Maths Age 14-16

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S3 Trigonometry

Maths Age 14-16


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S3.1 Right-angled triangles

S3 Trigonometry

Contents

S3.2 The three trigonometric ratios

S3.3 Finding side lengths

S3.4 Finding angles

S3.5 Angles of elevation and depression

S3.6 Trigonometry in 3-D


Right-angled triangles

A right-angled triangle contains a right angle.

The longest side opposite the right angle is called the hypotenuse.


The opposite and adjacent sides

The two shorter sides of a right-angled triangle are named with respect to one of the acute angles.

The side opposite the marked angle is called the opposite side.

The side between the marked angle and the right angle is called the adjacent side.

x


Label the sides


Similar right-angled triangles

If two right-angled triangles have an acute angle of the same size they must be similar.

For example, two triangles with an acute angle of 37° are similar.

The ratio of the side lengths in each triangle is the same.

34

=68

oppadj

=45

=810

adjhyp

=35

=610

opphyp

=

3 cm

4 cm

5 cm

37°

8 cm

6 cm10 cm

37°


Similar right-angled triangles


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Contents

S3 Trigonometry


S3.4 Finding angles





Trigonometry

The word trigonometry comes from the Greek meaning ‘triangle measurement’.

Trigonometry uses the fact that the side lengths of similar triangles are always in the same ratio to find unknown sides and angles.

For example, when one of the angles in a right-angled triangle is 30° the side opposite this angle is always half the length of the hypotenuse.

30°

8 cm?4 cm

6 cm12 cm

?30°


The sine ratio

The ratio of the length of the opposite sidethe length of the hypotenuse

is the sine ratio.

The value of the sine ratio depends on the size of the angles in the triangle.

θ

OPPOSITE

HY

PO

TE

NU

SE

We say:

sin θ =opposite

hypotenuse


The sine ratio

What is the value of sin 65°?

To work this out we can accurately draw a right-angled triangle with a 65° angle and measure the lengths of the opposite side and the hypotenuse.

In a right-angled triangle with an angle of 65°, what is the ratio of the opposite side to

the hypotenuse?

This is the same as asking:


The sine ratio


It doesn’t matter how big the triangle is because all right-angled triangles with an angle of 65° are similar.

The length of the opposite side divided by the length of the hypotenuse will always be the same value as long as the angle is the same.

In this triangle,

sin 65° =opposite

hypotenuse65°

10 cm

11 cm

= 1011

= 0.91 (to 2 d.p.)


The sine ratio using a table


It is not practical to draw a diagram each time.

Before the widespread use of scientific calculators, people would use a table of values to work this out.

Here is an extract from a table of sine values:

Angle in degrees

63

64

65

66

.0

0.891

0.899

0.906

0.914

.1

0.892

0.900

0.907

0.914

.2

0.893

0.900

0.908

0.915

.3

0.893

0.901

0.909

0.916

.4

0.894

0.902

0.909

0.916

.5

0.895

0.903

0.910

0.917


The sine ratio using a calculator


To find the value of sin 65° using a scientific calculator, start by making sure that your calculator is set to work in degrees.

Key in:

sin 6 5 =

Your calculator should display 0.906307787

This is 0.906 to 3 significant figures.


The cosine ratio

The ratio of the length of the adjacent sidethe length of the hypotenuse

is the cosine ratio.

The value of the cosine ratio depends on the size of the angles in the triangle.

θ

We say,

cos θ =adjacent

hypotenuse

A D J A C E N T

HY

PO

TE

NU

SE


The cosine ratio

In a right-angled triangle with an angle of 53°, what is the ratio of the adjacent side to

the hypotenuse?

To work this out we can accurately draw a right-angled triangle with a 53° angle and measure the lengths of the adjacent side and the hypotenuse.

What is the value of cos 53°?



The cosine ratio


The length of the opposite side divided by the length of the hypotenuse will always be the same value as long as the angle is the same.

In this triangle,

cos 53° =adjacent

hypotenuse

53°

6 cm

10 cm

= 610

= 0.6



The cosine ratio using a table


0.588

Here is an extract from a table of cosine values:

Angle in degrees

50

51

52

53

.0

0.643

0.629

0.616

0.602

.1

0.641

0.628

0.614

0.600

.2

0.640

0.627

0.613

0.599

.3

0.639

0.625

0.612

0.598

.4

0.637

0.624

0.610

0.596

.5

0.636

0.623

0.609

0.595

54

55

56

0.574

0.559

0.586

0.572

0.558

0.585

0.571

0.556

0.584

0.569

0.555

0.582

0.568

0.553

0.581

0.566

0.552


The cosine ratio using a calculator


To find the value of cos 25° using a scientific calculator, start by making sure that your calculator is set to work in degrees.

Key in:

cos 2 5 =




The tangent ratio

The ratio of the length of the opposite sidethe length of the adjacent side

is the tangent ratio.

The value of the tangent ratio depends on the size of the angles in the triangle.

θ

OPPOSITE

We say,

tan θ =oppositeadjacent

A D J A C E N T


The tangent ratio

What is the value of tan 71°?

In a right-angled triangle with an angle of 71°, what is the ratio of the opposite side to

the adjacent side?

To work this out we can accurately draw a right-angled triangle with a 71° angle and measure the lengths of the opposite side and the adjacent side.



The tangent ratio



The length of the opposite side divided by the length of the adjacent side will always be the same value as long as the angle is the same.

In this triangle,

tan 71° =oppositeadjacent71°

11.6 cm

4 cm

= 11.64

= 2.9


The tangent ratio using a table


Here is an extract from a table of tangent values:

Angle in degrees

70

71

72

73

.0

2.75

2.90

3.08

3.27

.1

2.76

2.92

3.10

3.29

.2

2.78

2.94

3.11

3.31

.3

2.79

2.95

3.13

3.33

.4

2.81

2.97

3.15

3.35

.5

2.82

2.99

3.17

3.38

74

75

76

3.49

3.73

4.01

3.51

3.76

4.04

3.53

3.78

4.07

3.56

3.81

4.10

3.58

3.84

4.13

3.61

3.87

4.17


The tangent ratio using a calculator


To find the value of tan 71° using a scientific calculator, start by making sure that your calculator is set to work in degrees.

Key in:

tan 7 1 =




Calculate the following ratios

Use your calculator to find the following to 3 significant figures.

1) sin 79° = 0.982 2) cos 28° = 0.883

3) tan 65° = 2.14 4) cos 11° = 0.982

5) sin 34° = 0.559 6) tan 84° = 9.51

7) tan 49° = 1.15 8) sin 62° = 0.883

9) tan 6° = 0.105 10) cos = 0.55956°


The relationship between sine and cosine

The sine of a given angle is equal to the cosine of the complement of that angle.

We can write this as,

sin θ = cos (90 – θ)

We can show this as follows,

θ

ab

sin θ = ab

ab

cos (90 – θ) = ab90 – θ


θ

OPPOSITE

HY

PO

TE

NU

SE

A D J A C E N T

The three trigonometric ratios

Sin θ =Opposite

Hypotenuse S O HS O H

Cos θ =Adjacent

Hypotenuse C A HC A H

Tan θ =OppositeAdjacent T O AT O A

Remember: S O HS O H C A HC A H T O AT O A


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Contents

S3 Trigonometry

S3.4 Finding angles






Finding side lengths

If we are given one side and one acute angle in a right-angled triangle we can use one of the three trigonometric ratios to find the lengths of other sides. For example,

56°

x12 cm

Find x to 2 decimal places.

We are given the hypotenuse and we want to find the length of the side opposite the angle, so we use:

sin θ =opposite

hypotenuse

sin 56° =x

12x = 12 × sin 56°

= 9.95 cm



A 5 m ladder is resting against a wall. It makes an angle of 70° with the ground.

5 m

70°x

What is the distance between the base of the ladder and the wall?

We are given the hypotenuse and we want to find the length of the side adjacent to the angle, so we use:

cos θ =adjacent

hypotenuse

cos 70° =x5

x = 5 × cos 70°= 1.71 m (to 2 d.p.)


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S3.4 Finding angles

Contents

S3 Trigonometry







The inverse of sin

sin θ = 0.5, what is the value of θ?

To work this out use the sin–1 key on the calculator.

sin–1 0.5 = 30°

sin–1 is the inverse of sin. It is sometimes called arcsin.

30° 0.5

sin

sin–1


The inverse of cos

Cos θ = 0.5, what is the value of θ?

To work this out use the cos–1 key on the calculator.

cos–1 0.5 = 60°

Cos–1 is the inverse of cos. It is sometimes called arccos.

60° 0.5

cos

cos–1


The inverse of tan

tan θ = 1, what is the value of θ?

To work this out use the tan–1 key on the calculator.

tan–1 1 = 45°

tan–1 is the inverse of tan. It is sometimes called arctan.

45° 1

tan

tan–1


Finding angles

We are given the lengths of the sides opposite and adjacent to the angle, so we use:

tan θ =oppositeadjacent

tan θ =85

= 57.99° (to 2 d.p.)

θ

5 cm

8 cm

θ = tan–1 (8 ÷ 5)

Find θ to 2 decimal places.


Finding angles


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Contents

S3 Trigonometry


S3.4 Finding angles





Angles of elevation


Angles of depression


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AS3.6 Trigonometry in 3-D

Contents

S3 Trigonometry


S3.4 Finding angles





Angles in a cuboid


Lengths in a square-based pyramid

Maths Age 14-16

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