Download Problem Set 6: Sum and Difference Formulas - Precalculus | MATH 1330 and more Assignments Pre-Calculus in PDF only on Docsity!
Math 1330, Precalculus
Simplify each of the following expressions.
1. sin ( π − x )
cos 2
x
3. cos 2
x
4. sin ( π + x )
5. sin 60( − θ ) + sin 60( + θ)
D D
6. cos 60( − θ) + cos 60( + θ)
D D
7. cos cos 4 4
x x
8. sin sin 6 6
x x
⎜ +^ ⎟ +^ ⎜ − ⎟
9. sin ( θ− 180 ) + sin ( θ+ 180 )
D D
10. cos 90( + θ) + cos 90( − θ)
D D
Answer the following.
11. Given that tan ( α )= − 4 , evaluate
tan 4
12. Given that tan ( β )= 2 , evaluate tan
13. Given that ( )
tan 3
x = , evaluate
tan 4
x
14. Given that ( )
tan 5
x = − , evaluate
tan 4
x
15. Given that tan ( x )= 5 and tan ( y )= 6 , evaluate
tan ( x + y ).
16. Given that tan ( x )= 4 and tan ( y )= − 2 ,
evaluate tan ( x − y ).
17. Given that tan ( α )= − 3 and ( )
tan 5
evaluate tan( α + β).
18. Given that ( )
tan 3
α = − and ( )
tan 2
evaluate tan( α − β).
Simplify each of the following expressions as much as
possible without a calculator.
19. cos 55( ) cos 10( ) +sin 55( ) sin 10( )
D D D D
20. cos 75( ) cos 15( ) −sin 75( ) sin 15( )
D D D D
21. sin 45( ) cos 15( ) −cos 45( ) sin 15( )
D D D D
22. sin 53( ) cos 7( ) +cos 53( ) sin 7( )
D D D D
23. cos cos sin sin 10 9 10 9
⎛ π^ ⎞ ⎛ π⎞ ⎛ π^ ⎞ ⎛ π⎞
24. sin cos cos sin 5 7 5 7
⎛ π^ ⎞ ⎛ π⎞ ⎛ π^ ⎞ ⎛ π⎞
sin cos cos sin 12 12 12 12
⎛ π^ ⎞ ⎛ π⎞ ⎛ π^ ⎞ ⎛ π⎞
cos cos sin sin 12 12 12 12
27. sin 2( A ) cos ( A ) −cos 2( A ) sin( A )
28. cos 3( α ) cos ( α) +sin 3( α ) cos( α)
tan 32 tan 2
1 tan 32 tan 2
D D
D D
tan 49 tan 4
1 tan 49 tan 4
D D
D D
Math 1330, Precalculus
tan tan 12 12
1 tan tan 12 12
tan tan 12 12
1 tan tan 12 12
( ) ( )
( ) ( )
tan tan
1 tan tan
a b b
a b b
( ) ( )
( ) ( )
tan 2 3 tan
1 tan 2 3 tan
c d d c
c d d c
Answer the following.
35. Rewrite each special angle below so that it has a
denominator of 12.
(a) 6
(b) 4
(c) 3
(d)
(e)
(f)
36. Use the answers from Exercise 35 to write each
fraction below as the sum of two special angles.
(Hint: Each of the solutions contains a multiple
of 4
π .)
(a)
(b)
(c)
(d)
37. Use the answers from Exercise 35 to write each
fraction below as the difference of two special
angles. (Hint: Each of the solutions contains a
multiple of 4
π .)
(a)
(b)
(c)
− (d)
38. For fractions with larger magnitude than those in
Exercises 36 and 37, it can be helpful to use
larger multiples of 4
π
. Rewrite each special
angle below so that it has a denominator of 12.
(a)
(b)
(c)
(d)
39. Use the answers from Exercises 35 and 38 to
write each fraction below as the sum of two
special angles. (Hint: Each of the solutions
contains a multiple of 4
π .)
(a)
(b)
(c)
(d)
40. Use the answers from Exercises 35 and 38 to
write each fraction below as the difference of
two special angles. (Hint: Each of the solutions
contains a multiple of 4
π .)
(a)
(b)
(c)
− (d)
41. Use the answers from Exercises 35 and 38, along
with their negatives to write each fraction below
as the difference, x − y , of two special angles,
where x is negative and y is positive.
(a)
− (b)
(c)
− (d)
42. Use the answers from Exercises 35 and 38, along
with their negatives to write each fraction below
as the difference, x − y , of two special angles,
where x is negative and y is positive.
(a)
− (b)
Math 1330, Precalculus
Evaluate the following.
cos tan tan 3 2
⎣ ⎝^ ⎠^ ⎝^ ⎠⎦
68. (^) ( )
sin tan 4 tan 4
⎣ ⎝^ ⎠⎦
tan cos sin 5 13
⎣ ⎝^ ⎠^ ⎝^ ⎠⎦
tan tan cos 24 5
⎣ ⎝^ ⎠^ ⎝^ ⎠⎦
Simplify the following.
71. cos (^) ( A ) (^) sin (^) ( B (^) ) ⎡cot (^) ( B (^) ) +tan( A )⎤ ⎣ ⎦ 72. tan (^) ( A ) (^) cos (^) ( B (^) ) ⎡cos (^) ( A (^) ) −cos (^) ( A ) (^) cot (^) ( A (^) ) tan( B )⎤ ⎣ ⎦ 73. sin (^) ( A − B (^) ) sin( A + B ) 74. cos (^) ( A + B (^) ) cos( A − B )
Prove the following.
75. sin (^) ( x − y (^) ) + sin (^) ( x + y (^) ) =2sin (^) ( x ) (^) cos( y ) 76. cos ( x − y ) + cos ( x + y ) =2 cos ( x ) cos( y )
( ) ( )
( ) ( )
( ) ( )
cos cos 2 cot cot sin sin
x y x y x y x y
( ) ( )
( ) ( )
( )
sin sin 2 tan cos cos
x y x y x x y
( )
( )
( ) ( )
( ) ( )
sin tan tan
sin tan tan
x y x y
x y x y
( )
( )
( ) ( )
( ) ( )
cos 1 tan tan
cos 1 tan tan
x y x y
x y x y
Use right triangle ABC below to answer the following
questions regarding cofunctions, in terms of side
lengths a , b , and c****.
81. (a) Find sin (^) ( A (^) ).
(b) Find cos (^) ( B (^) ).
(c) Analyze the answers for (a) and (b). What
do you notice?
(d) What is the relationship between angles A
and B? (i.e. If you knew the measure of one
angle, how would you find the other?) Write
answers in terms of degrees.
(e) Complete the following cofunction
relationships by filling in blank. Write your
answers in terms of A.
sin ( A ) =cos ________( )
cos ( A ) =sin ________( )
82. (a) Find tan (^) ( A (^) ).
(b) Find cot (^) ( B ).
(c) Analyze the answers for (a) and (b). What
do you notice?
(d) What is the relationship between angles A
and B? (i.e. If you knew the measure of one
angle, how would you find the other?) Write
answers in terms of degrees.
(e) Complete the following cofunction
relationships by filling in blank. Write your
answers in terms of A.
tan ( A ) =cot ________( )
cot ( A ) =tan ________( )
83. (a) Find sec (^) ( A ).
(b) Find csc (^) ( B (^) ).
(c) Analyze the answers for (a) and (b). What
do you notice?
Continued on the next page…
b
c
A
C a B
Math 1330, Precalculus
(d) What is the relationship between angles A
and B? (i.e. If you knew the measure of one
angle, how would you find the other?) Write
answers in terms of degrees.
(e) Complete the following cofunction
relationships by filling in the blank. Write
your answers in terms of A.
sec ( A ) =csc _______( )
csc ( A ) =sec _______( )
84. Use a sum or difference formula to prove that
sin 90 ( − θ) =cos( θ )
D .
Use cofunction relationships to solve the following for
acute angle x.
85. sin 75( ) =cos ( x )
D
cos sin 10
x
sec csc 5
x
88. csc 41( ) =sec ( x )
D
89. tan 72( ) =cot ( x )
D
90. cot tan( )
x
Simplify the following.
91. cos 90( − x ) ⋅csc( x )
D
92. sin sec( )
x x
93. sin tan( )
x x
94. csc sin( )
x x
⎜ −^ ⎟⋅
95. sec 90( − θ ) ⋅cos( θ)
D
96. tan 90( − x ) ⋅ csc 90( − x )
D D
97. − sec 20( ) sin 70( )
D D
98. cos csc 6 3
⎛ π^ ⎞ ⎛ π⎞
cot sec 12 12
2 2 cos 72 +cos 18
D D
