Chapter 5 - Trig Graphs and Equations NOTES.notebook

[Pages:6]Chapter 5 Trig Graphs and Equations NOTES.notebook

September 13, 2018

Starter

Chapter 5 - Trigonometry: Graphs and Equations

Period and Amplitude Today's Learning:

By the end of this lesson, you should be able to recognise sin, cos and tan graphs and understand terms such as period and amplitude.

A graph which has a repeated pattern is described as periodic. The horizontal extent of the basic graph is called the period. Half of the vertical extent is called the amplitude.

Period = 360o Amplitude = 1

Period = 360o Amplitude = 1

Period = 180o

Amplitude = cannot be measured

More Trig Graphs

Today's Learning: By the end of this lesson, you should be able to recognise and sketch

a trig graph in the form y = bsinax + c.

Examples

1) Sketch the graph y = 2 sin 3x

3 2 1

0

180o

-1 -2

360o

Now sketch the graph y = 2 sin 3x + 1.

Now sketch the graph y = 2 sin 3x + 1.

3

1

0

180o

-1

The period does not change. The amplitude does not change. The maximum value is now 3. The minimum value is now -1.

2) Sketch the graph y = sin (x 30).

Page 54

1 360o

Ex 4B

0 30o 180o 210o 360o Select questions from 1, 2 and 3

-1

Chapter 5 Trig Graphs and Equations NOTES.notebook

September 13, 2018

For each of the following: a) sketch the graph b) make at least two comments about the graph.

1) y = sinx + 2 2) y = 1 cos x

3) y = 2 sin 3x + 3 4) y = 3 cos 2x 1

Starter

State the trigonometric function of each of the following graphs:

a)

b)

c)

Radians

Today's Learning: By the end of this lesson, you should be able to re write angles measured in degrees to angles measured in radians.

We can measure angles in degrees or radians.

xo Arc length =360 x d

y r =360 x 2r

1 radian (y) r

r

r

r x 360 = y x 2r

Circumference is 2r

y = r x 360 2r

1 radian = 180

radians = 180o

To convert between degrees and radians:

Examples

1) Degrees Radians

180o

90o 2

45o 4 3

135o 4

2) Radians Degrees

180o 9 20o

5 9

100o

2 3

120o

ONES TO REMEMBER!

10o = 18

20o, 30o, 40o, 50o, 60o, .....

20o = 9

20o, 40o, 60o, 80o, 100o, .....

45o = 4

90o, 135o, 180o, 225o, 315o

Exact Values Today's Learning: By the end of this lesson, you should be able to find exact values of

sin, cos and tan.

Draw a sketch of an equilateral triangle with length 2 units. Fill in everything you know about this triangle.

Now split it in half and fill in everything you can.

Chapter 5 Trig Graphs and Equations NOTES.notebook

September 13, 2018

Draw a sketch of an isosceles right angled triangle with the two equal sides 1 unit long.

Fill in everything you know about this triangle.

Exact Values

You can use these two triangles to help you find exact values. MEMORISE THEM!

30o

36

2

3 60o

1

1

45o

2

4 45o

1

Table of Exact Values

Angles are normally measured anti clockwise from zero. Negative angles are normally measured clockwise from zero.

To find an exact value, first write the angle in terms of its associated acute angle.

Examples

Find the exact value of

1) sin 300o

S

A

T

C

2) cos (135)o

S

A

T

C

3) sin 5 3

S

A

T

C

Chapter 5 Trig Graphs and Equations NOTES.notebook

September 13, 2018

4) sin (135)o

Ex 4E Q1

S

A

T

C

Solving Problems using exact values

Today's Learning: By the end of this lesson, you should be able to use exact values to solve problems

Examples

1) Find the exact length of the side marked s in the diagram.

3

s

12 cm

sin x = opp hyp

sin 3

=

12 s

3 2

=

12 s

3 s = 12 2

3 s = 24

s = 24 3

= 24 x 3

3

3

= 24 3 3

= 8 3 cm

2) Calculate the exact length of the side marked x in the triangle

below: x

opp sin x =

hyp

60o 10m

x sin 60o =

10 x = 10 sin 60o

Ex 4E Q3 Q4

= 10 x 3 2

= 5 3 cm

Starter

x + 5 1. Simplify:

x2 - 25

1 2. Express with a rational denominator

2 + 5

3. Factorise

9x2 - 6x - 8

Today's Learning: By the end of this lesson, you should be able to solve more complex trig equations, including those in radians.

Examples

Trig Equations

1) Solve 3 tanx = 1 for tanx = 1 3

0 x 360o

SA

TC

x = 150o, 330o

2) Solve 4cos2x = 2

for

0 x 360o 0 2x 720o

SA TC

2x = 120o, 240o, 480o, 600o x = 60o, 120o, 240o, 300o

Chapter 5 Trig Graphs and Equations NOTES.notebook

3) Solve 2sin 4x + 3 = 0

2sin 4x = 3 sin 4x = 3 2

for 0 x 360o 0 4x 1440o

S

A

T

C

September 13, 2018

Starter

PP 2007 Calculator

4x = 240, 300, 600, 660, 960, 1020, 1320, 1380 x = 60o, 75o, 150o, 165o, 240o, 255o, 330o, 345o Ex 4H - Page 63 Q1

Fully factorise

1) d2 d 2

2) cos2x cosx 2

3) 2y2 + 5y 12

4) 2sin2x 5sinx 12

5) 18t2 + 33t 6

6) 18cos2x + 33cosx 6

4) Solve tan2x = 3

for 0 x 360

tanx = + 3

S

A

T

C

x = 60o, 120o, 240o, 300o

5) Solve 3tan2x = 1

for 0 x 2

tan2x = 1 3

S

A

tan x = +

1 3

T

C

x

=

6

,

5 , 6

7, 6

11 6

6) Solve 12cos2x 5cos x 2 = 0 for

(3cos x 2)(4cos x + 1) = 0

Either

3cos x 2 = 0

cos

x

=

2 3

0 x

SA

T

C

x = 0.84 , 5.44

Chapter 5 Trig Graphs and Equations NOTES.notebook

or 4cos x + 1 = 0 cos x = 1 4

SA

1.

TC

2.

x = 4.46 or 1.82 x = 0 .84, 5 . 44, 1. 82, 4 . 46

x = 0. 84, 1. 82

September 13, 2018

Starter

Today's Learning: By the end of this lesson, you should be able to solve compound angle trig equations.

Compound Angle Trig Equations

Examples

1) 2sin(2x + 30) + 1 = 0

0 x 360o

2sin(2x + 30) = 1 sin(2x + 30) = 1 2

0 2x 720o 30o 2x + 30 750o

S

A

T

C

2x + 30 = 210o, 330o, 570o, 690o 2x = 180o, 300o, 540o, 660o x = 90o, 150o, 270o, 330o

2) Solve

2sin(2x

3

)

=

1

sin(2x ) = 1 32

0 x 2

0 2x 4

3

2x

3

11 3

SA

TC

2x

3

=

6

,

56, 136, 176

2x

=

36,

7 6

, 15 , 19 66

x

=

3 12

,

7 12

,15 12

,

19 12

x =

4

, 7 ,15,19 12 12 12

3)

Solve 2cos(2x 100) + 1 = 2

0 x 360o

cos(2x

100)

=

1 2

0 2x 720o

100o 2x 100 620o

SA

TC

2x 100 = 60, 60, 300, 420,

2x = 40, 160, 400, 520

x = 20o, 80o, 200o, 260o

4)

Solve

3cos(2x

6

)

=

2

cos(2x

6

)

=

2 3

0 x

0 2x 2

6

2x

6

11 6

SA

TC

Ex 4I - page 65

Q1 a) non calc b) calc

Q2, Q3 calc

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