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Page 1: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.

created by Mr. Laffertycreated by Mr. Lafferty

Trigonometry EquationsTrigonometry Equationsw

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National 5

Solving Trig Equations

Negative Cosine

Trig Function and Circle Connection

Special Trig Relationships

Exam Type Questions

Page 2: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.

created by Mr. Laffertycreated by Mr. Lafferty

StarterStarterw

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National 5

Q1. How can we tell if two lines are parallel.

Q2. Write down the three ratios connecting

the circle , arc length and area of a sector.

Page 3: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.

created by Mr. Laffertycreated by Mr. Lafferty

Learning IntentionLearning Intention Success CriteriaSuccess Criteria

1.1. Understand the Understand the connection between the connection between the circle and sine, cosine circle and sine, cosine and tan functions.and tan functions.

2. Solve trig equations using graphically.

1. We are investigating the connect between the circle and trig functions.

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Solving Trig Equations Solving Trig Equations National 5

Page 4: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.

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Trig and Circle Connection Trig and Circle Connection

All +veSin +ve

Tan +ve Cos +ve

180o - xo

180o + xo 360o - xo

1 2 3 4

National 5

Sine Graph Construction

Cosine Graph Construction

Tan Graph Construction Demo

Page 5: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.

created by Mr. Laffertycreated by Mr. Lafferty

Learning IntentionLearning Intention Success CriteriaSuccess Criteria

1.1. Use the balancing method Use the balancing method to trig equationto trig equation

a sin xo + 1 = 0

2. Realise that there are many solutions to trig equations depending on domain.

1. We are learning how to solve trig equations of the form

a sin xo + 1 = 0

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Solving Trig Equations Solving Trig Equations National 5

Page 6: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.

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Solving Trig EquationsSolving Trig Equationsa sin xo + b = 0

Example :

Solving the equation sin xo = 0.5 in the range 0o to 360o

Graphically what are we

trying to solve

xo = sin-1(0.5)

xo = 30o

There is another solution

xo = 150o

(180o – 30o = 150o)

sin xo = (0.5)

1 2 3 4

National 5

Demo

Page 7: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.

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Solving Trig EquationsSolving Trig Equationsa cos xo + b = 0

Example :

Solving the equation cos xo = 0.625 in the range 0o to 360o

cos xo = 0.625

xo = 51.3o

(360o - 53.1o = 308.7o)

xo = cos -1 0.625

There is another solution

1 2 3 4

National 5

Graphically what are we

trying to solveDemo

Page 8: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.

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Solving Trig EquationsSolving Trig Equationsa tan xo + b = 0

Example :

Solving the equation tan xo – 2 = 0 in the range 0o to 360o

tan xo = 2

xo = 63.4o

x = 180o + 63.4o = 243.4o

xo = tan -1(2)

There is another solution

1 2 3 4

National 5

Graphically what are we

trying to solveDemo

Page 9: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.

Example :

Solving the equation 3sin xo + 1 = 0 in the range 0o to 360o

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Solving Trig EquationsSolving Trig Equationsa sin xo + b = 0

sin xo = -1/3

Calculate first Quad valuexo = 19.5o

x = 180o + 19.5o = 199.5o

( 360o - 19.5o = 340.5o)

There is another solution

1 2 3 4

National 5

Graphically what are we

trying to solveDemo

Page 10: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.

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Solving Trig EquationsSolving Trig Equationsa sin xo + b = 0

Example :

Solving the equation 2sin xo + 1 = 0 in the range 0o to 720o

sin xo = -1/2

Calculate first Quad valuexo = 30o

xo = 210o and 330o

360o + 210o = 570o

There are further solutions at

National 5

360o + 330o = 690o

Graphically what are we

trying to solveDemo

Page 11: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.

created by Mr. Laffertycreated by Mr. Lafferty

Now try N5 TJ Ex20.1 Q4 to Q10

Ch 20 (Page 198)

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Solving Trig EquationsSolving Trig Equations

National 5

Page 12: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.
Page 13: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.
Page 14: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.

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Outcome 3Higher

created by Mr. Laffertycreated by Mr. Lafferty

Solving Trig Equations

Example

Solving the equation cos2x = 1 in the range 0o to 360o

Graphically what are we

trying to solve

cos xo = ± 1

cos xo = 1

cos2 xo = 1

xo = 0o and 360o

C

AS

T0o180

o

270o

90o

cos xo = -1xo = 180o

Page 15: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.

created by Mr. Laffertycreated by Mr. Lafferty

Now try N5 TJ Ex20.1 Q4 onwards

Ch 20 (Page 198)

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Solving Trig EquationsSolving Trig Equations

National 5

Page 16: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.

19 Apr 202319 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.

Starter QuestionsStarter Questionsw

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2

1. I f lines have the same gradient

What is special about them.

2. Factorise x +4x -12

3. Find the missing angles.

Nat 5

54o

Page 17: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.

19 Apr 202319 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.ww

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1.1. Know what a negative Know what a negative cosine ratio means.cosine ratio means.

1. We are learning what a negative cosine ratio means with respect to the angle.

Cosine RuleCosine RuleNat 5

2. 2. Solve Solve REAL LIFE problems problems that involve finding an that involve finding an angle of a triangle.angle of a triangle.

Page 18: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.

C

B

A19 Apr 202319 Apr 2023 Created by Mr Lafferty Maths DeptCreated by Mr Lafferty Maths Dept

Cosine RuleCosine Rulew

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Nat 5

a

b

c

The Cosine Rule can be used with ANY triangle as long as we have been given enough information.

Works for any Triangle

cos2 2 2a =b +c - 2bc A

Page 19: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.

19 Apr 202319 Apr 2023 Created by Mr Lafferty Maths DeptCreated by Mr Lafferty Maths Dept

Cosine RuleCosine Rulew

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Nat 5

How to determine when to use the Cosine Rule.

Works for any Triangle

1. Do you know ALL the lengths.

2. Do you know 2 sides and the angle in between.

SASOR

If YES to any of the questions then Cosine Rule

Otherwise use the Sine Rule

Two questions

Page 20: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.

Finding Angles Finding Angles Using The Cosine RuleUsing The Cosine Rule

Consider the Cosine Rule again:We are going to change the subject of the formula to cos Ao

Turn the formula around:b2 + c2 – 2bc cos Ao = a2

Take b2 and c2 across.-2bc cos Ao = a2 – b2 – c2

Divide by – 2 bc.2 2 2

cos2

o a b cA

bc

Divide top and bottom by -12 2 2

cos2

o b c aA

bc

You now have a formula for finding an angle if you know all three sides of the triangle.w

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Nat 5 Works for any Triangle

Page 21: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.

Write down the formula for cos Fo

2 2 2

cos2

od e f

Fde

Label and identify Fo and d , e and f.

Fo = ? f = 8e = 10d = 12

Substitute values into the formula.

2 2 212 10 8cos

2 12 10

oF

Calculate cos Fo .Cos Fo =0.75

Use cos-1 0.75 to find FoFo = 41.4o

Example : Calculate the

unknown angle Fo .

Finding Angles Finding Angles Using The Cosine RuleUsing The Cosine Rule

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Nat 5 Works for any Triangle

E

D

F

e

d

f

Page 22: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.

Example : Find the unknown

Angle in the triangle:

Write down the formula.

2 2 2

cos2

o b c aA

bc

Identify the sides and angle.

Ao = yo a = 26 b = 15 c = 13

2 2 215 13 26cos

2 15 13oA

Find the value of cosAo

cosAo = - 0.723 The negative tells you

the angle is obtuse.Ao = 136.3ow

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Nat 5

Finding Angles Finding Angles Using The Cosine RuleUsing The Cosine Rule

Works for any Triangle A

BC

c

a

b

Page 23: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.

created by Mr. Laffertycreated by Mr. Lafferty

Now try N5 TJ Ex20.2

Ch 20 (Page 200)

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Solving Trig EquationsSolving Trig Equations

National 5

Page 24: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.

created by Mr. Laffertycreated by Mr. Lafferty

StarterStarter

2

1. Make a the subject of the f ormula

5 = 10b + a

2. Use the quadratic f ormula to solve

x + 5x + 1

3. Sketch the f unction y = 4sin3x

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National 5

Page 25: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.

created by Mr. Laffertycreated by Mr. Lafferty

Learning IntentionLearning Intention Success CriteriaSuccess Criteria

1.1. Know and learn the two Know and learn the two special trig relationships.special trig relationships.

2. Apply them to solve problems.

1. To explain some special trig relationships

sin 2 xo + cos 2 xo = ?

and

tan xo and sin x cos x

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Solving Trig Equations Solving Trig Equations National 5

Page 26: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.

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Solving Trig EquationsSolving Trig Equations

Lets investigate

sin 2xo + cos 2 xo = ?

Calculate value for x = 10, 20, 50, 250

sin 2xo + cos 2 xo = 1

Learn !

National 5

sin 2xo = 1 - cos 2 xo

cos2xo = 1 - sin2 xo

Page 27: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.

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Solving Trig EquationsSolving Trig Equations

Lets investigate

tan xo

Calculate value for x = 10, 20, 50, 250

Learn !

sin xo cos xo

and

tan xo sin xo cos xo =

National 5

Page 28: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.

cos2xo = 2516

cos2xo = 1 - sin2xo

Given that sin xo = .35

Find cos xo .

cos2xo = 1 - 35(

(2

cos2xo = 1 - 925

√ cosxo = 54

Page 29: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.

sin2xo = 10064

sin2xo = 1 - cos2xo

Given that cos xo = .6

10Find sin xo and tanxo.

(

(

sin2xo = 1 - 610

2

sin2xo = 1 - 10036

√ sinxo = 108

tan xo sin xo cos xo =

810

610

=

8

6=

4

3=tan xo

Page 30: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.

= sinxsin2x + sinxcos2x

= sinx(sin2x + sinxcos2x)

(sin2x + cos2x) = 1

LHS

= sinx = RHS

Page 31: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.

sinx cosx

cosx

sinx =

sinx

sinx =

= 1

Page 32: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.

LHS 1 – sin2A = cos2A

sin2A

cos2A =

= tan2A = RHS

Page 33: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.
Page 34: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.

created by Mr. Laffertycreated by Mr. Lafferty

Now try N5 TJ Ex20.3

Ch 20 (Page 201)

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Solving Trig EquationsSolving Trig Equations

National 5

Page 35: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.

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Higher

Are you on Target !

• Update you log book

• Make sure you complete and correct

ALL of the Trigonometry questions in the past paper booklet.

Page 36: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.
Page 37: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.
Page 38: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.
Page 39: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.
Page 40: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.
Page 41: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.
Page 42: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.
Page 43: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.
Page 44: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.
Page 45: Created by Mr. Lafferty Trigonometry Equations  National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connection.