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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
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
ww
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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
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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Starter QuestionsStarter Questionsw
ww
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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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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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Starter QuestionsStarter Questionsw
ww
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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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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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Starter QuestionsStarter Questionsw
ww
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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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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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Starter QuestionsStarter Questionsw
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o o
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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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