Transcript Slide 5- 1 Homework, Page 451

Homework, Page 451
Evaluate without using a calculator. Use the Pythagorean
identities rather than reference angles.
3
1. Find sin  and cos if tan   and sin   0.
4
3
9
2
2
tan    tan   1  sec    1  sec 2 
4
16
25
5
4
sec  
 sec   cos 
16
4
5
16
25
2
2
2
1  cot   csc   1   csc  
 csc 2 
9
9
5
3
csc   sin  
3
5
2
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Slide 5- 1
Homework, Page 451
Use identities to find the value of the expression.


5. If sin   0.45, find cos  2   .
cos     sin 
2




If sin   0.45, then cos     0.45
2
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Slide 5- 2
Homework, Page 451
Use basic identities to simplify the expression.
9. tan x cos x
sin x
tan x cos x 
cos x  sin x
cos x
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Slide 5- 3
Homework, Page 451
Use basic identities
to simplify the expression.
2
1  tan x
13.
csc 2 x
1
1  tan 2 x sec 2 x cos 2 x
1 sin 2 x
2




tan
x
2
2
2
1
csc x
csc x
cos x 1
sin 2 x
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Slide 5- 4
Homework, Page 451
Simplify the expression to either 1 or –1.
17. sin x csc   x 
1
sin x csc   x   sin x   csc x    sin x
 1
sin x
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Slide 5- 5
Homework, Page 451
Simplify the expression to either 1 or –1.
21.
sin 2   x   cos 2   x 
sin 2   x   cos 2   x     sin x    cos x   sin 2 x  cos 2 x  1
2
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2
Slide 5- 6
Homework, Page 451
Simplify the expression to either a constant or a basic
trigonometric function. Support your answer graphically.
25.


 
sec2 x  csc2 x  tan 2 x  cot 2 x
 
sec 2 x  csc 2 x  tan 2 x  cot 2 x


 sec 2 x  csc 2 x  tan 2 x  cot 2 x
 sec 2 x  tan 2 x  csc 2 x  cot 2 x

 
 sec 2 x  tan 2 x  csc 2 x  cot 2 x

 11  2
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Slide 5- 7
Homework, Page 451
Use basic identities to change the expression to one involving
only sines and cosines. Then simplify to a basic trigonometric
function.
29. sin x cos x tan x sec x csc x
sin x 1
1
sin x cos x tan x sec x csc x  sin x cos x
cos x cos x sin x
sin x

 tan x
cos x
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Slide 5- 8
Homework, Page 451
Combine the fractions and simplify to a multiple of a power of a
basic trigonometric function.
33.
1
sec 2 x

2
sin x tan 2 x
1
2
1
sec 2 x
1
1
1
cos
x




 2
2
2
2
2
2
sin x tan x sin x sin x sin x sin x
cos 2 x
2
2


2csc
x
2
sin x
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Slide 5- 9
Homework, Page 451
Combine the fractions and simplify to a multiple of a power of a
basic trigonometric function.
37.
sec x sin x

sin x cos x
sec x sin x sec x cos x sin x sin x



sin x cos x sin x cos x cos x sin x
1  sin 2 x
cos 2 x
cos x



 cot x
sin x cos x sin x cos x sin x
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Slide 5- 10
Homework, Page 451
Write each expression in factored form as an algebraic expression
of a single trigonometric function.
41. 1  2sin x  1  cos 2 x 


1  2sin x  1  cos x  1  2sin x  sin x  1  sin x 
2
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2
2
Slide 5- 11
Homework, Page 451
Write each expression in factored form as an algebraic expression
of a single trigonometric function.
4
4 tan x 
 sin x csc x
cot x
4
sin x
2
2
4 tan x 
 sin x csc x  4 tan x  4 tan x 
cot x
sin x
 4 tan 2 x  4 tan x  1
45.
2
  2 tan x  1 2 tan x  1
  2 tan x  1
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2
Slide 5- 12
Homework, Page 451
Write each expression as an algebraic expression of a single
trigonometric function.
2
sin
x
49.
1  cos x
sin 2 x
1  cos 2 x 1  cos x 1  cos x 


 1  cos x
1  cos x 1  cos x
1  cos x
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Slide 5- 13
Homework, Page 451
Find all solutions to the equation in the interval (0, 2π).
53.
tan x sin 2 x  tan x
3
sin
x
sin
x
sin
x sin x
2
2
tan x sin x  tan x 
sin x 


cos x
cos x
cos x cos x


sin 3 x  sin x  sin 3 x  sin x  0  sin x sin 2 x  1  0
3 

x   , , 
2 
2
 
 3
tan    undefined  tan 
2
 2

  undefined  sin  0

x 
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Slide 5- 14
Homework, Page 451
Find all solutions to the equation.
2
4cos
x  4cos x  1  0
57.
4 cos 2 x  4 cos x  1  0   2 cos x  1 2 cos x  1  0
1
5


2 cos x  1  cos x   x    2n ,
 2n 
2
3
3

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Slide 5- 15
Homework, Page 451
Find all solutions to the equation.
61.
cos  sin x   1
cos  sin x   1  cos y  1  y  0  sin x  0  x  n
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Slide 5- 16
Homework, Page 451
Find all solutions to the equation.
65. sin x  0.30
sin x  0.30  sin 1 0.30  0.305, 2.837
x  0.305  n2 , 2.837  n2 
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Slide 5- 17
Homework, Page 451
Write the function as a multiple of a basic trigonometric function.
69.
1  x 2 , x  cos
1  x 2  1  cos2   sin 2   sin 
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Slide 5- 18
Homework, Page 451
Write the function as a multiple of a basic trigonometric function.
73.
x 2  81, x  9 tan 
81tan 2   81  9 tan 2   1  9 sec2   9sec
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Slide 5- 19
Homework, Page 451
77. Which of the following could not be set equal to
sin x as an identity?
b.
 2  x
cos  x   
2
c.
1  cos2 x
d.
tan x sec x
a.
e.
cos 
 sin   x 
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Slide 5- 20
Homework, Page 451
81. Write all six trigonometric functions in terms of
sin x.
1
sin x  sin x  csc x 
sin x


cos x  sin   x  sec x 
2
tan x 
sin x

sin   x
2

1


sin   x 
2
 cot x 
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sin   x
2
sin x
Slide 5- 21
Homework, Page 451
85. Because its orbit is elliptical, the distance from the Moon to
the Earth in miles, measured from the center of the Moon to the
center of the Earth, varies periodically. On Monday, January 18,
2002, the Moon was at its apogee. The distance from the Moon
to the Earth each Friday from January 23 to March 27 is recorded
in the table.
Date Day
1/23 0
1/30 7
2/6 14
Distance
251,966
238,344
225,784
Date Day
2/27 35
3/6 42
3/13 49
Distance
236,315
226,101
242,390
2/13
2/20
240,385
251,807
3/20
3/27
251,333
234,347
21
28
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56
63
Slide 5- 22
Homework, Page 451
85. a. Draw a scatterplot of the data using day as x and distance as y.
b. Do a sine regression and plot the resulting curve on the
scatterplot.
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Slide 5- 23
Homework, Page 451
85. c. What is the approximate number of days from apogee to
apogee?  28 days
d. Approximately how far is the Moon from the Earth at perigee?
perigee  238,354.9926  13,111.0422  225,244 miles
e. Use a cofunction to write a cosine curve that fits the data.


d  t   13,111sin  0.22997 x  1.571  238,855  sin   cos    
d  t   13,111cos  0.22997 x   238,855
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2

Slide 5- 24
5.2
Proving Trigonometric Identities
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What you’ll learn about




A Proof Strategy
Proving Identities
Disproving Non-Identities
Identities in Calculus
… and why
Proving identities gives you excellent insights into the
was mathematical proofs are constructed.
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Slide 5- 26
General Strategies I for Proving an Identity
1. The proof begins with the expression on one
side of the identity.
2. The proof ends with the expression on the
other side.
3. The proof in between consists of showing a
sequence of expressions, each one easily
seen to be equivalent to its preceding
expression.
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Slide 5- 27
Example Proving an Algebraic Identity
1 1 2 x
Prove the identity:  
.
x 2
2x
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Slide 5- 28
General Strategies II for Proving an Identity
1. Begin with the more complicated expression
and work toward the less complicated
expression.
2. If no other move suggests itself, convert the
entire expression to one involving sines and
cosines.
3. Combine fractions by combining them over a
common denominator.
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Slide 5- 29
Example Proving a Trigonometric
Identity
sin x
1  cos x
Prove the identity:

.
1  cos x
sin x
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Slide 5- 30
General Strategies III for Proving an
Identity
1. Use the algebraic identity (a+b)(a–b) = a2 – b2
to set up applications of the Pythagorean
identities.
2. Always be mindful of the “target” expression,
and favor manipulations that bring you closer
to your goal.
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Slide 5- 31
Example Proving a Trigonometric
Identity
sin x
1  cos x 2 1  cos x 
Prove the identity:


.
1  cos x
sin x
sin x
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Slide 5- 32
Identities in Calculus


x  1  tan x  sec x 
1. cos3 x  1  sin 2 x  cos x 
2. sec 4
2
2
1 1
3. sin x   cos 2 x
2 2
1 1
2
4. cos x   cos 2 x
2 2
5. sin 5 x  1  2cos 2 x  cos 4 x  sin x 
2




6. sin 2 x cos5 x  sin 2  2sin 4 x  sin 6 x  cos x 
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Slide 5- 33
Homework



Homework Assignment #36
Read Section 5.3
Page 460, Exercises: 1 – 77 (EOO)
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Slide 5- 34
5.3
Sum and Difference Identities
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What you’ll learn about





Cosine of a Difference
Cosine of a Sum
Sine of a Difference or Sum
Tangent of a Difference or Sum
Verifying a Sinusoid Algebraically
… and why
These identities provide clear examples of how different the
algebra of functions can be from the algebra of real numbers.
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Slide 5- 36
Cosine of a Sum or Difference
cos  u  v   cos u cos v  sin u sin v
cos  u  v   cos u cos v  sin u sin v
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Slide 5- 37
Example Using the Cosine-of-aDifference Identity
Find the exact value of cos 75 without using a
calculator.
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Slide 5- 38
Sine of a Sum or Difference
sin  u  v   sin u cos v  sin v cos u
sin  u  v   sin u cos v  sin v cos u
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 39
Example Using the Sum and Difference
Formulas
Write the following expression as the sine or cosine of
an angle: sin

3
cos

4
 sin

4
cos

3
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Slide 5- 40
Tangent of a Difference of Sum
sin  u  v 
sin u cos v  sin v cos u sec u sec v
tan  u  v  

cos  u  v  cos u cos v  sin u sin v sec u sec v
tan u  tan v

1  tan u tan v
sin  u  v  sin u cos v  sin v cos u sec u sec v
tan  u  v  

cos  u  v  cos u cos v  sin u sin v sec u sec v
tan u  tan v

1  tan u tan v
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 41
Example Proving an Identity
 


Prove the identity: cos    x   y   sin  x  y 

 2

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Slide 5- 42
Example Proving a Reduction Formula


sec   u   csc u
2

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Slide 5- 43
Example Expressing a Function as a
Sinusoid
y  5sin x  12cos x
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Slide 5- 44
Example Proving an Identity
2
3
sin 3u  3cos u sin u  sin u
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Slide 5- 45