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Appendix: Answers to Odd Homework Problems
Lecture 1: Angle Measurement
#1 3.75 radians
#7 20
#3 1 radian
#9 28π inches  88 inches
Lecture 2: Trigonometric Functions Defined
#1 30
#3 π3 radians
2
3
#7 0
#9
Lecture 3: Winding Function
#1  0,1
#3 

#7 
1
2
,
1
2


#9  
or
#5
3π
4
#5 
radians
1
2
or 
2
2
2 3
3
3
2
,  12
1
2
,
3
2


#5  1, 0 
Lecture 4: Right Triangle Trigonometry
#1 ≈528.8 feet
#3 ≈39.5 additional feet
Lecture 5: Graphing Trigonometric Functions
1
10
1
#1
#3
#5
10
1
#7
Lecture 6: Dilations and Reflections
#1
#3
1
#5
1
π
#7
π
π
π
#9
1
1

π
2
1
π
2
Lecture 7: Translations
#1
2
#3
#5
1
1

π
2
π
2
π
2
π
1
1
π
π
1
1

#7
#9
Lecture 8: Inverse Functions
#1  π2
#3
π
6
π
6
π
2
π
6
#5
π
3
Lecture 9: Basic Identities
sin x
#1 LHS  sin1 x  cos1 x  cos
x  tan x  RHS
#3 RHS  tan 2 λ  sec2 λ  1   sec λ  1 sec λ 1  LHS
#5
tan x sin x
sec x 1
sec x 1
 sec
x 1 
tan x sin x sec x 1
sec2 x 1

tan x sin x sec x 1
tan 2 x
#7 LHS  csc2 x  1  cot 2 x  1  tan12 x 
sin x sec x 1
tan x

tan 2 x
tan 2 x
 tan12 x 
x
 sin x  sec x  1  cos
sin x   sec x  1 cos x  1  cos x
tan 2 x 1
tan 2 x
 RHS
sin θ
cos θ
sin θ
cos θ sin θ
#9 LHS  1  tan θ  1  cos
 RHS
θ  cos θ  cos θ 
cos θ
2
2
sin x
cos x
sin x
cos x
1
1
#11 RHS  sin x1cos x  tan x1 cot x  sin x1cos x   cos
x  sin x   sin x cos2 x  sin 2 x cos x  cos2 x  sin 2 x  sec x  csc x  LHS
Lecture 10: Sum and Difference Identities
#1 2 4 6 or 12 23
#3 6 4
2
or
3 1
2 2
#5 LHS  cos  α  180  cos α cos180  sin α sin180  cos α  1  sin α  0   cos α  RHS
π  tan x
0 tan x
#7 RHS  tan  π  x   1tan
 tan π tan x  10tan x 
tan x
10

tan x
1
 tan x  LHS
#9 RHS  12 cos x cos y  sin x sin y  cos x cos y  sin x sin y   12 2cos x cos y   cos x cos y  LHS
#11 cos 2 x  cos  x  x   cos x cos x  sin x sin x  cos 2 x  sin 2 x  1  sin 2 x  sin 2 x  1  2sin 2 x
Lecture 11-12: Double-Angle & Half-Angle Identities
#1
1
2
#3
2 2
2
x
1 2cos x 1
2cos x
cos x
#5 RHS  1sincos2
2 x  2sin x cos x  2sin x cos x  sin x  cot x  LHS
2
2
#7 RHS   sin x  cos x    sin x  cos x  sin x  cos x   sin 2 x  2sin x cos x  cos 2 x  1  sin 2 x  LHS
2
x
#9 LHS  1sincos2
2x 

1 1 2sin 2 x
2sin x cos x

1 2sin x
2sin x
sin x
 12sin
x cos x  2sin x cos x  cos x  tan x  RHS
Lecture 13: Conditional Identities
#1 x  π6 , 56π
#7 θ  arctan  12   kπ
2
2
#3 x  π3 , 56π
#5 x  π6 , 32π ( 56π is extraneous)
#9 x   π6  2kπ , x  76π  2kπ
Lecture 14: Law of Sines
#1 ≈2.5 km
#3 h  9.5  c no triangle
Lecture 15: Law of Cosines
#1 BC  21.7 m
#3 B  114.5 , C  45.9 , A  19.6
Lecture 16: Parabolas
2
#1 y   x  4   13 , vertex:  4, 13 , focus:  4, 12.75 , directrix: y  13.25
#3 y  2  x  3  13 , vertex:  3, 13 , focus:  3, 12 78  , directrix: y  13 18
2
#5 x   12  y  1  72 , vertex:   72 , 1 , focus:  4, 1 , directrix: x  3
2
#7 LHS 
1
1
1  sin x 1  sin x 1  sin x




 RHS
1  sin x 1  sin x 1  sin x 1  sin 2 x cos 2 x
Lecture 17: Circles and Ellipses
2
2
#1  x  2    y  3  4 ,  2,3 , r  2
#3 center: 1, 2  , foci: 1, 2  5
Lecture 18: Hyperbolas
#1 y   32 x
#3 center:  0, 0  , foci:  2, 0 , asymptotes: y   x

#5 center:  1, 2  , vertices:
#7 LHS 


#5 e 

 3, 2   5, 2  , foci:  4, 2   6, 2 , asymptotes:
y   34  x  1  2
csc x  sin x csc x sin x


 csc x  csc x  1  csc2 x  1  cot 2 x  RHS
sin x
sin x sin x
Lecture 19: Vectors
#1 r  2, θ  π6 or 30
#7 2
#13 LHS 


 1 
1  cos 2  2x 
1  sin
2

x
2
1
Lecture 20: Complex Numbers
#1 z  2, θ  56π or 150
1 cos x
2
1 cos x
2


#3 r  1, θ   π2 or  90
#9 0
#5 r  5, θ  43π or 240
#11  95
2
2

x
1 cos x
2
1  1cos
2
2 
2


x
1 cos x
2
1  1cos

2
2
2
1 cos x
2
1 cos x
2

1  cos x
 RHS
1  cos x
#3 z  13, θ  arctan  23   180  213.7
#5 trigonometric form: 3 2cis  π4  ; exponential form: 3 2e 4 ; standard form: 3  3i
iπ
#7 trigonometric form: 48cis  56π  ; standard form: 24 3  24i
#9 trigonometric form:
9
16
cis  π  ; standard form:  169
Lecture 21: Roots
#1 trigonometric form: cis  π4  , cis  54π  ; standard form:
#3 standard form:
#5
2
3
2 2
2
3  i,  1  3i,  3  i,1  3i
2
2

2
2
i, 
2
2

2
2
i
Lecture 22: Polar Coordinates
#1 3 5,arctan  2   3 5, 63.4

#5  3
3, 3
 


#3  2, 56π  or  2,150  or  2, 210
#7


3,1
#9  x  1  y 2  1
#11 y  1  x
#13 r  4
#17
#15 r 2  3r cos θ  4  0
#19
θ  π3
2
r  4cos3θ
Lecture 23: Parametric Equations
#1
y  x  2
x2  y 2  16
#3
#5
1
1
1
1
1
1
Lecture 24: Sequences
n1
#1 12 ,1, 2, 4,
#3 621 13 
#5 1, 3, 7,17, 41,
#7 x  kπ, x  π3  2πk , x  53π  2πk
Lecture 25: Series
 8 
7
#1
k 1

1 k 1
2


#3 91
#5 27
cos  2 x  1  2sin 2 x
1
2sin 2 x



 csc2 x  2  RHS
sin 2 x
sin 2 x
sin 2 x sin 2 x
30, 000
.
 3k  1  15, 250 is greater than 15, 000 
sec   π3 
#7 LHS 

100
#9
k 1
y  2x  3
Lecture 26: Combinations and Binomial Theorem
n  n  1!
n  n  1!
 n 
n!
n
n



 n
#1 

 n  1  n  1! n   n  1 !  n  1! n  n  1!  n  1!  0  1! 1! 1
#3 110
4
#5  2 x  3  16 x 4  96 x3  216 x 2  216 x  81