Copyright 2011 Pearson Canada Inc. Trigonometry T - 1

Copyright 2011 Pearson Canada Inc.

Trigonometry

T - 1


Angles and Radian Measure

§ 1

T - 2


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.

An intercepted arc is the distance along the circumference of the circle between the initial and terminal side of a central angle.

Intercepted arc

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:

radians.s

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?

20cm4

5cm

s

r

The radian measure of θ is 4.

5 cm

20 cm

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Converting Between Degrees and Radians

The measure of one complete rotation in radians is:2

2 radianss r

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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Conversion Between Degrees and RadiansUsing the basic relationship π radians = 180˚,

1. To convert degrees to radians, multiply degrees by

2. To convert radians to degrees, multiply radians by

radians

180

180

radians

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Example:Convert each angle in degrees to radians.135˚ 120˚

radians 135 3135 135 radians radians

180 180 4

radians 120 2120 120 radians radians

180 180 3

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Example:Convert each angle in radians to degrees.

5 5 radians 180 5 180radians 150

6 6 radians 6

5

6

3

radians 180 180radians 60

3 3 radians 3

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Angles and the Cartesian Plane

§ 2

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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.

x

y

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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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The xy-plane is divided into four quadrants.

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y

Quadrant I

x

Quadrant II

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.


Angles Formed by Revolution of Terminal Sides

18



Example:Draw and label each angle in standard position.

3

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y

x

Terminal side

Initial sideVertex3



Example:Draw and label each angle in standard position.

2

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y

x

Terminal side

Initial sideVertex

2


Degree and Radian Measures of Common Angles

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Coterminal Angles

22

Two angles with the same initial and terminal side but possibly different rotations are called coterminal angles.

Coterminal Angles Measured in DegreesAn 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˚.


Coterminal Angles

23


Coterminal Angles

Example:Assume the following angle is in standard position.

Find a positive angle less than 360˚ that is coterminal with it.

460˚

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460˚ – 360˚ = 100˚

Angles of 460˚ and 100˚ are coterminal.


Coterminal Angles


Find a positive angle less than 360˚ that is coterminal with it.

– 60˚

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– 60˚ + 360˚ = 300˚

Angles of – 60˚ and 300˚ are coterminal.


Coterminal Angles

26

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.


Coterminal Angles


Find a positive angle less than 2π that is coterminal with it.

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Angles of and are coterminal.

7

2

7 7 4 32

2 2 2 2

7

2

3

2


Right Triangle Trigonometry

§ 3

T - 28


Leg

Leg

Hypotenuse

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.

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Angle C is always taken to be the right angle, making side c the hypotenuse.


Leg

Leg

Hypotenuse

The Pythagorean Theorem

The Pythagorean Theorem in terms of the standard labelling of a right triangle is given by

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2 2 2c a b


Hypotenuse

a=3 cm

b=4 cm

The Pythagorean Theorem

Example:Find the length of the hypotenuse c where a = 3 cm

and b = 4 cm.

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2 2 2c a b 2 2 23 4c 2 25c

25 5c The length of the hypotenuse is 5 cm.


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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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 sin

length of hypotenuse

a

c

length of side adjacent to angle cos

length of hypotenuse

b

c

length of side opposite angle tan

length of side adjacent to angle

a

b

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Trigonometry values for a given angle are always the same no matter how large the triangle is.

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Example:Find the value of each of the three primary

trigonometric ratios of θ.

4b

2 5c 2 2 2c a b 2 2 2(2 5) 4a 220 16a

2 4a 2a

Example continues.T - 35



Example:Find the value of each of the three primary

trigonometric ratios of θ.

4b

2 5c

opposite 2 1sin

hypotenuse 2 5 5

Example continues.

adjacent 4 2cos

hypotenuse 2 5 5

opposite 2 1tan

adjacent 4 2

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Primary Trigonometric Ratios of Special Angles

2sin 45

2

2cos 45

2

tan 45 1

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Primary Trigonometric Ratios of Special Angles

1sin 30

2

3cos30

2

3tan 30

3

1cos 60

2

3sin 60

2

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.cos 1.2

Function Mode Keystrokes Display, rounded to four decimal places

cos 1.2 Radian

Radian

COS 1.2 =

sin3

sin3

SIN ( π ÷ ) =3

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0.3624

0.8660


Solving Applied Problems Involving Trigonometry

§ 4

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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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40˚

Solving Right Triangles

Example:

Solve the given triangle, rounding lengths to two decimal places.

90 90 40 50B A

tan 4012

a

12 tan 40 10.07a cos 40 12c

12cos 40

c

1215.66

cos 40c

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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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1200 m

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?

sin 241200

a

1200sin 24 488a

The distance across the pond is 488 m.T - 45


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.

40tan

36

The angle of elevation is 48˚.

40m

36m

1 40tan 48

36

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Copyright 2011 Pearson Canada Inc. Trigonometry T - 1