Download Slide 5- 3 Homework, Page 468

Homework, Page 468
Use a sum or difference identity to find an exact value.
1. sin15
sin15  sin  45  30   sin 45 cos30  cos 45 sin 30
 2 3  2 1
6
2





4
 2 2   2 2 4
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
6 2
4
Slide 5- 1
Homework, Page 468
Use a sum or difference identity to find an exact value.
5. cos


12




  
cos  cos     cos cos  sin sin
12
3
4
3
4
3 4
1 2  3 2
2
6





4
2 2   2 2  4
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
2 6
4
Slide 5- 2
Homework, Page 468
Use a sum or difference identity to find an exact value.
7
9. cos
12
7




  
cos
 cos     cos cos  sin sin
12
3
4
3
4
3 4
1 2  3 2
2
6





4
2 2   2 2  4
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
2 6
4
Slide 5- 3
Homework, Page 468
Write the expression as the sine, cosine, or tangent of an angle.


5
2
13. sin cos
sin

5
cos

2
 sin
 sin

2

2
cos
cos

5

5
 sin

5
cos



 cos sin
2
5
2
  
 sin   
5 2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 4
Homework, Page 468
Write the expression as the sine, cosine, or tangent of an angle.


7
7
17. cos cos x  sin sin x




cos cos x  sin sin x  cos   x 
7
7
7

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 5
Homework, Page 468
Write the expression as the sine, cosine, or tangent of an angle.
tan 2 y  tan 3 x
21.
1  tan 2 y tan 3 x
tan 2 y  tan 3 x
 tan  2 y  3 x 
1  tan 2 y tan 3 x
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Slide 5- 6
Homework, Page 468
Prove the identity.


25. cos  x    sin x
2



sin x  cos  x  
2

 cos x cos

 sin x sin
2
 cos x  0   sin x 1

2
 sin x
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Slide 5- 7
Homework, Page 468
Prove the identity.
29.
  1  tan 

tan    
4  1  tan 

1  tan 


 tan   
1  tan 
4


tan   tan

1  tan  tan
tan   1

1  tan  1
4

4
1  tan 

1  tan 
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Slide 5- 8
Homework, Page 468
Match each graph with a pair of the equations.
33.
 d . y  sin  2 x  5
 h. y  sin 2 x cos5  cos 2 x sin 5
sin  2 x  5   sin 2 x cos5  cos 2 x sin 5
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 9
Homework, Page 468
Prove the reduction formula.


37. sin   u   cos u
2



cos u  sin   u 
2

 sin


cos u  cos sin u
2
2
 1 cos u   0  sin u
 cos u
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 10
Homework, Page 468
Prove the reduction formula.


41. csc   u   sec u
2

1
1


 cos u
sin   u 
2



cos u  sin   u 
2

 sin

cos u  cos

2
2
 1 cos u   0  sin u
sin u
 cos u
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 11
Homework, Page 468
Express the function as a sinusoid in the form y = a sin (bx + c).
45. y  cos3x  2sin 3x
y  cos3 x  2sin 3 x  y  a sin  bx  c 
a sin  bx  c   a  sin bx cos c  cos bx sin c 
2sin 3 x  cos3 x  a cos c sin bx  a sin c cos bx
b  3  a cos c  2  a sin c  1
2
2
2
a
cos
c

a
sin
c

2

1

a
5a  5

 

2
2
2
4
1 2
cos c 
;sin c 
 c  cos
 0.464
5
5
5

1 2 
cos3 x  2sin 3 x  5 sin  3 x  cos
  2.236sin  3 x  0.464 
5

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 12
Homework, Page 468
Prove the identity.
49.
cos3x  cos3 x  3sin 2 x cos x
cos3 x  3sin 2 x cos x  cos3 x
 cos 2 x cos x  sin 2 x sin x
 cos x  cos x cos x  sin x sin x  
sin x  sin x cos x  cos x sin x 
 cos3 x  cos x sin 2 x 
sin 2 x cos x  sin 2 x cos x
 cos3 x  3sin 2 x cos x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 13
Homework, Page 468
Prove the identity.
tan 2 x  tan 2 y
53. tan  x  y  tan  x  y  
1  tan 2 x tan 2 y
tan x  tan y tan x  tan y
tan  x  y  tan  x  y  
1  tan x tan y 1  tan x tan y
tan 2 x  tan 2 y

1  tan 2 x tan 2 y
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 14
Homework, Page 468
57. If cos A + cos B = 0, then A and B are
supplementary angles.
False.
If A and B are supplementary angles, then A  B  180º.
For example, cos   cos 2  0, but  + 2  180º.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 15
Homework, Page 468
f 1  f  2 
61. A function with the property f 1  2  
is
1  f 1 f  2 
A. f  x   sin x
f  x   tan x
B.
C. f  x   sec x
D.
E.
f  x   ex
f  x   1
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 16
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- 18
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- 19
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- 20
cos 2 x 
Deriving Double-Angle Identities
sin 2 x 
tan 2 x 
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Slide 5- 21
Double Angle Identities
sin 2u  2sin u cos u
cos 2 u  sin 2 u

2
cos 2u  2cos u  1
1  2sin 2 u

2 tan u
tan 2u 
2
1  tan u
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 22
Example Solving a Problem Using double
Angle Identities
Find all solutions in the interval  0,2 
sin 2 x  sin x
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Slide 5- 23
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- 24
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- 25
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- 26
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- 27



Homework
Homework Assignment #13
Read Section 5.5
Page 475, Exercises: 1 – 57 (EOO)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 28
What you’ll learn about




Deriving the Law of Sines
Solving Triangles (AAS, ASA)
The Ambiguous Case (SSA)
Applications
… and why
The Law of Sines is a powerful extension of the triangle
congruence theorems of Euclidean geometry.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 29
Deriving the Law of Sines
Consider the two triangles ABC.
C
b
a
h
A
B
c
C
b
a
A
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
c
h
B
Slide 5- 30
Law of Sines
In ABC with angles A, B, and C opposite sides a, b,
and c, respectively, the following equation is true:
sin A sin B sin C


.
a
b
c
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 31
Example Solving a Triangle Given Two
Angles and a Side
Solve ABC given that A  38, B  46, and a  9.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 32
Example Solving a Triangle Given Two Sides
and an Angle (The Ambiguous Case)
Solve ABC given that a  5, b  6, and A  30.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 33
Example Finding the Height of a Pole
A road slopes 15 above the horizontal, and a vertical telephone
pole stands beside the road. The angle of elevation of the Sun is
65, and the pole casts a 15 foot shadow downhill along the road.
A
65
Find the height of the pole.
º
x
B
15º
C
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 34