of 45 /45
created by Mr. Lafferty created by Mr. Lafferty Trigonometry Equations Trigonometry Equations www.mathsrevision.com National 5 Solving Trig Equations Negative Cosine Trig Function and Circle Connectio Special Trig Relationships Exam Type Questions

dorothy-ross
• Category

## Documents

• view

224

0

### Transcript of Created by Mr. Lafferty Trigonometry Equations National 5 Solving Trig Equations Negative Cosine...

created by Mr. Laffertycreated by Mr. Lafferty

Trigonometry EquationsTrigonometry Equationsw

ww

.math

srevis

ion

.com

National 5

Solving Trig Equations

Negative Cosine

Trig Function and Circle Connection

Special Trig Relationships

Exam Type Questions

created by Mr. Laffertycreated by Mr. Lafferty

StarterStarterw

ww

.math

srevis

ion

.com

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.

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.

ww

w.m

ath

srevis

ion

.com

Solving Trig Equations Solving Trig Equations National 5

created by Mr. Laffertycreated by Mr. Laffertyww

w.m

ath

srevis

ion

.com

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

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

ww

w.m

ath

srevis

ion

.com

Solving Trig Equations Solving Trig Equations National 5

created by Mr. Laffertycreated by Mr. Laffertyww

w.m

ath

srevis

ion

.com

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

created by Mr. Laffertycreated by Mr. Laffertyww

w.m

ath

srevis

ion

.com

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

created by Mr. Laffertycreated by Mr. Laffertyww

w.m

ath

srevis

ion

.com

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

Example :

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

created by Mr. Laffertycreated by Mr. Laffertyww

w.m

ath

srevis

ion

.com

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

ww

w.m

ath

srevis

ion

.com

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

created by Mr. Laffertycreated by Mr. Lafferty

Now try N5 TJ Ex20.1 Q4 to Q10

Ch 20 (Page 198)

ww

w.m

ath

srevis

ion

.com

Solving Trig EquationsSolving Trig Equations

National 5

ww

w.m

ath

srevis

ion

.com

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

created by Mr. Laffertycreated by Mr. Lafferty

Now try N5 TJ Ex20.1 Q4 onwards

Ch 20 (Page 198)

ww

w.m

ath

srevis

ion

.com

Solving Trig EquationsSolving Trig Equations

National 5

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

Starter QuestionsStarter Questionsw

ww

.math

srevis

ion

.com

2

1. I f lines have the same gradient

2. Factorise x +4x -12

3. Find the missing angles.

Nat 5

54o

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

w.m

ath

srevis

ion

.com Learning IntentionLearning Intention Success CriteriaSuccess Criteria

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.

C

B

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

Cosine RuleCosine Rulew

ww

.math

srevis

ion

.com

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

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

Cosine RuleCosine Rulew

ww

.math

srevis

ion

.com

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

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

ww

.math

srevis

ion

.com

Nat 5 Works for any Triangle

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

ww

w.m

ath

srevis

ion

.com

Nat 5 Works for any Triangle

E

D

F

e

d

f

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

ww

.math

srevis

ion

.com

Nat 5

Finding Angles Finding Angles Using The Cosine RuleUsing The Cosine Rule

Works for any Triangle A

BC

c

a

b

created by Mr. Laffertycreated by Mr. Lafferty

Now try N5 TJ Ex20.2

Ch 20 (Page 200)

ww

w.m

ath

srevis

ion

.com

Solving Trig EquationsSolving Trig Equations

National 5

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

ww

w.m

ath

srevis

ion

.com

National 5

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

ww

w.m

ath

srevis

ion

.com

Solving Trig Equations Solving Trig Equations National 5

created by Mr. Laffertycreated by Mr. Laffertyww

w.m

ath

srevis

ion

.com

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

created by Mr. Laffertycreated by Mr. Laffertyww

w.m

ath

srevis

ion

.com

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

cos2xo = 2516

cos2xo = 1 - sin2xo

Given that sin xo = .35

Find cos xo .

cos2xo = 1 - 35(

(2

cos2xo = 1 - 925

√ cosxo = 54

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

= sinxsin2x + sinxcos2x

= sinx(sin2x + sinxcos2x)

(sin2x + cos2x) = 1

LHS

= sinx = RHS

sinx cosx

cosx

sinx =

sinx

sinx =

= 1

LHS 1 – sin2A = cos2A

sin2A

cos2A =

= tan2A = RHS

created by Mr. Laffertycreated by Mr. Lafferty

Now try N5 TJ Ex20.3

Ch 20 (Page 201)

ww

w.m

ath

srevis

ion

.com

Solving Trig EquationsSolving Trig Equations

National 5

ww

w.m

ath

srevis

ion

.com

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.