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Trigonometry Graphs . Int 2. Graphs of the form y = a sin x o. Graphs of the form y = a sin bx o. Phase angle. www.mathsrevision.com. Solving Trig Equations. Special trig relationships. Int 2. Starter. www.mathsrevision.com. Sine Graph . Int 2. Learning Intention. Success Criteria.

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Presentation Transcript
slide1

Trigonometry Graphs

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

starter

Int 2

Starter

www.mathsrevision.com

created by Mr. Lafferty

slide3

Sine Graph

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

slide4

Key Features

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

slide5

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

slide6

Sine Graph

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

slide7

y = 5sinxo

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

slide8

Key Features

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

slide9

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

slide10

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

slide11

Key Features

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

slide12

Tangent Graphs

Int 2

www.mathsrevision.com

created by Mr. Lafferty

slide13

Tangent Graph

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

slide14

Combination Graphs

Int 2

Revision Booklet

All questions

www.mathsrevision.com

created by Mr. Lafferty

starter1

Int 2

Starter

www.mathsrevision.com

created by Mr. Lafferty

slide16

Trig Graphs

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

slide17

Period of a Function

Int 2

When a pattern repeats itself over and over,

it is said to be periodic.

Sine function has a period of 360o

www.mathsrevision.com

Let’s investigate the function

y = sin bx

created by Mr. Lafferty

slide18

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

slide19

Trigonometry Graphs

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

slide20

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

slide21

Trigonometry Graphs

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

slide22

Trigonometry Graphs

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

slide23

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

slide24

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

slide25

Combination Graphs

Int 2

Revision Booklet

All questions

www.mathsrevision.com

created by Mr. Lafferty

starter2

Int 2

Starter

www.mathsrevision.com

created by Mr. Lafferty

slide27

Phase Angle

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

slide28

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

www.mathsrevision.com

45o

90o

180o

270o

360o

-1

created by Mr. Lafferty

slide29

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

slide30

Phase Angle

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

slide31

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

slide32

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

slide33

Summary of work So far

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

slide34

Sketch Graph

y = a cos (x – b)

Int 2

a =3

b =30

y = 2 cos (x - 30)

www.mathsrevision.com

created by Mr. Lafferty

slide35

Combination Graphs

Int 2

Revision Booklet

All questions

www.mathsrevision.com

created by Mr. Lafferty

starter3

Int 2

Starter

www.mathsrevision.com

created by Mr. Lafferty

slide37

Solving Trig Equations

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

slide38

Solving Trig Equations

Int 2

Sin +ve

All +ve

180o - xo

180o + xo

360o - xo

www.mathsrevision.com

Cos +ve

Tan +ve

1

2

3

4

created by Mr. Lafferty

slide39

Solving Trig Equations

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)

www.mathsrevision.com

xo = 30o

There is another solution

xo = 150o

1

2

3

4

(180o – 30o = 150o)

created by Mr. Lafferty

slide40

Solving Trig Equations

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

www.mathsrevision.com

x = 180o + 19.5o = 199.5o

There is another solution

1

2

3

4

( 360o - 19.5o = 340.5o)

created by Mr. Lafferty

slide41

Solving Trig Equations

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

www.mathsrevision.com

xo = 51.3o

There is another solution

(360o - 53.1o = 308.7o)

1

2

3

4

created by Mr. Lafferty

slide42

Solving Trig Equations

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)

www.mathsrevision.com

xo = 63.4o

There is another solution

x = 180o + 63.4o = 243.4o

1

2

3

4

created by Mr. Lafferty

slide43

Solving Trig Equations

Int 2

Now try MIA Ex6

First Column Only

(page 249)

www.mathsrevision.com

created by Mr. Lafferty

starter4

Int 2

Starter

www.mathsrevision.com

created by Mr. Lafferty

slide45

Solving Trig Equations

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

slide46

Solving Trig Equations

Int 2

Lets investigate

sin 2xo + cos 2 xo = ?

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

www.mathsrevision.com

Learn !

sin 2xo + cos 2 xo = 1

created by Mr. Lafferty

slide47

sin xo

sin xo

cos xo

cos xo

Solving Trig Equations

Int 2

Lets investigate

tan xo

and

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

www.mathsrevision.com

Learn !

tan xo

=

created by Mr. Lafferty

slide48

Solving Trig Equations

Int 2

Now try MIA Ex7

(page 252)

www.mathsrevision.com

created by Mr. Lafferty