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Transcript Slide 1

Homework, Page 381

Identify the one angle that is not coterminal with the others.

210 , 450 , 870  150  510   360   150   210   360   150  450   360 150  870   360   510   360   150  Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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Homework, Page 381

Evaluate the six trigonometric functions of the angle

y 5.

 (-1, -1)

 x

2 

2  2  2 sin    1   2  csc   2 2 cos    1   2  2 2 tan    1  1 cot Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley  sec  1  1 2  1 2  1     2 2

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Homework, Page 381

Point P is on the terminal side of angle

. Evaluate the six trigonometric functions for

. If the function is undefined, write undefined.

 

  90   sin   2 csc   1 cos  tan  0 sec  undefined 0 undefined  cot  1 0

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Homework, Page 381

State the sign (+ or –) of (a) sin

, (b) cos t (c) tan

for values of

13.

 0,  2

     

sin cos tan

t t t

     

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Homework, Page 381

Determine the sign (+ or –) of the given value without a calculator.

17.

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Homework, Page 381

Choose the point on the terminal side of

21.

  45  (a) (2, 2) (b)   (c)   Choice (a) as tan 45º = 1.

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Homework, Page 381

Evaluate without using a calculator by using ratios in a reference triangle.

25. cos120  120   180 60 cos is negative in Quadrant II cos120    cos 60    1 2

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Homework, Page 381

Evaluate without using a calculator by using ratios in a reference triangle.

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Homework, Page 381

Evaluate without using a calculator by using ratios in a reference triangle.

33. cos 23  6 cos 23  6  cos 24   6 6  cos   6 Cosine is positive in Quadrant IV cos   6  2 3

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Homework, Page 381

Find (a) sin θ, (b) cos θ, and (c) tan θ for the given quadrantal angle. If the value is undefined, write undefined.

37.  450   450   360 90 360  

     

sin 270    1 cos 270   tan 270   0 undefined 270 

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Homework, Page 381

Find (a) sin θ, (b) cos θ, and (c) tan θ for the given quadrantal  angle. If the value is undefined, write undefined.

41.  7 2  7   4  3  2 2   2  4   2 2

     

sin cos tan  2   2 2    1 0 undefined

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Homework, Page 381

Evaluate without using a calculator.

45.    2 5 and cos   0

 5 2  2 2  21 sec  tan    1 cos   1 21  5  2 21   2 21 21 5 21  5 21 21

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Homework, Page 381

Evaluate by using the period of the function.

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Homework, Page 381

53. Use your calculator to evaluate the expressions in Exercises 49 – 52. Does your calculator give the correct answer. Many miss all four. Give a brief explanation why.

The calculator algorithms apparently do recognize large multiples of pi and end up evaluating at nearby values.

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Homework, Page 381

57. A weight suspended from a spring is set into motion. Its displacement d from equilibrium is

 0.4

cos 4

d is the displacement in inches and t is the time in seconds. Find the displacement at the given time.

(a) t = 0  0.4

  0.4 1 1  0.4

(b)

t = 3  0.4

    0.185

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Homework, Page 381

61. If θ is an angle in a triangle such that cos θ < 0, then θ is an obtuse angle. Justify your answer.

True. An obtuse angle in the standard position would have its terminal side in the second quadrant and cosine is negative in the second quadrant.

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Homework, Page 381

65. The range of the function

  

sin

a. [1] b. [-1, 1] c. [0, 1] d. [0, 2] e. [0, ∞] cos



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Homework, Page 381

Find the value of the unique real number θ between 0 and 2π that satisfies the two given conditions.

69. tan    1 and sin   0 If tan and sin are negative, cos must be positive. The angle must be in the fourth quadrant and the reference angle is π/4, so θ = 2π – π/4 = 7π/4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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4.4

Graphs of Sine and Cosine: Sinusoids

What you’ll learn about  The Basic Waves  Sinusoids and Transformations  Modeling Periodic Behavior with Sinusoids … and why Sine and cosine gain added significance when used to model waves and periodic behavior.

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Leading Questions  A function is a sinusoid if it can be written in the form

y = a

sin (

bx + c

) +

, where

and

≠ 0.

 The function

y = a

cos (

bx + c

) +

is not a sinusoid.

 The amplitude of a sinusoid is |

 The period of a sinusoid is |

|/2π.

 The frequency of a sinusoid is |

|/2π.

 Sinusoids are often used to model the behavior of periodic occurrences.

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Sinusoid A function is a

sinusoid

if it can be written in the form 

sin(

bx c d

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Amplitude of a Sinusoid The

amplitude



sin(



cos(

bx c d d

Graphically, the amplitude is half the distance between the trough and the crest of the wave .

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Example

Finding Amplitude

Find the amplitude of each function and use the language of transformations to describe how the graphs are related.

 sin

x y

  2sin

x y

 1 3 sin

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The

period

  is

  full cycle of the wave.

Period of a Sinusoid 

sin(

bx d

is 

cos(

bx c d

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Example

Finding Period and Frequency

Find the period and frequency of each function and use the language of transformations to describe how the graphs are related.

1  sin

x y

 

 

3  3sin

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Example

Horizontal Stretch or Shrink

Find the period of



and Period

sin   and use the language of transformations to describe how the graph relates to

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Frequency of a Sinusoid The

frequency

   1 

cos(



sin(

bx c d p

. Similarly, the frequency of

c d

 | | / 2   1

. Graphically, the frequency is the number of complete cycles the wave c ompletes in a unit interval.

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Example

Combining a Phase Shift with a Period Change

that goes through (2,0).

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Graphs of Sinusoids The graphs of

(where





)) 

and





)) 

 0 and

 0) have the following characteristics: period = 2  ; frequency = 2  .

When compared to the graphs of



sin

and



cos

, respectively, they also have the following characteristics:

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Constructing a Sinusoidal Model using Time 1. Determine the maximum value

M A





, and 2 the vertical shift will be





2 2. Determine the period , the time interval of a single cy cle of the periodic function. The horizontal shrink (or stretch) will be

 2 

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Constructing a Sinusoidal Model using Time 3. Choose an appropriate sinusoid based on behavior  

 

A A



)) ))  )) 



attains a maximum value;

attains a minimum value; is halfway between a minimum and a maximum value;  

)) 

is halfway between a maximum and a minimum value.

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Example

Constructing a Sinusoidal Model

On a certain day, high tide occurs at 7:12 AM and the water depth is measured at 15 ft. On the same day, low tide occurs at 1:24 PM and the water depth measures 8 ft.

(a) Write a sinusoidal function modeling the tide.

(b) What is the approximate depth of water at 11:00 AM?

At 3:00 PM?

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Example

Constructing a Sinusoidal Model

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Following Questions  The period of the tangent function is 2π.

 Tangent is an odd function.

 Cotangent is an even function.

 The graph of cosecant has relative minimum values, but no absolute minimum value.

 Some trig equations may be solved algebraically.

 Most trig equations may be solved graphically.

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Homework

 Homework Assignment #29  Read Section 4.5

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4.5

Graphs of Tangent, Cotangent, Secant, and Cosecant

Quick Review State the period of the function.

 cos 4

 sin 1 4

Find the zeros and the vertical asymptotes of the function.

 

 1 

x x

 1  2

  1

 3  5. Tell whether



2  4 is odd, even, or neither.

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Quick Review Solutions State the period of the function.

  /2 2.

 sin 1 4

8  Find the zeros and the vertical asymptotes of the function.

  

x x x

 1  1   2

  1

 3   1  1;

 3,

  2 5. Tell whether



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What you’ll learn about     The Tangent Function The Cotangent Function The Secant Function The Cosecant Function … and why This will give us functions for the remaining trigonometric ratios.

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Example

Analyzing Trigonometric Functions

Analyze the function for domain, range, continuity, increasing or decreasing, symmetry, boundedness, extrema, asymptotes, and end behavior  sec

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Example

Transformations of Trigonometric Functions

Describe the transformations required to obtain the graph of the given function from a basic trigonometric function.

  2 sec 1 2

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Example

Solving Trigonometric Equations

Solve the equation for x in the given interval.

sec

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Example

Solving Trigonometric Equations With a Calculator

Solve the equation for

csc

  1.5,  3  in the given interval.

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Example

Solving Trigonometric Word Problems

A hot air balloon is being blow due east from point P and traveling at a constant height of 800 ft. The angle y is formed by the ground and the line of vision from point P to the balloon. The angle changes as the balloon travels.

a. Express the horizontal distance x as a function of the angle y.

b. When the angle is , what is the horizontal distance from P?

 20

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Slide 1

Transcript Slide 1

Homework, Page 381

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 

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     

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     

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     

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  



Homework, Page 381

Graphs of Sine and Cosine: Sinusoids

Finding Amplitude

Finding Period and Frequency

 

Horizontal Stretch or Shrink

and Period

Combining a Phase Shift with a Period Change

Constructing a Sinusoidal Model

Constructing a Sinusoidal Model

Homework

Graphs of Tangent, Cotangent, Secant, and Cosecant

Analyzing Trigonometric Functions

Transformations of Trigonometric Functions

Solving Trigonometric Equations

Solving Trigonometric Equations With a Calculator

Solving Trigonometric Word Problems

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