles of 460˚ and 100˚ are coterminal.
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Coterminal Angles
Example:
Assume the following angle is in standard position.
Find a positive angle less than 360˚ that is
coterminal with it.
– 60˚
– 60˚ + 360˚ = 300˚
Angles of – 60˚ and 300˚ are coterminal.
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Coterminal Angles
Coterminal Angles Measured in Radians
An angle of θ radians (an angle measured in radians)
is coterminal with angles of θ + 2πk, where k is an
integer.
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Coterminal Angles
Example:
Assume the following angle is in standard position.
Find a positive angle less than 2π that is coterminal
with it.
7
2
7
7 4 3
 2 


2
2
2
2
Angles of 7 and 3 are coterminal.
2
2
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§3
Right Triangle
Trigonometry
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Labelling a Right Triangle
Using the standard labelling of a right triangle,
we label its sides and angles so that side a is
opposite to angle A, side b is opposite to angle B,
and side c is opposite to angle C.
Hypotenuse
Leg
Leg
Angle C is always taken to be the right angle, making
side c the hypotenuse.
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The Pythagorean Theorem
The Pythagorean Theorem in terms of the standard
labelling of a right triangle is given by
c  a b
2
2
2
Hypotenuse
Leg
Leg
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The Pythagorean Theorem
Example:
Find the length of the hypotenuse c where a = 3 cm
and b = 4 cm.
Hypotenuse
c  a b
2
2
c 3 4
2
c  25
2
2
2
2
a=3 cm
b=4 cm
c  25  5
The length of the hypotenuse is 5 cm.
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Primary Trigonometric Ratios
The three primary trigonometric ratios and their
abbreviations are
Name
Abbreviation
Sine
sin
Cosine
cos
Tangent
tan
Consider a right triangle with one of its acute
angles labelled θ.
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Primary Trigonometric Ratios
Right Triangle Definitions of Sine, Cosine, and
Tangent
The three primary trigonometric ratios of the acute angle θ are
defined as follows:
length of side opposite angle  a
sin  

length of hypotenuse
c
length of side adjacent to angle  b
cos  

length of hypotenuse
c
tan  
length of side opposite angle 
a

length of side adjacent to angle  b
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Primary Trigonometric Ratios
Trigonometry values for a given angle are
always the same no matter how large the
triangle is.
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Primary Trigonometric Ratios
Example:
Find the value of each of the three primary
trigonometric ratios of θ.
c  a b
2
2
2
(2 5)  a  4
2
2
2
c2 5
b4
20  a  16
2
a 4
a2
2
Example continues.
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Primary Trigonometric Ratios
Example:
Find the value of each of the three primary
trigonometric ratios of θ.
c2 5
opposite
2
sin  


hypotenuse 2 5
adjacent
4
cos  


hypotenuse 2 5
opposite 2 1
tan  
 
adjacent 4 2
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1
5
2
5
b4
Example continues.
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Primary Trigonometric Ratios of Special
Angles
2
sin 45 
2
2
cos 45 
2
tan 45  1
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Primary Trigonometric Ratios of Special
Angles
1
sin 30 
2
3
sin 60 
2
3
cos 30 
2
3
tan 30 
3
1
cos 60 
2
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tan 60  3
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Primary Trigonometric Ratios Using a
Calculator
Example:
Use a calculator to find the value to four decimal
places.

sin
cos 1.2
3
Function
Mode
cos 1.2
Radian

sin
3
Radian
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Display,
rounded to four
decimal places
Keystrokes
COS 1.2
SIN
(
0.3624
=
π
÷
3
)
=
0.8660
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§4
Solving Applied Problems
Involving Trigonometry
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Solving Right Triangles
Solving a right triangle means finding the missing
lengths of its sides and the measurements of its
angles.
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Solving Right Triangles
Example:
Solve the given triangle, rounding lengths to
two decimal places.
B  90  A  90  40  50
a
tan 40 
12
a  12 tan 40  10.07
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40˚
12
c
c  cos 40  12
12
c
 15.66
cos 40
cos 40 
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Applied Problems
An angle formed by a horizontal line and the line
of sight to an object that is above the horizontal
line is called the angle of elevation.
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Applied Problems
The angle formed by a horizontal line and the line
of sight to an object that is below the horizontal
line is called the angle of depression.
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Applied Problems
Example:
The irregular blue shape is a pond. The distance
across the pond, a, is unknown. To find this
distance a surveyor took the measurements shown in
the figure. What is the distance across the pond?
a
sin 24 
1200
a  1200sin 24  488
1200 m
The distance across the pond is 488 m.
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Applied Problems
Example:
A building is 40 metres high and it casts a shadow
36 metres long. Find the angle of elevation of the
sun to the nearest degree.
40
tan  
36
1  40 
  tan    48
 36 
The angle of elevation is 48˚.
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40m
36m
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