Download Slide 5- 1 Homework, Page 460

Homework, Page 460
Prove 3the algebraic
identity.
2
x x
  x  1 x  1  1  x
1.
x
x3  x 2
1 x 
  x  1 x  1
x


x x2  x

x



x2  1
 

 x2  x  x2  1
 x2  x  x2  1
 x 1
 1 x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 1
Homework, Page 460
Tell whether or not f (x) = sin x is an identity.
5.
sin 2 x  cos 2 x
f  x 
csc x
sin 2 x  cos 2 x
f  x 
csc x
1

csc x
1

1
sin x
 sin x  Yes, f  x   sin x is an identity.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 2
Homework, Page 460
Tell whether or not f (x) = sin x is an identity.
3
2
f
x

sin
x
1

cot
x
 
9.




f  x   sin 3 x 1  cot 2 x


2

cos
x
2
  sin x  sin x 1 

2
sin
x





  sin x  sin 2 x  cos 2 x

  sin x 1
 sin x  Yes, f  x   sin x is an identity.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 3
Homework, Page 460
Prove the identity.
2
2
1

tan
x

sec
x  2 tan x


13.
2
1

tan
x

sec
x  2 tan x


2
 1  tan 2 x  2 tan x
 1  2 tan x  tan 2 x
 1  tan x 1  tan x 
 1  tan x 
2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 4
Homework, Page 460
Prove the identity.
cos 2 x  1
  tan x sin x
17.
cos x
cos 2 x  1
  tan x sin x
cos x
sin x

sin x
cos x
 sin 2 x

cos x


 1  cos 2 x

cos x
cos 2 x  1

cos x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 5
Homework, Page 460
Prove the identity.
21.  cos t  sin t    cos t  sin t   2
2
2
2   cos t  sin t    cos t  sin t 
2
2
 cos 2 t  2cos t sin t  sin 2 t  cos 2 t  2cos t sin t  sin 2 t
 2cos 2 t  2cos t sin t  2sin 2 t  2cos t sin t

 2  cos
 2 cos 2 t  sin 2 t  cos t sin t  cos t sin t
2
t  sin 2 t


21
2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 6
Homework, Page 460
Prove the identity.
cos 
1  sin 
25. 1  sin   cos 
cos 
1  sin  1  sin 

1  sin 
cos  1  sin 
1  sin 2 

cos  1  sin  
cos 2 

cos  1  sin  
cos 

1  sin 
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 7
Homework, Page 460
Prove the identity.
29. cot 2 x  cos2 x  cos2 x cot 2 x
cot 2 x  cos 2 x  cos 2 x cot 2 x


 cos 2 x csc 2 x  1
 1

 cos x  2  1
 sin x 
cos 2 x
2


cos
x
2
sin x
 cot 2 x  cos 2 x
2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 8
Homework, Page 460
Prove the identity.
2
2
 x sin   y cos     x cos   y sin    x 2  y 2
33.
x 2  y 2   x sin   y cos     x cos   y sin  
2
2
 x 2 sin 2   2 xy sin  cos   y 2 cos 2 
 x 2 cos 2   2 xy cos  sin   y 2 sin 2 


 y  cos   sin  
 x 2 sin 2   cos 2   2 xy  sin  cos   cos  sin  
2
2
2
 x2  y 2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 9
Homework, Page 460
Prove the identity.
sin x  cos x
2sin 2 x  1

37.
sin x  cos x 1  2sin x cos x
2sin 2 x  sin 2 x  cos 2 x

sin 2 x  cos 2 x  2sin x cos x




sin 2 x  cos 2 x

sin 2 x  2sin x cos x  cos 2 x
sin x  cos x  sin x  cos x 


 sin x  cos x  sin x  cos x 
sin x  cos x

sin x  cos x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 10
Homework, Page 460
Prove the identity.
2
3
2
4
sin
x
cos
x

sin
x

sin
x  cos x 
41.




sin 2 x cos3 x  sin 2 x  sin 4 x  cos x 


x  cos x   cos x 
 sin 2 x 1  sin 2 x  cos x 
 sin 2
2
 sin 2 x cos3 x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 11
Homework, Page 460
Prove the identity.
tan x
cot x
45. 1  cot x  1  tan x  1  sec x csc x
tan x sin x
cot x cos x
1  sec x csc x 

1  cot x sin x 1  tan x cos x
sin 2 x
cos 2 x
cos x
sin x


sin x  cos x cos x  sin x
sin 2 x
cos 2 x
cos x
sin x


sin x  cos x sin x  cos x
sin 2 x cos 2 x

sin x cos x
cos
x
sin
x

sin x  cos x sin x cos x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 12
Homework, Page 460
Prove the identity.
45.
sin 2 x cos 2 x

sin x cos x
1  sec x csc x  cos x sin x
sin x  cos x sin x cos x
sin 3 x  cos3 x

 sin x  cos x  sin x cos x 
sin 2 x  sin x cos x  cos 2 x

sin x cos x
1  sin x cos x

sin x cos x
 1  sec x csc x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 13
Homework, Page 460
Prove the identity.
49. cos3 x  1  sin 2 x   cos x 


  cos x   cos x 
cos3 x  1  sin 2 x  cos x 
2
 cos3 x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 14
Homework, Page 460
Match the function with an equivalent expression, then confirm
the matching with a proof.
53. 1  sec x 1  cos x 
d.
1  sec x 1  cos x   1  cos x  sec x  1
1
  cos x 
cos x
 cos 2 x  1

cos x
sin 2 x

cos x
 tan x sin x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 15
Homework, Page 460
Match the function with an equivalent expression, then confirm
the matching with a proof.
1
1

1
1
sin x
sec x  tan x
57.
b.

sec x  tan x
cos x cos x
cos x 1  sin x

1  sin x 1  sin x
cos x 1  sin x 

1  sin 2 x
cos x 1  sin x 

cos 2 x
1  sin x

cos x
 sec x  tan x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 16
Homework, Page 460
61. Which of the following is a good first step in
proving: sin x  1  cos x
1  cos x
sin x


cos   x 
sin x
sin x

A. sin x
B.
2



1  cos x sin 2 x  cos 2 x  cos x
1  cos x
1  cos x
sin x
sin x csc x
sin x
sin x 1  cos x
C. 1  cos x  1  cos x csc x D. 1  cos x  1  cos x 1  cos x
sin x
sin x 1  cos x

E. 1  cos x 1  cos x 1  cos x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 17
Homework, Page 460
Identify a simple function that has the same graph. Then confirm
your answer with a proof.
65. cos x tan x
y  sin x
sin x
cos x tan x  cos x
cos x
 sin x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 18
Homework, Page 460
Identify a simple function that has the same graph. Then confirm
your answer with a proof.


2
2
sec
x
1

sin
x
69.

y 1




1
2
sec x 1  sin x 
cos
x
2
cos x
1
2
2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Slide 5- 19
Homework, Page 460
Confirm the identity.
73. 1  sin t  1  sin t
1  sin t
cos t
1  sin t
1  sin t 1  sin t

1  sin t
1  sin t 1  sin t


1  sin t 
2
1  sin 2 t
1  sin t 
2
cos 2 t
1  sin t

cos t
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 20
Homework, Page 460
Confirm the identity.
77.
ln tan x  ln sin x  ln cos x
ln tan x  ln sin x  ln cos x
sin x
 ln
cos x
 ln tan x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 21
5.3
Sum and Difference Identities
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Quick Review
Express the angle as a sum or difference of special angles (multiples
of 30 , 45 ,  /6, or  /4). Answers are not unique.
1. 15
2.  /12
3. 75
Tell whether or not the identity f ( x  y )  f ( x)  f ( y ) holds for the function f .
4. f ( x)  16 x
5. f ( x)  2e
x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 23
Quick Review Solutions
Express the angle as a sum or difference of special angles (multiples
of 30 , 45 ,  /6, or  /4). Answers are not unique.
1. 15
45  30

2.  /12


3 4
3. 75
30  45
Tell whether or not the identity f ( x  y )  f ( x)  f ( y ) holds for the function f .
4. f ( x)  16 x yes
5. f ( x)  2e
no
x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 24
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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 25
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 26
Example Using the Cosine-of-aDifference Identity
Find the exact value of cos 75 without using a
calculator.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 27
Sine of a Sum or Difference
sin  u  v   sin u cos v  cos u sin v
sin  u  v   sin u cos v  cos u sin v
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 28
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 29
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- 30
Example Proving an Identity



Prove the identity: cos    x   y   sin  x  y 

 2

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 31
Example Proving a Reduction Formula


sec   u   csc u
2

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 32
Example Expressing a Function as a
Sinusoid
y  5sin x  12cos x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 33
Example Proving an Identity
2
3
sin 3u  3cos u sin u  sin u
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 34
Homework




Homework Assignment #12
Read Section 5.4
Page 468, Exercises: 1 – 61 (EOO)
Quiz next time
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 35
5.4
Multiple-Angle Identities
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Quick Review
Find the general solution of the equation.
1. cot x  1  0
2. (sin x)(1  cos x)  0
3. cos x  sin x  0


4. 2sin x  2  2sin x  1  0
5. Find the height of the isosceles triangle with base length 6
and leg length 4.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 37
Quick Review Solutions
Find the general solution of the equation.
3
1. cot x  1  0
x
n
4
2. (sin x)(1  cos x)  0 x   n
3. cos x  sin x  0

x


4
n
4. 2sin x  2  2sin x  1  0 x 
5
 2 n
6
6
5. Find the height of the isosceles triangle with base length 6
x

5
7
 2 n, x 
 2 n,
4
4
 2 n, x 
and leg length 4.
7
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 38
What you’ll learn about




Double-Angle Identities
Power-Reducing Identities
Half-Angle Identities
Solving Trigonometric Equations
… and why
These identities are useful in calculus courses.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 39
cos 2 x 
Deriving Double-Angle Identities
sin 2 x 
tan 2 x 
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 40
Double Angle Identities
sin 2u  2sin u cos u
cos u  sin u

cos 2u  2 cos u  1
1  2sin u

2
2
2
2
2 tan u
tan 2u 
1  tan u
2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 41
Example Solving a Problem Using double
Angle Identities
Find all solutions in the interval  0,2 
sin 2 x  sin x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 42
Power-Reducing Identities
1  cos 2u
sin u 
2
1  cos 2u
2
cos u 
2
1  cos 2u
2
tan u 
1  cos 2u
2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 43
Example Reducing a Power of 4
Rewrite sin 4 x in terms of trigonometric functions
with no power greater than 1.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 44
Half-Angle Identities
u
1  cos u
sin  
2
2
u
1  cos u
cos  
2
2
 1  cos u

1

cos
u

u 1  cos u
tan  
2  sin u
 sin u
1  cos u

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 45
Example Using a Double Angle Identity
Solve algebraically in the interval [0, 2 ) : sin 2 x  sin x.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 46