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Graphs of the form y = a sin x oPowerPoint Presentation

Graphs of the form y = a sin x o

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Int 2

Graphs of the form y = a sin xo

Graphs of the form y = a sin bxo

Phase angle

www.mathsrevision.com

Solving Trig Equations

Special trig relationships

created by Mr. Lafferty

Int 2

Learning Intention

Success Criteria

- Identify the key points for various graphs.

- To investigate graphs of the form
- y = a sin xo
- y = a cos xo
- y = tan xo

www.mathsrevision.com

created by Mr. Lafferty

Sine Graph

Zeros at 0, 180o and 360o

Max value at x = 90o

Int 2

Minimum value at x = 270o

Key Features

www.mathsrevision.com

Domain is 0 to 360o

(repeats itself every 360o)

Maximum value of 1

Minimum value of -1

created by Mr. Lafferty

What effect does the number at the front have on the graphs ?

y = sinxo

y = 2sinxo

y = 3sinxo

y = 0.5sinxo

y = -sinxo

Sine Graph

Int 2

3

2

1

0

www.mathsrevision.com

90o

180o

270o

360o

-1

-2

-3

created by Mr. Lafferty

Int 2

y = a sin (x)

www.mathsrevision.com

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.

created by Mr. Lafferty

y = 5sinx ?o

y = 4sinxo

y = sinxo

y = -6sinxo

Sine Graph

Int 2

6

4

2

0

www.mathsrevision.com

90o

180o

270o

360o

-2

-4

-6

created by Mr. Lafferty

Cosine Graphs

Zeros at 90o and 270o

Max value at x = 0o and 360o

Int 2

Minimum value at x = 180o

Key Features

www.mathsrevision.com

Domain is 0 to 360o

(repeats itself every 360o)

Maximum value of 1

Minimum value of -1

created by Mr. Lafferty

What effect does the number at the front have on the graphs ?

y = cosxo

y = 2cosxo

y = 3cosxo

y = 0.5cosxo

y = -cosxo

Cosine

Int 2

3

2

1

0

www.mathsrevision.com

90o

180o

270o

360o

-1

-2

-3

created by Mr. Lafferty

y = ?cosxo

y = 4cosxo

y = 6cosxo

y = cosxo

y = -cosxo

Cosine Graph

Int 2

6

4

2

0

www.mathsrevision.com

90o

180o

270o

360o

-2

-4

-6

created by Mr. Lafferty

Tangent Graphs

Zeros at 0 and 180o

Int 2

Key Features

www.mathsrevision.com

Domain is 0 to 180o

(repeats itself every 180o)

created by Mr. Lafferty

Int 2

y = a tan (x)

www.mathsrevision.com

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.

created by Mr. Lafferty

Int 2

Revision Booklet

All questions

www.mathsrevision.com

created by Mr. Lafferty

Int 2

Learning Intention

Success Criteria

- Identify the key points for more complicated Trig graphs.

- To investigate graphs of the form
- y = a sin bxo
- y = a cos bxo
- y = tan bxo

www.mathsrevision.com

created by Mr. Lafferty

Int 2

When a pattern repeats itself over and over,

it is said to be periodic.

Sine function has a period of 360o

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Let’s investigate the function

y = sin bx

created by Mr. Lafferty

What effect does the number in front of x have on the graphs ?

y = sinxo

y = sin2xo

y = sin4xo

y = sin0.5xo

Sine Graph

Int 2

3

2

1

0

www.mathsrevision.com

90o

180o

270o

360o

-1

-2

-3

created by Mr. Lafferty

Int 2

y = a sin (bx)

How many times

it repeats

itself in 360o

www.mathsrevision.com

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.

created by Mr. Lafferty

What effect does the number at the front have on the graphs ?

y = cosxo

y = cos2xo

y = cos3xo

Cosine

Int 2

3

2

1

0

www.mathsrevision.com

90o

180o

270o

360o

-1

-2

-3

created by Mr. Lafferty

Int 2

y = a cos (bx)

How many times

it repeats

itself in 360o

www.mathsrevision.com

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.

created by Mr. Lafferty

Int 2

y = a tan (bx)

How many times

it repeats

itself in 180o

www.mathsrevision.com

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.

created by Mr. Lafferty

Write down equations for graphs shown ? ?

y = 0.5sin2xo

y = 2sin4xo

y = 3sin0.5xo

Trig Graph

Combinations

Int 2

3

2

1

0

www.mathsrevision.com

90o

180o

270o

360o

-1

-2

-3

created by Mr. Lafferty

Write down equations for the graphs shown? ?

y = 1.5cos2xo

y = -2cos2xo

y = 0.5cos4xo

Cosine

Combinations

Int 2

3

2

1

0

www.mathsrevision.com

90o

180o

270o

360o

-1

-2

-3

created by Mr. Lafferty

Int 2

Revision Booklet

All questions

www.mathsrevision.com

created by Mr. Lafferty

Int 2

Learning Intention

Success Criteria

- Understand the term phase angle / phase shift.
- Read off the values
- for a and b for a graph of the form.
- y = a sin( x – c )o

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

www.mathsrevision.com

created by Mr. Lafferty

By how much do we have to move the standard sine curve so it fits on the other sine curve?

Sine Graph

Int 2

y = sin(x - 45)o

1

To the right “-”

45o

0

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45o

90o

180o

270o

360o

-1

created by Mr. Lafferty

By how much do we have to move the standard sine curve so it fits on the other sine curve?

Sine Graph

Int 2

y = sin(x + 60)o

1

To the left “+”

60o

0

www.mathsrevision.com

-60o

90o

180o

270o

360o

-1

created by Mr. Lafferty

Phase Angle fits on the other sine curve?

Int 2

y = sin (x - c)

Moves graph

along x - axis

www.mathsrevision.com

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

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

created by Mr. Lafferty

By how much do we have to move the standard cosine curve so it fits on the other cosine curve?

Cosine Graph

Int 2

y = cos(x - 70)o

1

To the right “-”

70o

0

160o

www.mathsrevision.com

90o

180o

270o

360o

-1

created by Mr. Lafferty

By how much do we have to move the standard cosine curve so it fits on the other cosine curve?

Cosine Graph

Int 2

y = cos(x + 56)o

1

To the left “+”

56o

0

34o

www.mathsrevision.com

90o

180o

270o

360o

-1

created by Mr. Lafferty

Summary of work So far it fits on the other cosine curve?

Int 2

y = a sin (x - b)

For a > 1 stretches graph

in the y-axis direction

For b > 0 moves graph to

the right along x – axis

For a < 1 compresses graph

in the y - axis direction

For b < 0 moves graph to

the left along x – axis

www.mathsrevision.com

For a - negative flips graph

in the x – axis.

created by Mr. Lafferty

Sketch Graph it fits on the other cosine curve?

y = a cos (x – b)

Int 2

a =3

b =30

y = 2 cos (x - 30)

www.mathsrevision.com

created by Mr. Lafferty

Combination Graphs it fits on the other cosine curve?

Int 2

Revision Booklet

All questions

www.mathsrevision.com

created by Mr. Lafferty

Solving Trig Equations it fits on the other cosine curve?

Int 2

Learning Intention

Success Criteria

- Use the rule for solving any ‘ normal ‘ equation
- Realise that there are many solutions to trig equations depending on domain.

- To explain how to solve
- trig equations of the form
- a sin xo + 1 = 0

www.mathsrevision.com

created by Mr. Lafferty

Solving Trig Equations it fits on the other cosine curve?

Int 2

Sin +ve

All +ve

180o - xo

180o + xo

360o - xo

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Cos +ve

Tan +ve

1

2

3

4

created by Mr. Lafferty

Solving Trig Equations it fits on the other cosine curve?

Graphically what are we trying to solve

a sin xo + b = 0

Int 2

Example 1 :

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

sin xo = (0.5)

xo = sin-1(0.5)

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xo = 30o

There is another solution

xo = 150o

1

2

3

4

(180o – 30o = 150o)

created by Mr. Lafferty

Solving Trig Equations it fits on the other cosine curve?

Graphically what are we trying to solve

a sin xo + b = 0

Int 2

Example 1 :

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

sin xo = -1/3

Calculate first Quad value

xo = 19.5o

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x = 180o + 19.5o = 199.5o

There is another solution

1

2

3

4

( 360o - 19.5o = 340.5o)

created by Mr. Lafferty

Solving Trig Equations it fits on the other cosine curve?

Graphically what are we trying to solve

a cos xo + b = 0

Int 2

Example 1 :

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

cos xo = 0.625

xo = cos -1 0.625

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xo = 51.3o

There is another solution

(360o - 53.1o = 308.7o)

1

2

3

4

created by Mr. Lafferty

Solving Trig Equations it fits on the other cosine curve?

Graphically what are we trying to solve

a tan xo + b = 0

Int 2

Example 1 :

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

tan xo = 2

xo = tan -1(2)

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xo = 63.4o

There is another solution

x = 180o + 63.4o = 243.4o

1

2

3

4

created by Mr. Lafferty

Solving Trig Equations it fits on the other cosine curve?

Int 2

Now try MIA Ex6

First Column Only

(page 249)

www.mathsrevision.com

created by Mr. Lafferty

Solving Trig Equations it fits on the other cosine curve?

Int 2

Learning Intention

Success Criteria

- Know and learn the two special trig relationships.
- Apply them to solve problems.

- To explain some special trig relationships
- sin 2 xo +cos 2xo = ?
- and
- tan xo and sin x
- cos x

www.mathsrevision.com

created by Mr. Lafferty

Solving Trig Equations it fits on the other cosine curve?

Int 2

Lets investigate

sin 2xo + cos 2 xo = ?

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

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Learn !

sin 2xo + cos 2 xo = 1

created by Mr. Lafferty

sin x it fits on the other cosine curve?o

sin xo

cos xo

cos xo

Solving Trig Equations

Int 2

Lets investigate

tan xo

and

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

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Learn !

tan xo

=

created by Mr. Lafferty

Solving Trig Equations it fits on the other cosine curve?

Int 2

Now try MIA Ex7

(page 252)

www.mathsrevision.com

created by Mr. Lafferty

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