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www.mathsrevision.com Trigonomet Trigonomet ry ry Let’s Investigate Extension The Tangent Ratio The Tangent Angle The Sine Ratio The Sine Angle The Cosine Ratio The Cosine Angle Mixed Problems

Trigonometry Let’s Investigate Extension The Tangent Ratio The Tangent Angle The Sine Ratio The Sine Angle The Cosine Ratio The

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TrigonomeTrigonometrytryLet’s Investigate

Extension

The Tangent RatioThe Tangent Angle

The Sine RatioThe Sine Angle

The Cosine RatioThe Cosine Angle

Mixed Problems

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Starter QuestionsStarter Questionsw

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1. Find the missing value

2. Calculate of 6000

3. What is the next three numbers in the sequence 9, 15, 21, 27, ...., ...., .... 4. Round 72 to the nearest 10

3 ?=4 20

20%

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Let’s Investigate!Let’s Investigate!

TrigonomeTrigonometrytry

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TrigonomeTrigonometrytry

Trigonometry means “triangle” and “measurement”.

AdjacentO

pp

osit

e

x°x°

hypotenuse

We will be using right-angled triangles.

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TrigonomeTrigonometrytry

30°

Adjacent

Op

posit

e

hypotenuse

OppositeAdjacent

= 0.6

Mathemagic!

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TrigonomeTrigonometrytry

45°

Adjacent

Op

posit

e

hypotenuse

OppositeAdjacent

= 1

Try another!

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TrigonomeTrigonometrytry

For an angle of 30°, OppositeAdjacent

= 0.6

We write tan 30° = 0.6

OppositeAdjacent

is called the tangent of an angle.

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TrigonomeTrigonometrytry

Tan 25Tan 25°° 0.4660.466

Tan 26Tan 26°° 0.4880.488

Tan 27Tan 27°° 0.5100.510

Tan 28Tan 28°° 0.5320.532

Tan 29Tan 29°° 0.5540.554

Tan 30Tan 30°° 0.5770.577

Tan 31Tan 31°° 0.6010.601

Tan 32Tan 32°° 0.6250.625

Tan 33Tan 33° ° 0.6490.649

Tan 34Tan 34°° 0.6750.675

Tan 30° = 0.577

Accurate to 3 decimal places!

The ancient Greeks discovered this and repeated this for possible angles.

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TrigonomeTrigonometrytry

Now-a-days we can use calculators instead of tables to find the Tan of an angle.

TanOn your calculator press

Notice that your calculator is incredibly accurate!!

Followed by 30, and press

=

Accurate to 9 decimal places!

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TrigonomeTrigonometrytry

What’s the point of all this???

Don’t worry, you’re about to find out!

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TrigonomeTrigonometrytry

12 m

How high is the tower?

Opp

60°

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TrigonomeTrigonometrytry

60°

12 mAdjacent

Op

posit

e

hypotenuse Copy this!

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TrigonomeTrigonometrytry

Tan x° =

Opp

Adj

Tan 60° =

Opp12

= Opp

12 x Tan 60°Opp =

12 x Tan 60°= 20.8m (1 d.p.)

Change side, change sign!

Copy this!

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TrigonomeTrigonometrytry

So the tower’s 20.8 m high!

Don’t worry, you’ll be trying plenty of examples!!

20.8m

?

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Starter QuestionsStarter Questionsw

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.com 1. Find the perimeter of the shape

2. Calculate 30% of 900

3. Find the area of the rectangle

6cm in length by 4 cm wide.

4. Name the shape.

3cm

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TrigonomeTrigonometrytry

Adj

x°x°

Tan x° =O

pp

osit

e

Opp

Adjacent

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TrigonomeTrigonometrytry

Example Example

65°65°

Tan x° =

Opp Opp

Adj

Hyp cc

8m8m Tan 65° =

c8

= c

8 x Tan 65°

c =

8 x Tan 65° = 17.2m (1 d.p.)

AdjChange side, change sign!

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TrigonomeTrigonometrytry

Now tryExercise 1.

(HSDU Support Materials)

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Starter QuestionsStarter Questionsw

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.com 1. Name the part of the circle.

2. Calculate 60% of 300

o3. I f I am f acing North and turn 90 clockwise, which direction am I f acing

4. How many lines of symmetry has the shape.

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Using Tan to calculate anglesUsing Tan to calculate angles

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TrigonomeTrigonometrytry

ExampleExample

x°x°Tan x° =

Opp

Opp

Adj

Hyp

SOH CAH TOA

12m12mTan x° = 18

12

= 1.5 Tan x°

Adj

18m18m

?

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TrigonomeTrigonometrytry

= 1.5Tan x°How do we find x°?

We need to use Tan ⁻¹on the calculator.

2nd

Tan ⁻¹is written above Tan

Tan ⁻¹

To get this press TanFollowed by

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TrigonomeTrigonometrytry

x =

Tan ⁻¹1.5 = 56.3° (1 d.p.)

= 1.5Tan x°

2nd Tan

Tan ⁻¹

Press

Enter =1.5

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TrigonomeTrigonometrytry

Now tryExercise 2.

(HSDU Support Materials)

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Starter QuestionsStarter Questionsw

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1. 13.9 x 7

2. Calculate 23.34 x 10

3.

4. Find the missing number

3 of 804

1, 1, 2, 3, 5, 8, ...., ...., ....

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TrigonomeTrigonometrytryThe Sine RatioThe Sine Ratio

x°x°

Sin x° =O

pp

osit

e

OppHyp

hypotenuse

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TrigonomeTrigonometrytry

ExampleExample

34°34°Sin x° =

OppOpp

Hyp

HypOO

11c11cmm

Sin 34° =

O11= O

11 x Sin 34°

O =

11 x Sin 34°

= 6.2cm (1 d.p.)

Change side, change sign!

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TrigonomeTrigonometrytry

Now tryExercise 3.

(HSDU Support Materials)

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Starter QuestionsStarter Questionsw

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.com 1. 320 8

2. Calculate 20% of 360

3. Calculate 72 - 58

4. Calculate the value of the missing angle.

57o

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Using Sin to calculate anglesUsing Sin to calculate angles

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TrigonomeTrigonometrytry

ExampleExample

x°x°

Sin x° =

Opp

Opp

Hyp

Hyp

SOH CAH TOA6m6m 9m9m

Sin x° =

69

= 0.667 (3 d.p.)

Sin x°

?

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TrigonomeTrigonometrytry

=0.667 (3 d.p.)Sin x°How do we find x°?

We need to use Sin ⁻¹on the calculator.

2nd

Sin ⁻¹is written above Sin

Sin ⁻¹

To get this press SinFollowed by

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TrigonomeTrigonometrytry

x =

Sin ⁻¹0.667 = 41.8° (1 d.p.)

= 0.667 (3 d.p.)

Sin x°

2nd Sin

Sin ⁻¹

Press

Enter =0.667

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TrigonomeTrigonometrytry

Now tryExercise 4.

(HSDU Support Materials)

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Starter QuestionsStarter Questionsw

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1. 2.39 - 1.58 + 3.2

2. Calculate 15 of 380

3. What is the next three numbers in the sequence 2, 15, 28, 41, ...., ...., .... 4. Round 3.25 to the nearest 0.1

%

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TrigonomeTrigonometrytry

The Cosine The Cosine RatioRatio

Cos x° =

Adjacent

Adj

x°x°

Hyphypotenuse

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TrigonomeTrigonometrytry

ExampleExample

40°40°Cos x° =

Opp

Adj

Hyp Hyp

bb

35m35mmm

Cos 40° =

b35

= b

35 x Cos 40°

b =

35 x Cos 40°

= 26.8mm (1 d.p.)

Adj

Change side, change sign!

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TrigonomeTrigonometrytry

Now tryExercise 5.

(HSDU Support Materials)

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Starter QuestionsStarter Questionsw

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.com 75% of £ 200Q1. Calculate

Q2. Round to 1 decimal place 2.354.

Q3. How many minutes in 3hours

Q4. The answer to the question is 180. What is thequestion.

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Using Cos to calculate anglesUsing Cos to calculate angles

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TrigonomeTrigonometrytry

ExampleExample

x°x°Cos x° =

Opp

Adj

Hyp Hyp

SOH CAH TOA

45cm45cm

Cos x° = 3445= 0.756 (3 d.p.)Cos

x°x =

Cos ⁻¹0.756 =40.9° (1 d.p.)

Adj34cm34cm

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TrigonomeTrigonometrytry

Now tryExercise 6.

(HSDU Support Materials)

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Starter QuestionsStarter Questionsw

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o o

2. What kind of angle is this

3. 56.98 10

4. Name the angle that is between 180 and 360

1. Calculate 14 x 100

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The Three RatiosThe Three Ratios

Cosine

Sine

Tangent

Sine

Sine

Tangent

Cosine

Cosine

Sine

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TrigonomeTrigonometrytryThe Three RatiosThe Three Ratios

Sin x° =Opp

HypCos x° =

Adj

HypTan x° =

Opp

Adj

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TrigonomeTrigonometrytry

Sin x° =Opp

HypCos x° =

Adj

HypTan x° =

Opp

Adj

CAH TOASOH

AC H

OT A

OS H

Copy this!

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Mixed ExamplesMixed Examples

Cos 12°

Sin 60°

Tan 27°

Sin 30°

Sin 35°

Tan 40°

Cos 20°

Cos 79°

Sin 36°

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TrigonomeTrigonometrytry

Example 1Example 1

40°40°Sin x° =

OppOpp

Hyp

HypSOH CAH TOAOO

1515mm

Sin 40° =

O15= O

15 x Sin 40°

O = 15 x Sin 40°

= 9.6m (1 d.p.)

Change side, change sign!

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TrigonomeTrigonometrytry

Example 2Example 2

35°35°

Cos x° =

OppAdj

Hyp Hyp

SOH CAH TOA bb

23cm23cm

Cos 35° =

b23= b

23 x Cos 35°

b =

23 x Cos 35°= 18.8cm (1 d.p.)

Adj

Change side, change sign!

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TrigonomeTrigonometrytry

Example 3Example 3

60°60°

Tan x° =

Opp Opp

Adj

Hyp

SOH CAH TOA

cc

15m15m Tan 60° =

c15

= c

15 x Tan 60°c =

15 x Tan 60°= 26.0m (1 d.p.)

AdjChange side, change sign!

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TrigonomeTrigonometrytry

Now tryExercise 7.

(HSDU Support Materials)

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Starter QuestionsStarter Questionsw

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2. What kind of angle is this

3. 1.268 100

4. Name the angle that is between 0 and 90

1. Calculate 41.9 x 100

Level E

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ExtensionExtensionw

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TrigonomeTrigonometrytry

Example 1Example 1

30°30°

Sin x° =

Opp

Opp

Hyp

Hyp

SOH CAH TOA23cm23cm bb

Sin 30° =

23b

?

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TrigonomeTrigonometrytry

Sin 30° =

23b

Change sides, change signs!

Sin 30°23b

=

(This means b = 23 ÷ Sin 30º)

b=

46 cm

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TrigonomeTrigonometrytry

Example 2Example 2

50°50°

Cos x° =

OppAdj

HypHyp

SOH CAH TOA 7m7m

pp

Cos 50° =

7p

p=p= 10.9m (1 d.p.)

Adj

Change sides, change signs!

Cos 50°

7

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TrigonomeTrigonometrytry

Example 3Example 3

55°55°

Tan x° =

Opp Opp

Adj

Hyp

SOH CAH TOA

9m9m

ddAdj

Tan 55° =

9d

d=

d= 6.3m (1 d.p.)

Change sides, change signs!

Tan 55°

9