Warm-up:

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Warm-up

• Find the six trig ratios for a 240 angle.

2
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?

3
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?

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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
  p.
• Radians are unitless. No unit or symbol is used.

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Degree/Radian Conversions
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Converting back and forth

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

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Learning the standard angles in radians
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45º- 45º- 90º
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30º- 60º- 90º
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The Unit Circle
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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.

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Evaluate each expression
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Arc Length(s)

• s r?
• ? must be a central angle measured in radians

15
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

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Area of a Sector

• s ½ r2?
• ? must be a central angle measured in
  radians

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

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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.

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Homework

• p 348 17 55 odd and 59.
• Portfolio 6 due on Thursday
• Unit 7 Test probably next Tuesday

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Homework
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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?

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What does it mean for a function to be periodic?
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Periodic Functions

• If the values of a function are repeated over
  each given interval of the domain, the function
  is said to be PERIODIC.

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What do we know about sine and cosine?
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Sine and Cosine as Functions

• Lets graph sine!

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Properties of the sine function

• Period
• Domain
• Range
• x-intercepts
• y-intercept
• Maximum value
• Minimum value

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Using the graph to determine a function value

• Find using the graph of the sine
  function.

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Using the graph to determine a function value

• Find all the values of ? for which
  .

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Using the graph to determine a function value
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Using the graph to determine a function value
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Now lets graph cosine!
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Properties of the cosine function

• Period
• Domain
• Range
• x-intercepts
• y-intercept
• Maximum value
• Minimum value

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How are sine and cosine alike? Different?
36
Using the graph to determine a function value

• Find

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Using the graph to determine a function value
38
How do I tell whether a graph is a sine or cosine
function?
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Using sine and cosine functions

• p 365 53

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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

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Homework

• P 363 13 39 odd, 53 and 55
• Portfolio 6 due Thursday.
• Unit 6 reassessments due on Friday.
• Unit 7 Test Tuesday.

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Homework
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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
• 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?

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Lets sketch our functions
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Lets graph y sin x on our calculatorsin
radians!
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Check y cos x in degrees!
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Amplitude
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Lets move the constant
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Period
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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.

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

• A
• Period or
• Frequency

53
State the amplitude, period and frequency for
each function then sketch the graph.

• A
• Period or
• Frequency

54
State the amplitude, period and frequency for
each function then sketch the graph.

• A
• Period or
• Frequency

55
Okay, think about this

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

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Build your own function

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

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Build your own function

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

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Now build the equationfrom a graph!

• p 374

59
Group Work

• You will receive cards with 3 different
  categories
• Type of graph sine or cosine
• Amplitude and Reflection about x-axis
• Period
• Choose one card from each category.
• Build an equation that meets the specifications.
• Sketch the graph.

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Homework

• P 373 17 53 odd, 57, 59
• Quiz!

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Warm-up
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Homework
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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?

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Adding or Subtracting a Constant from the
Function
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Lets sketch a few
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What if we have a constant inside the function
with ??
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Sketch some
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then put it all together!
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Build an equation
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Compound Functions

• The sum or products of trig functions.

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Homework

• P383 15 41 odd
• Quiz!
• Test! Tuesday

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Warm-up

• Graph 2 periods of each

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Homework
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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?

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Can trig functions be used to model real world
situations?

• Of course! Would have been a mighty short
  section if they couldnt!
• When would I use them?
• Whenever data shows fairly strong periodic
  behavior of some kind, try to fit it to a Trig
  Function.

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How would I translate data into a function?
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How accurate will my predictions be?
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Lets Do It!!!
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Homework

• p 391 to 393 7, 9, 11, 15

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Homework
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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?

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The Tangent Function

• Period
• Domain
• Range
• X-intercepts (zeros)
• Y-intercept
• Asymptotes

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Lets graph a couple
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The Cotangent Function

• Period
• Domain
• Range
• X-intercepts (zeros)
• Y-intercept
• Asymptotes

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Graph one
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The Cosecant Function

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

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Try this
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The Secant Function

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

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Last one
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Homework

• P 400 13 41 odd and 47
• Unit 7 Test Tuesday
• Portfolio 7 due on Friday