3 2 3sin( 240 ) csc( 240 )

2 31

cos( 240 ) sec( 240 ) 22

3tan( 240 ) 3 cot( 240 )

3

Warm-up:

Find the six trig ratios for a –240˚ angle.

Unit 7: “A Little Triggier…” Chapter 6: Graphs of Trig Functions

In this chapter we will answer…

What exactly is a radian? How are radians related to degrees?

How do I draw and use the graphs of trig functions and their inverses?

What do I do to find the amplitude, period, phase shift and vertical shift for trig functions?

When trig functions be used to model a given situation?

7.1: find exact values of trigonometric functions (6-1)7.2: find length of intercepted arcs and area of sectors (6-1)

In this section we will answer… What exactly is a radian and why the pi? Can I switch between radians and degrees? If they both measure angles why do I need to

learn radians at all? How can I determine the length of an arc and the

area of a sector?

What exactly is a radian and why the pi? What is a degree?

Radians are based on the circumference of the circle.

Radian measurements are usually shown in terms of π.

Radians are unitless. No unit or symbol is used.

Degree/Radian Conversions

1801 radian

or approximately 57.3

1 degree 180

or approximately 0.017 radians

p

p

°=

°

=°

Converting back and forth…

Change 115º to a radian measure in terms of pi.

Change radians to degree measure. 78p

Learning the standard angles in radians:

45º- 45º- 90º

30º- 60º- 90º

The Unit Circle

Finding Trig Ratios with Radian Measures: Memorize the radian measures.

Force yourself to think in and recognize radian measure without having to convert to degrees.

Evaluate each expression:

7sin

3tan5

3cos

4

p

p

p

1 sin 2 cos 3 tan 4 csc 5 sec 6 cot

111 0 9 17 2 25

27 2

2 10 18 266 6 6 3

53 11 19

4 4

pp p

p p p p

p p

= = = = = =

= = = = -

= = = - = -

= = 27 64

4 3 134 12 20 28

3 3 4 63 22

5 13 21 292 2 62 5 11 13

6 14 22 303 3 6 43

7 4

pp

p p p p

p p pp

p p p p

p

= - = -

= = = - = -

= = = - =

= = = - = -

=7

15 23 54

5 11 178 16 24

6 6 6

pp

p p p

= = -

= = =

Arc Length(s):

s = rθ

θ must be a central angle measured

in radians

θ

sr

Try one…

The Swiss have long been highly regarded as the makers of fine watches. The central angle formed by the hands of a watch on “12” and “5” is 150º. The radius of the minute hand is cm. Find the distance traversed by the end of the minute hand to the nearest hundredth of a cm.

1.96 cm

34

Area of a Sector:

s = ½ r2θ

θ must be a central angle measured

in radians

θ

sr

Find the area of the sector with the following central angle and radius:

7, 3 cm

12

270 , 5 in

r

r

pq

q

= =

= ° =

A sector has an arc length of 15 feet and a central angle of radians.

Find the radius of the circle.

Find the area of the sector.

34p

A Mechanics Problem:

A single pulley is being used to pull up a weight. Suppose the diameter of the pulley is 2.5 feet.

How far will the weight rise if the pulley turns 1.5 rotations?

Find the number of degrees the pulley must be rotated to raise the weight 4.5 feet.

Homework:

p 348 #17 – 55 odd and 59.

Portfolio 6 due on Thursday

Unit 7 Test probably next Tuesday

1 sin 2 cos 3 tan 4 csc 5 sec 6 cot11

1 0 9 17 2 252

7 22 10 18 26

6 6 6 35

3 11 194 4

pp p

p p p p

p p

= = = = = =

= = = =-

= = =- =-

= = 27 64

4 3 134 12 20 28

3 3 4 63 22

5 13 21 292 2 62 5 11 13

6 14 22 303 3 6 4

37

4

pp

p p p p

p p pp

p p p p

p

=- =-

= = =- =-

= = =- =

= = =- =-

=7

15 23 54

5 11 178 16 24

6 6 6

pp

p p p

= =-

= = =

Homework:

7.3: use the language of trigonometric graphing to describe a graph (6-3)7.4: graph sine and cosine functions from equations (6-3)

In this section we will answer…

What does it mean for a function to be periodic? How do we determine the period of a function? How are sine and cosine functions alike? Different? How can I use a periodic graph to determine the

value of the function for a particular domain value? How do I tell whether a graph is a sine or cosine

function?

What does it mean for a function to be periodic?

Periodic Functions: If the values of a function are repeated

over each given interval of the domain, the function is said to be PERIODIC.

A f unction is periodic if , f or some real number ,

( ) ( ) f or each in the domain of .

The smallest positive value of f or which

( ) ( ) is the period of the f unction.

f x f x x f

f x f x

a

a

a

a

+ =

+ =

What do we know about sine and cosine?

Sine and Cosine as Functions:

Let’s graph sine!

sin cosy x y x= =

let 2 2 , in increments of 4

xp

p p- £ £

Properties of the sine function: Period:

Domain:

Range:

x-intercepts:

y-intercept:

Maximum value:

Minimum value:

Using the graph to determine a function value:

Find using the graph of the sine function.

sin3p

Using the graph to determine a function value:

Find all the values of θ for which .

sin 1q= -

Using the graph to determine a function value:Graph sin f or 3 x 5y x p p= £ £

Using the graph to determine a function value:

7 9Graph sin f or x

2 2y x

p p= £ £

Now let’s graph cosine!

Properties of the cosine function: Period:

Domain:

Range:

x-intercepts:

y-intercept:

Maximum value:

Minimum value:

How are sine and cosine alike? Different?

Using the graph to determine a function value: Find cos

2p

Using the graph to determine a function value:Graph cos f or 3 x 5y x p p= £ £

How do I tell whether a graph is a sine or cosine function?

Using sine and cosine functions:

p 365 #53

Partner Work:

All work done on one piece of paper. 1st person solves a problem. The 2nd person coaches or encourages as

needed. When the 2nd person agrees with the solution they initial the problem.

Now 2nd person solves and 1st coaches, encourages and initials.

p 363 #1-12 all

Homework: P 363 #13 – 39 odd, 53 and 55

Portfolio 6 due Thursday.

Unit 6 reassessments due on Friday.

Unit 7 Test Tuesday.

1 sin 2 cos 3 tan 4 csc 5 sec 6 cot

111 0 9 17 2 25

22

2 10 390 18 266 6 3

53 11 1

4 4

pp p

p p p

p p

= = = = = =

= ° = = = -

= = ° = - = -

= = 9 45 27 6

4 34 60 12 20 28 210

3 422

5 13 270 21 292 62 5 11

6 14 22 30 5853 3 6

7 135

p

p p

p pp

p p p

= - ° = -

= ° = = - = - °

= = ° = - =

= = = - = - °

= 15 315 23 5

5 118 16 24 780

6 6

p

p p

° = ° = -

= = = - °

Homework:

7.3: use the language of trigonometric graphing to describe a graph (6-4)7.4: graph sine and cosine functions from equations (6-4)

In this section we will answer…In this section we will answer… Can the period of a function change? How can I determine the period of a function

from its equation? What is amplitude? What causes a change in amplitude? If I know the type of function, its period and

amplitude, how do I find the equation? Can I find the equation for a function from just its

graph?

Let’s sketch our functions…

Let’s graph y = sin x on our calculators…in radians!

Check y = cos x …in degrees!

Amplitude:

For the f unctions:

will be the amplitude of the f unction.

sin and cos

A

y A y Aq q= =

Let’s move the constant…

Period

For the f unctions:

2Period

sin and cos

k

y k y k

p

q q

=

= =

Did you know? Frequency is related to period.

Period is the amount of time to complete one cycle. Frequency is the number of cycles per unit of time.

1 1Period Frequency

f requency Period= =

State the amplitude, period and frequency for each function then sketch the graph.

2siny q= A =

Period = or

Frequency =

State the amplitude, period and frequency for each function then sketch the graph.

2cos4

yq

= A =

Period = or

Frequency =

State the amplitude, period and frequency for each function then sketch the graph.

8sin2y q= A =

Period = or

Frequency =

Okay, think about this…

3siny q= -

A negative multiplying the function will reflect the function about the x-axis.

Build your own function…

Write the equation of the sine function with the given amplitude and period.

2

Period 4

A

p

=

=

Build your own function…

Write the equation of the cosine function with the given amplitude and period.

6

2Period

3

A

p

=

=

Now build the equation…from a graph! p 374

Group Work: You will receive cards with 3 different

categories:Type of graph: sine or cosineAmplitude and Reflection about x-axis:Period:

Choose one card from each category. Build an equation that meets the

specifications. Sketch the graph.

Homework:

P 373 #17 – 53 odd, 57, 59

Quiz! Quiz!

Warm-up:

Homework:

7.3: use the language of trigonometric graphing to describe a graph (6-5)7.4: graph sine and cosine functions from equations (6-5)

In this section we will answer… Can we shift our functions vertically?

Horizontally? If I move a function horizontally how do I

tell whether it is sine or cosine? What is a compound function? How do I

sketch one?

Adding or Subtracting a Constant from the Function:

For sin and cos

will cause a vertical shif t

in the same direction as its sign.

y z y z

z

q q= + = +

Let’s sketch a few…

What if we have a constant inside the function with θ?

For sin( ) and cos( )

will cause a horizontal shif t

in the opposite direction as its sign.

y c y c

c

q q= - = -

Sketch some…

…then put it all together! ( ) sin ( )

and

( ) cos ( )

( ) refl ection

2 360Amplitude Period

Phase Shif t Vertical Shif t

y A k c z

y A k c z

Ak k

c z

q

q

p

= - - +

= - - +

- =

°= = =

= =

Build an equation:

Compound Functions:

The sum or products of trig functions.

siny x x

cos cos2y x x

Homework:

P383 #15 – 41 odd

Quiz!

Test! Tuesday

Warm-up:

Graph 2 periods of each:

2sin 3 in degrees

4cos(3 ) in radians

y

y

Homework:

7.5: use sine and cosine graphs to model real-world data (6-6)

In this section we will answer… Can trig functions be used to model real

world situations? How would I translate data into a function? How accurate will my predictions be?

Can trig functions be used to model real world situations? Of course! Would have been a mighty short

section if they couldn’t!

When would I use them?

Whenever data shows fairly strong periodic behavior of some kind, try to fit it to a Trig Function.

How would I translate data into a function?

Highest Value Lowest Value1. Amplitude

2

Highest Value + Lowest Value2. Vertical Shift

2

3. Period = How long until cycle begins to repeat.

24. (for sine and cosine)

periodk

5. Phase Shift = read off graph or plug everything in for one point

and then solve back to get "c".

How accurate will my predictions be?

Let’s Do It!!!

Homework:

p 391 to 393 # 7, 9, 11, 15

1 sin 2 cos 3 tan 4 csc 5 sec 6 cot

111 0 9 17 2 25

22

2 10 390 18 266 6 3

53 11 1

4 4

pp p

p p p

p p

= = = = = =

= ° = = = -

= = ° = - = -

= = 9 45 27 6

4 34 60 12 20 28 210

3 422

5 13 270 21 292 62 5 11

6 14 22 30 5853 3 6

7 135

p

p p

p pp

p p p

= - ° = -

= ° = = - = - °

= = ° = - =

= = = - = - °

= 15 315 23 5

5 118 16 24 780

6 6

p

p p

° = ° = -

= = = - °

Homework:

7.6: graph secant, cosecant, tangent, and cotangent functions from equations (6-7)

In this section we will answer… What about the other trig functions? How do they resemble sine and cosine?

How do they differ? How do I write equations based on the

other trig functions?

The Tangent Function

Period:

Domain:

Range:

X-intercepts (zeros):

Y-intercept:

Asymptotes:

Let’s graph a couple…

12tan 1 (in degrees)

3y

tan 23

y

The Cotangent Function Period:

Domain:

Range:

X-intercepts (zeros):

Y-intercept:

Asymptotes:

Graph one…

cot 22 4

y

The Cosecant Function

Period: Domain: Range: X-intercepts (zeros): Y-intercept: Asymptotes: Maximum: Minimum:

Try this…

3 csc(3 180 )y

The Secant Function

Period: Domain: Range: X-intercepts (zeros): Y-intercept: Asymptotes: Maximum: Minimum:

Last one…

3sec 2y

Homework:

P 400 #13 – 41 odd and 47

Unit 7 Test Tuesday

Portfolio 7 due on Friday