Created by Mr. Lafferty Graphs of the form y = a sin x o Trigonometry Graphs Int 2 Graphs of the form y = a sin bx o Phase angle

created by Mr. Laffertycreated by Mr. Lafferty

Graphs of the form y = a sin xo

Trigonometry Graphs Trigonometry Graphs w

ww

.math

srevis

ion

.com

Int 2

Graphs of the form y = a sin bxo

Phase angle

Solving Trig Equations

Special trig relationships


StarterStarter

2 2x +7x +6

Q3. Solve (2x -1)(x -1) = 0

1. Factorise the following.

2. A TV is reduced by 20% to £200.

What was the original price.

ww

w.m

ath

srevis

ion

.com

Int 2


Learning IntentionLearning Intention Success CriteriaSuccess Criteria

1.1. Identify the key points Identify the key points for various graphs.for various graphs.

1. To investigate graphs of the form

y = a sin xo

y = a cos xo

y = tan xo

ww

w.m

ath

srevis

ion

.com

Int 2

Sine Graph Sine Graph

created by Mr. Laffertycreated by Mr. Laffertyww

w.m

ath

srevis

ion

.com

Int 2Sine Graph Sine Graph

Key Features

Domain is 0 to 360o

(repeats itself every 360o)

Maximum value of 1

Minimum value of -1

Key Features

Zeros at 0, 180o and 360o

Max value at x = 90o

Minimum value at x = 270o


w.m

ath

srevis

ion

.com

Int 2


1

2

3

-3

-2

-1

090o 180o 270o 360o

y = sinxo

y = 2sinxo

y = 3sinxo

y = 0.5sinxo

y = -sinxo

What effect does the

number at the front have on the graphs ?


Sine Graph Sine Graph w

ww

.math

srevis

ion

.com

Int 2

y = a sin (x)

For a > 1 stretches graph in the y-axis direction

For a < 1 compresses graph in the y - axis direction

For a - negative flips graph in the x – axis.


w.m

ath

srevis

ion

.com

Int 2


2

4

6

-6

-4

-2

090o 180o 270o 360o

y = 5sinxo

y = 4sinxo

y = sinxo

y = -6sinxo


w.m

ath

srevis

ion

.com

Int 2

Cosine Graphs Cosine Graphs

Key Features

Key Features

Domain is 0 to 360o


Maximum value of 1

Minimum value of -1

Zeros at 90o and 270o

Max value at x = 0o and 360o

Minimum value at x = 180o


w.m

ath

srevis

ion

.com

Int 2

CosineCosine

1

2

3

-3

-2

-1

090o 180o 270o 360o

y = cosxo

y = 2cosxo

y = 3cosxo

y = 0.5cosxo

y = -cosxo




w.m

ath

srevis

ion

.com

Int 2

Cosine Graph Cosine Graph

2

4

6

-6

-4

-2

090o 180o 270o 360o

y = cosxo

y = 4cosxo

y = 6cosxo

y = cosxo

y = -cosxo


w.m

ath

srevis

ion

.com

Int 2

Tangent Graphs Tangent Graphs

Key Features

Key Features

Domain is 0 to 180o


Zeros at 0 and 180o


w.m

ath

srevis

ion

.com

Int 2

Tangent Graphs Tangent Graphs


Tangent Graph Tangent Graph w

ww

.math

srevis

ion

.com

Int 2

y = a tan (x)





Revision BookletAll questions

ww

w.m

ath

srevis

ion

.com

Int 2

Combination GraphsCombination Graphs


StarterStarter

6

+ 36x

4 w3.

w 2w

4 - 2 5

2

1. Calculate y (y +y )

2. Factorise 4ab

ww

w.m

ath

srevis

ion

.com

Int 2



1.1. Identify the key points Identify the key points for more complicated for more complicated Trig graphs.Trig graphs.

1. To investigate graphs of the form

y = a sin bxo

y = a cos bxo

y = tan bxo

ww

w.m

ath

srevis

ion

.com

Int 2

Trig Graphs Trig Graphs


w.m

ath

srevis

ion

.com

Int 2

When a pattern repeats itself over and over, it is said to be periodic.

Sine function has a period of 360o

Period of a FunctionPeriod of a Function

Let’s investigate the function

y = sin bx


w.m

ath

srevis

ion

.com

Int 2


1

2

3

-3

-2

-1

090o 180o 270o 360o

y = sinxo

y = sin2xo

y = sin4xo

y = sin0.5xo


number in front of x have on the graphs ?


w.m

ath

srevis

ion

.com

Int 2

Trigonometry Graphs Trigonometry Graphs

y = a sin (bx)




How many times it repeats

itself in 360o


w.m

ath

srevis

ion

.com

Int 2

CosineCosine

1

2

3

-3

-2

-1

090o 180o 270o 360o

y = cosxo

y = cos2xo

y = cos3xo




w.m

ath

srevis

ion

.com

Int 2


y = a cos (bx)





itself in 360o


w.m

ath

srevis

ion

.com

Int 2


y = a tan (bx)





itself in 180o

y = 0.5sin2xo

y = 2sin4xo

y = 3sin0.5xo


w.m

ath

srevis

ion

.com

Int 2

Trig Graph Trig Graph

1

2

3

-3

-2

-1

090o 180o 270o 360o

Write down equations for

graphs shown ?

Combinations


w.m

ath

srevis

ion

.com

Int 2

CosineCosine

1

2

3

-3

-2

-1

090o 180o 270o 360o

Write down equations for the graphs shown?

Combinations

y = 1.5cos2xo

y = -2cos2xo

y = 0.5cos4xo



ww

w.m

ath

srevis

ion

.com

Int 2



StarterStarter

2

2

3 12 ÷

s s

3. Sketch the f unction y = (x +5) + 1

1. Make x the subject of the formula

4 y =

x

2.

ww

w.m

ath

srevis

ion

.com

Int 2



1.1. Understand the term Understand the term phase angle / phase shift.phase angle / phase shift.

2. Read off the values for a and b for a graph of the form.

y = a sin( x – c )o

1. To explain what phase angle / phase shift is using knowledge from quadratics.

ww

w.m

ath

srevis

ion

.com

Int 2

Phase Angle Phase Angle


w.m

ath

srevis

ion

.com

Int 2


1

-1

090o 180o 270o 360o

y = sin(x - 45)o

45o

To the right “-”45o

By how much do we have to move the

standard sine curve so it fits on the other sine

curve?


w.m

ath

srevis

ion

.com

Int 2


1

-1

0 90o 180o 270o 360o-60o

y = sin(x + 60)o

To the left “+”60o


standard sine curve so it fits on the other sine

curve?


w.m

ath

srevis

ion

.com

Int 2

Phase Angle Phase Angle

y = sin (x - c)

For c > 0 moves graph to the right along x – axis

For c < 0 moves graph to the left along x – axis

Moves graph along x - axis

70o


w.m

ath

srevis

ion

.com

Int 2


1

-1

090o 180o 270o 360o

y = cos(x - 70)o

160o

To the right “-”


standard cosine curve so it fits on the other

cosine curve?

56o


w.m

ath

srevis

ion

.com

Int 2


1

-1

090o 180o 270o 360o

y = cos(x + 56)o

34o

To the left “+”


standard cosine curve so it fits on the other

cosine curve?


w.m

ath

srevis

ion

.com

Int 2

Summary of work So farSummary of work So far

y = a sin (x - b)

For b > 0 moves graph to the right along x – axis

For b < 0 moves graph to the left along x – axis





w.m

ath

srevis

ion

.com

Int 2

Sketch GraphSketch Graphy = a cos (x – b)

a =3 b =30y = 2 cos (x - 30)



ww

w.m

ath

srevis

ion

.com

Int 2



StarterStarter

2

x + 6x + 2

3. Sketch the f unction y = 2sin4x

1. Make b the subject of the formula

b c =

a

2. Use the quadratic formula to solve

ww

w.m

ath

srevis

ion

.com

Int 2



1.1. Use the rule for solving Use the rule for solving any ‘ normal ‘ equation any ‘ normal ‘ equation

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

1. To explain how to solvetrig equations of the form

a sin xo + 1 = 0

ww

w.m

ath

srevis

ion

.com

Int 2

Solving Trig Equations Solving Trig Equations


w.m

ath

srevis

ion

.com

Int 2


All +veSin +ve

Tan +ve Cos +ve

180o - xo

180o + xo 360o - xo

1 2 3 4


w.m

ath

srevis

ion

.com

Int 2

Solving Trig EquationsSolving Trig Equationsa sin xo + b = 0

Example 1 :

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


w.m

ath

srevis

ion

.com

Int 2

Solving Trig EquationsSolving Trig Equationsa sin xo + b = 0

Example 1 :

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


trying to solve

sin xo = -1/3

Calculate first Quad valuexo = 19.5o

x = 180o + 19.5o = 199.5o

( 360o - 19.5o = 340.5o)


1 2 3 4


w.m

ath

srevis

ion

.com

Int 2

Solving Trig EquationsSolving Trig Equationsa cos xo + b = 0

Example 1 :

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


trying to solve

cos xo = 0.625

xo = 51.3o

(360o - 53.1o = 308.7o)

xo = cos -1 0.625


1 2 3 4


w.m

ath

srevis

ion

.com

Int 2

Solving Trig EquationsSolving Trig Equationsa tan xo + b = 0

Example 1 :

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


trying to solve

tan xo = 2

xo = 63.4o

x = 180o + 63.4o = 243.4o

xo = tan -1(2)


1 2 3 4


Now try MIA Ex6 First Column Only

(page 249)

ww

w.m

ath

srevis

ion

.com

Int 2

Solving Trig EquationsSolving Trig Equations


StarterStarter

2

x + 5x + 1

3. Sketch the f unction y = 4sin3x

1. Make a the subject of the formula

b c =

a

2. Use the quadratic formula to solve

ww

w.m

ath

srevis

ion

.com

Int 2



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

Int 2



w.m

ath

srevis

ion

.com

Int 2


Lets investigate

sin 2xo + cos 2 xo = ?

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

sin 2xo + cos 2 xo = 1

Learn !


w.m

ath

srevis

ion

.com

Int 2


Lets investigate

tan xo

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

Learn !

sin xo cos xo

and

tan xo sin xo cos xo =


Now try MIA Ex7

(page 252)

ww

w.m

ath

srevis

ion

.com

Int 2


Documents

Created by Mr. Lafferty Graphs of the form y = a sin x o Trigonometry Graphs Int 2 Graphs of the form y = a sin bx o Phase angle