# Columbia University Analytic Trigonometry Questionnaire

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section 5.1(f) ?5.2?5.4?5.5

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(Exercises for Chapter 5: Analytic Trigonometry) E.5.1
CHAPTER 5:
Analytic Trigonometry
(A) means “refer to Part A,” (B) means “refer to Part B,” etc.
(Calculator) means “use a calculator.” Otherwise, do not use a calculator.
Write units in your final answers where appropriate. Try to avoid rounding intermediate
results; if you do round off, do it to at least five significant digits.
SECTION 5.1:
FUNDAMENTAL TRIGONOMETRIC IDENTITIES
Ignore domain issues in these problems.
1) Complete the Identities. Fill out the table below so that, for each row, the left
side is equivalent to the right side, based on the type of identity given in the last
column. (A)
Left Side
Right Side
Type of Identity (ID)
csc ( x )
tan ( x )
sinlD
Reciprocal ID
tan ( x )
⎛π

tan ⎜ − x ⎟
⎝2

Quotient ID
sscosycotysinED
cos ( x )
( )
Reciprocal ID
Cofunction ID
Cofunction ID
( )
as
( )
tank
Even / Odd (Negative-Angle)
ID
Even / Odd (Negative-Angle)
ID
Even / Odd (Negative-Angle)
ID
sin 2 ( x ) + cos 2 ( x )
1
Pythagorean ID
tan 2 ( x ) + 1
sea
Pythagorean ID
⽐比 凶
Pythagorean ID
sin − x
cos − x
tan − x
1+ cot 2 ( x )
s.in x
(Exercises for Chapter 5: Analytic Trigonometry) E.5.2
2) Simplify the following. Find the most “compact” equivalent expression. (A-F)
a)
1 − sec ( − x )
1 − cos ( − x )
; b)
tan (θ ) + cot (θ )
; c) sec 2 ( x ) − sec 2 ( x ) sin 2 ( x ) ;
cot (θ )
⎛π ⎞
sin ⎜ − t ⎟
⎝2 ⎠
d) cot 4 ( x ) + 2cot 2 ( x ) + 1; e)
; f) ⎡⎣ csc (α ) + cot ( − α ) ⎤⎦ ⎡⎣1+ cos ( − α ) ⎤⎦
cot ( t )
3) Use the given trigonometric substitution to rewrite the given algebraic expression as a
trigonometric expression in θ , where θ is acute. Simplify. (These types of
substitutions are used in an advanced integration technique in calculus.) (G)
a) Substitute x = 4 sin (θ ) in the expression 16 − x 2 .
b) Substitute x = 6 tan (θ ) in the expression
x 2 + 36 .
c) Substitute x = 3sec (θ ) in the expression
x2 − 9 .
SECTION 5.2: VERIFYING TRIGONOMETRIC IDENTITIES
Ignore domain issues in these problems.
1) Verifying the following identities. (A-C)
1− sin ( − x )
a)
= sec ( x ) + tan ( x )
cos ( − x )
b)
sin ( u ) cos ( u ) − cos ( u )
= − cot ( u )
sin ( u ) − sin 2 ( u )
c)
1
− tan ( − α ) = csc (α ) sec (α )
tan (α )
d)
1
1
+
= 2sec ( x ) tan ( x )
csc ( x ) + 1 csc ( x ) − 1
e)
f)
1− cos (θ )
1+ cos (θ )
=
1− cos (θ )
sin (θ )
1
= csc 2 ( β ) + csc ( β ) cot ( β )
1− cos ( β )
()
()
()
()
g) tan5/2 x + tan1/2 x = ⎡⎣sec 2 x ⎤⎦ tan x
2
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(Exercises for Chapter 5: Analytic Trigonometry) E.5.3
SECTION 5.3: SOLVING TRIGONOMETRIC EQUATIONS
Use radian measure for angles in the following problems. Give exact solutions.
1) Find all real solutions, and find the particular solutions in the interval [ 0, 2π ) .
a) 2sin ( x ) − 3 = 0 . (A)
b) 6 cos (θ ) + 3 2 = 0 . (A)
c) cos (α ) = − π . (A)
d) sin ( u ) = −1 . (A)
e) cos ( u ) = 0 . (A)
f) 2 + csc ( u ) = 0 . (A)
g) sec ( x ) = 2 . (A)
1
. (A)
2
i) 3tan ( x ) = 3 . (A, B)
h) csc ( x ) =
j) cot (θ ) = 0 . (A, B)
k) cot 2 (θ ) = 3 . (A-C)
l) tan 2 (θ ) + tan (θ ) = 0 . (A, B, D)
m) csc 4 ( x ) − 3csc 3 ( x ) + 2 csc 2 ( x ) = 0 . (A, D)
n) 2 cos 3 ( x ) + sin 2 ( x ) = 1+ cos ( − x ) . (A, D, E)
o) 2 cos ( 4x ) − 1 = 0 . (A, F)
p) csc ( 3x ) = 1 . (A, F)
q) tan 2 ( 3x ) = 3 . (B, C, F)
2) Consider the equation tan ( x ) = 2 . (A, B, G)
a) Find the solutions of the equation in the interval [ 0, 2π ) .
b) Approximate the solutions you found in a) to four significant digits.
(Calculator)
c) Find all real solutions of the equation.
3) Consider the equation 5 cos 2 ( x ) − 14 cos ( x ) − 3 = 0 . (A, D, G)
a) Find the solutions of the equation in the interval [ 0, 2π ) .
b) Approximate the solutions you found in a) to four significant digits.
(Calculator)
c) Find all real solutions of the equation.
section 5.3
7 a 25 以好 冷 0
25灬 B
sinx 浯
(Exercises for Chapter 5: Analytic Trigonometry) E.5.4
SECTIONS 5.4 and 5.5:
MORE TRIGONOMETRIC IDENTITIES
1) Complete the Identities. Fill out the table below so that, for each row, the left
side is equivalent to the right side, based on the type of identity given in the last
column. (A, Handout)
Left Side
(
sin u + v
Right Side
)
Sum ID
sinwnwswztcoscussinwscoscyoswts.in
(
)
(
)
sin u − v
(
)
(
)
wslujsincyoscuscoslnt5
(
)
tanuttmM
cos u + v
tan u + v
cos u − v
tan u − v
⼼心 5⼼心⼼心
tuuttanlbze
l
tanlujtmussinmzoscn
nu Sims
( )
lttanluHanlyzsinlujosuboszun
( )
sinaui.rswhy
sin 2u
cos 2u
tan ( 2u )
zoszcuH2tl
sin 2 ( u )
tmiujltoswnz
cos ( u )
2
⎛θ⎞
sin ⎜ ⎟
⎝ 2⎠
⎛θ⎞
cos ⎜ ⎟
⎝ 2⎠
⎛θ ⎞
tan ⎜ ⎟
⎝ 2⎠
Type of Identity (ID)
iiicoslzujltwsczuj
z.it Ì cos⼝口以
⼟土
faith

tri

Sum ID
Sum ID
Difference ID
Difference ID
Difference ID
Double-Angle ID
Double-Angle ID
(write all three versions)
Double-Angle ID
Power-Reducing ID (PRI)
Power-Reducing ID (PRI)
Half-Angle ID
Half-Angle ID
Half-Angle ID
(write all three versions)
(Exercises for Chapter 5: Analytic Trigonometry) E.5.5
2) Use the Sum Identities to find the exact values of the following. (A, B)
( )
cos ( 75 )
tan ( 75 ) . Remember to rationalize the denominator.
a) sin 75
b)
c)

⎛π⎞
3) Find the exact value of sin 22.5 , or sin ⎜ ⎟ . (A, B)
⎝ 8⎠
(
)
⎛π⎞
4) Find the exact value of cos 22.5 , or cos ⎜ ⎟ . (A, B)
⎝ 8⎠
(
)
5) Find the exact values of the following expressions. (A, B)
( ) ( )
( ) ( )
a) sin 80 cos 20 − cos 80 sin 20
( ) ( )
( )
b) cos 55 cos 5 − sin ( 55 ) sin 5

⎛ 3π ⎞
⎛ 3π ⎞
c) 2 sin ⎜ ⎟ cos ⎜ ⎟
⎝ 8 ⎠
⎝ 8 ⎠
⎛π⎞
d) 2 cos 2 ⎜ ⎟ − 1
⎝ 12 ⎠
()
()
6) Simplify cos 4 θ − sin 4 θ . (A, B)
7) Simplify
sin ( 2x ) cos ( 2x )
. (A, B)
3cos 2 ( 2x ) − 3sin 2 ( 2x )
8) Verifying the following identities. (A, B)
(
)
()
a) sin θ + π = − sin θ . (Also, try to see why this is true using the Unit Circle.)
b)
sin ( 2x )
1+ cot ( x )
2
= 2sin 3 ( x ) cos ( x )
c) Verify the Sum Identity for tangent, tan ( u + v ) =
Sum Identities for sine and cosine.
tan ( u ) + tan ( v )
, by using the
1− tan ( u ) tan ( v )
(Exercises for Chapter 5: Analytic Trigonometry) E.5.6.
9) For each of the equations below, find all real solutions, and find the particular
solutions in the interval [ 0, 2π ) .
a) 4 sin ( x ) cos ( x ) − 1 = 0
b) cos ( 2x ) + 3cos ( x ) + 2 = 0
c) sin ( 2x ) = sin ( x )
()
10) Rewrite sin ⎡⎣ 2arcsin x ⎤⎦ as an equivalent algebraic expression. Assume x is in
⎡⎣ −1, 1⎤⎦ .
11) Use the Power-Reducing Identities (PRIs) to rewrite the expression cos 4 x using
only the first power of cosine expressions (and no other powers). Fill in the blanks
below with real numbers:
cos 4 ( x ) =
+
cos ( 2x ) +
cos ( 4x )
12) Rewrite each expression below using either a Product-to-Sum Identity or a
Sum-to-Product Identity. Use Even / Odd (Negative-Angle) Identities where
appropriate.
a) cos ( 3θ ) cos ( 5θ )
b) cos ( 5α ) + cos ( 3α )
c) sin ( 3x ) + sin ( x )
d) sin ( 9θ ) cos (10θ )
e) sin ( 4x ) sin ( x )
f) cos ( 7x ) − cos ( x )
g) sin ( 8α ) − sin ( 2α )
h) cos ( 5α ) sin ( 4α )
YOU DO NOT HAVE TO MEMORIZE THE PRODUCT-TO-SUM IDENTITIES;
THEY WILL BE PROVIDED TO YOU IF NECESSARY ON THE EXAM. THE
SAME GOES FOR THE SUM-TO-PRODUCT IDENTITIES.
(Answers for Chapter 5: Analytic Trigonometry) A.5.1
CHAPTER 5:
Analytic Trigonometry
SECTION 5.1:
FUNDAMENTAL TRIGONOMETRIC IDENTITIES
1)
Left Side
csc ( x )
tan ( x )
tan ( x )
Right Side
1
sin ( x )
1
cot ( x )
sin ( x )
cos ( x )
Type of Identity (ID)
Reciprocal ID
Reciprocal ID
Quotient ID
⎛π

tan ⎜ − x ⎟
⎝2

cot ( x )
Cofunction ID
cos ( x )
⎛π

sin ⎜ − x ⎟
⎝2

Cofunction ID
( )
cos ( − x )
tan ( − x )
− sin ( x )
Even / Odd (Negative-Angle) ID
cos ( x )
Even / Odd (Negative-Angle) ID
− tan ( x )
Even / Odd (Negative-Angle) ID
1
Pythagorean ID
sec 2 ( x )
Pythagorean ID
csc 2 ( x )
Pythagorean ID
sin − x
sin 2 ( x ) + cos 2 ( x )
tan 2 ( x ) + 1
1+ cot 2 ( x )
()
2) a) − sec ( x ) ; b) sec 2 (θ ) ; c) 1; d) csc 4 x ; e) sin ( t ) ; f) sin (α )
3) a) 4 cos (θ ) ; b) 6sec (θ ) ; c) 3tan (θ )
SECTION 5.2: VERIFYING TRIGONOMETRIC IDENTITIES
1) Solutions will vary.
(Answers for Chapter 5: Analytic Trigonometry) A.5.2
SECTION 5.3: SOLVING TRIGONOMETRIC EQUATIONS
1)

π

⎧ π 2π ⎫
+ 2π n n ∈ ⎬ . In [ 0, 2π ) : ⎨ ,
a) ⎨ x ∈ x = + 2π n or x =
⎬.
3
3
⎩3 3 ⎭
⎪⎩
⎪⎭

+ 2π n n ∈ ⎬ , or, equivalently,
b) ⎨ θ ∈ θ = ±
4
⎪⎩
⎪⎭

⎧ 3π 5π ⎫
θ
∈
θ
=
+
2
π
n
or
θ
=
+
2
π
n
n
∈

⎬ . In [ 0, 2π ) : ⎨ ,
⎬.
4
4
⎩4 4 ⎭
⎪⎩
⎪⎭
c) No real solutions; the solution set is ∅ . No real solutions in [ 0, 2π ) .
(
(
)
)
(
)

⎧ 3π ⎫
+ 2π n n ∈ ⎬ . Solutions in [ 0, 2π ) : ⎨ ⎬ .
d) ⎨ u ∈ u =
2
⎩2 ⎭
⎪⎭
⎩⎪

π
⎧ π 3π ⎫
e) ⎨ u ∈ u = + π n n ∈ ⎬ . Solutions in [ 0, 2π ) : ⎨ ,
⎬.
2
⎩2 2 ⎭
⎪⎩
⎪⎭

11π
+ 2π n or u =
+ 2π n n ∈ ⎬ , or, equivalently,
f) ⎨ u ∈ u =
6
6
⎩⎪
⎭⎪

⎧ 7π 11π ⎫

π
+ 2π n or u = − + 2π n n ∈ ⎬ . In [ 0, 2π ) : ⎨ ,
⎨ u ∈ u =
⎬.
6
6
6 ⎭
⎩ 6
⎩⎪
⎭⎪

π
g) ⎨ x ∈ x = ± + 2π n n ∈ ⎬ , or, equivalently,
3
⎪⎩
⎪⎭

⎧ π 5π ⎫
π

+ 2π n n ∈ ⎬ . In [ 0, 2π ) : ⎨ , ⎬ .
⎨ x ∈ x = + 2π n or x =
3
3
⎩3 3 ⎭
⎩⎪
⎭⎪
(
)
(
)
(
)
(
(
)
)
(
)
h) No real solutions; the solution set is ∅ . No real solutions in [ 0, 2π ) .

⎧ π 7π ⎫
π
i) ⎨ x ∈ x = + π n n ∈ ⎬ . Solutions in [ 0, 2π ) : ⎨ ,
⎬.
6
⎪⎩
⎪⎭
⎩6 6 ⎭

π
⎧ π 3π ⎫
j) ⎨ θ ∈ θ = + π n n ∈ ⎬ . Solutions in [ 0, 2π ) : ⎨ ,
⎬.
2
⎩2 2 ⎭
⎪⎩
⎪⎭

π
k) ⎨ θ ∈ θ = ± + π n n ∈ ⎬ , or, equivalently,
6
⎪⎩
⎪⎭
π

⎧ π 5π 7π 11π ⎫
+ π n ( n ∈ ) ⎬ . In [ 0,2π ) : ⎨ , ,
,
⎨ θ ∈ θ = + π n or θ =
⎬.
6
6
⎩6 6 6 6 ⎭

(
)
(
)
(
)
(Answers for Chapter 5: Analytic Trigonometry) A.5.3

+ π n n ∈ ⎬ , or, equivalently,
l) ⎨ θ ∈ θ = π n or θ =
4
⎩⎪
⎭⎪

π
7π ⎫
⎧ 3π
, π,
⎨ θ ∈ θ = π n or θ = − + π n n ∈ ⎬ . In [ 0, 2π ) : ⎨0,
⎬.
4
4
4

⎩⎪
⎭⎪
(
)
(
)

π
π

+ 2π n n ∈ ⎬ .
m) ⎨ x ∈ x = + 2π n or x = + 2π n or x =
6
2
6
⎪⎩
⎪⎭
⎧ π π 5π ⎫
Solutions in [ 0, 2π ) : ⎨ , ,
⎬.
⎩6 2 6 ⎭

π
2π n
n ∈ ⎬ , by rotational symmetry. Less
n) ⎨ x ∈ x = + π n or x =
2
3
⎪⎩
⎪⎭

π

+ 2π n n ∈
efficiently: ⎨ x ∈ x = + π n or x = 2π n or x = ±
2
3
⎪⎩
(
(
)
)
(
⎧ π 2π 4π 3π ⎫
Solutions in [ 0, 2π ) : ⎨0, ,
,
,
⎬.
2
3
3
2

π πn
+
n ∈ ⎬ . The following form may
o) ⎨ x ∈ x = ±
12 2
⎩⎪
⎭⎪
(

)⎬⎪ .

)

π πn
5π π n
+
or x =
+
n ∈ ⎬ .
be more useful for later: ⎨ x ∈ x =
12 2
12 2
⎩⎪
⎭⎪
⎧ π 5π 7π 11π 13π 17π 19π 23π ⎫
Solutions in [ 0, 2π ) : ⎨ ,
,
,
,
,
,
,
⎬.
⎩12 12 12 12 12 12 12 12 ⎭

π 2π n
⎧ π 5π 3π ⎫
n ∈ ⎬ . In [ 0, 2π ) : ⎨ ,
p) ⎨ x ∈ x = +
,
⎬.
6
3
⎩6 6 2 ⎭
⎩⎪
⎭⎪
(
(
)
)

π πn
n ∈ ⎬ . The following form may be more useful for
q) ⎨ x ∈ x = ± +
9
3
⎪⎩
⎪⎭

π πn
2π π n
or x =
+
n ∈ ⎬ . Solutions in [ 0, 2π ) :
later: ⎨ x ∈ x = +
9
3
9
3
⎩⎪
⎭⎪
⎧ π 2π 4π 5π 7π 8π 10π 11π 13π 14π 16π 17π ⎫
,
,
,
,
,
,
,
,
,
,
⎨ ,
⎬.
9
9
9
9
9
9 ⎭
⎩9 9 9 9 9 9
(
)
(
2) a)
{arctan 2,
)
π + arctan 2} ; equivalently, { tan −1 2, π + tan −1 2} .
b) Approximately: {1.107, 4.249} . (Make sure your calculator is in radian mode.)
c)
{ x ∈ x = arctan 2 + π n ( n ∈)} , or { x ∈ x = tan
−1
(
)}
2 + π n n ∈ .
(Answers for Chapter 5: Analytic Trigonometry) A.5.4
3)

⎛ 1⎞
⎛ 1⎞ ⎫
a) Solutions in [ 0, 2π ) : ⎨arccos ⎜ − ⎟ , π + arccos ⎜ ⎟ ⎬ . Equivalent forms:
⎝ 5⎠
⎝ 5⎠ ⎭

⎧ −1 ⎛ 1 ⎞
⎛ 1⎞
⎛ 1⎞ ⎫
−1 ⎛ 1 ⎞ ⎫ ⎧
⎨cos ⎜⎝ − ⎟⎠ , π + cos ⎜⎝ ⎟⎠ ⎬ , ⎨π − arccos ⎜⎝ ⎟⎠ , π + arccos ⎜⎝ ⎟⎠ ⎬ , and
5
5 ⎭ ⎩
5
5 ⎭

⎛ 1⎞
⎛ 1⎞ ⎫
⎨arccos ⎜⎝ − ⎟⎠ , 2π − arccos ⎜⎝ − ⎟⎠ ⎬ .
5
5 ⎭

b) Approximately: {1.772, 4.511} . (Make sure your calculator is in radian mode.)
⎧⎪
⎫⎪
⎛ 1⎞
c) ⎨ x ∈ x = ± arccos ⎜ − ⎟ + 2π n n ∈ ⎬ , or, equivalently,
⎝ 5⎠
⎩⎪
⎭⎪
⎧⎪
⎫⎪
1⎞
−1 ⎛
x
∈
x
=
±
cos

+
2
π
n
n
∈

⎬ , or, equivalently,
⎜⎝ 5 ⎟⎠
⎩⎪
⎭⎪
⎧⎪
⎫⎪
⎛ 1⎞
⎨ x ∈ x = ± arccos ⎜ ⎟ + 2n + 1 π n ∈ ⎬ .
⎝ 5⎠
⎪⎩
⎪⎭
(
)
(
(
) (
)
)
SECTIONS 5.4 and 5.5:
MORE TRIGONOMETRIC IDENTITIES
1)
Left Side
Right Side
Type of Identity (ID)
sin u + v
(
)
sin ( u ) cos ( v ) + cos ( u ) sin ( v )
Sum ID
(
)
cos ( u ) cos ( v ) − sin ( u ) sin ( v )
Sum ID
(
)
sin u − v
(
)
sin ( u ) cos ( v ) − cos ( u ) sin ( v )
Difference ID
(
)
cos ( u ) cos ( v ) + sin ( u ) sin ( v )
Difference ID
(
)
cos u + v
tan u + v
cos u − v
tan u − v
( )
sin 2u
tan ( u ) + tan ( v )
1− tan ( u ) tan ( v )
tan ( u ) − tan ( v )
1+ tan ( u ) tan ( v )
2 sin ( u ) cos ( u )
Sum ID
Difference ID
Double-Angle ID
(Answers for Chapter 5: Analytic Trigonometry) A.5.5
Left Side
Right Side
cos ( u ) − sin ( u ) , 1− 2sin ( u ) , and
2
( )
cos 2u
2
2cos ( u ) − 1
2
2 tan ( u )
tan ( 2u )
1 − cos ( 2u )
2
1 + cos 2u
( )
cos 2 ( u )
2
⎛θ⎞
sin ⎜ ⎟
⎝ 2⎠
or
1 1
− cos ( 2u )
2 2
or
1 1
+ cos 2u
2 2
( )
±
1− cos (θ )
±
1+ cos (θ )
2
(Choose the sign appropriately.)
⎛θ⎞
cos ⎜ ⎟
⎝ 2⎠
2
(Choose the sign appropriately.)
±
1− cos (θ )
1+ cos (θ )
=
1− cos (θ )
sin (θ )
Type of Identity (ID)
Double-Angle ID
(write all three versions)
Double-Angle ID
1− tan 2 ( u )
sin 2 ( u )
⎛θ ⎞
tan ⎜ ⎟
⎝ 2⎠
2
=
sin (θ )
1+ cos (θ )
(Choose the sign appropriately.)
Power-Reducing ID
(PRI)
Power-Reducing ID
(PRI)
Half-Angle ID
Half-Angle ID
Half-Angle ID
(write all three versions)
2)
a)
2+ 6
; b)
4
3)
2− 2
2
4)
2+ 2
2
6− 2
; c)
4
3 + 2 (rationalize the denominator in
3+3
).
3− 3
1
3
2
3
; b) ; c)
; d)
2
2
2
2
6) cos ( 2θ )
5) a)
7)
tan ( 4x )
6
8)
a) Hint: Use a Sum Identity.
b) Hints: Use a Double-Angle Identity and a Pythagorean Identity.
c) Hints: Use the Sum Identities for sine and cosine, and then divide the numerator
and the denominator by cos ( u ) cos ( v ) .
(Answers for Chapter 5: Analytic Trigonometry) A.5.6.
9)

π

+ π n or x =
+ π n n ∈
a) All real solutions: ⎨ x ∈ x =
12
12
⎩⎪
⎧ π 5π 13π 17π ⎫
Solutions in [ 0, 2π ) : ⎨ ,
,
,

⎩12 12 12 12 ⎭
(

)⎬⎪ .

+ 2π n or x = π + 2π n n ∈ ⎬ , or,
b) All real solutions: ⎨ x ∈ x = ±
3
⎪⎩
⎪⎭

+ 2π n or x =
+ 2π n or x = π + 2π n n ∈
equivalently, ⎨ x ∈ x =
3
3
⎪⎩
(
)
(
4π ⎫
⎧ 2π
Solutions in [ 0, 2π ) : ⎨ , π ,

3 ⎭
⎩ 3

π
c) All real solutions: ⎨ x ∈ x = π n or x = ± + 2π n n ∈ ⎬ , or,
3
⎪⎩
⎪⎭

π

+ 2π n n ∈ ⎬ ,
equivalently, ⎨ x ∈ x = π n or x = + 2π n or x =
3
3
⎪⎭
⎩⎪

π 2π n
or, equivalently, ⎨ x ∈ x = 2π n or x = +
n ∈ ⎬ .
3
3

(
)
(
(
)
)
5π ⎫
⎧ π
Solutions in [ 0, 2π ) : ⎨0, , π ,

3 ⎭
⎩ 3
10) 2x 1 − x 2
11) cos 4 ( x ) =
3
8
+
1
cos ( 2x ) +
2
1
cos ( 4x )
8
12)
a)
1
1
⎡⎣ cos ( 2θ ) + cos ( 8θ ) ⎤⎦ , which is simplified from ⎡⎣ cos ( − 2θ ) + cos ( 8θ ) ⎤⎦ ;
2
2
b) 2 cos ( 4α ) cos (α ) ; c) 2sin ( 2x ) cos ( x )
d)
1
1
⎡⎣sin (19θ ) − sin (θ ) ⎤⎦ , which is simplified from ⎡⎣sin (19θ ) + sin ( − θ ) ⎤⎦ ;
2
2
e)
1
⎡cos 3x − cos 5x ⎤ ; f) − 2sin 4x sin 3x ; g) 2cos 5α sin 3α ;

2⎣
h)
1
⎡sin ( 9α ) − sin (α ) ⎤⎦
2⎣
( )
( )
( ) ( )
( ) ( )

)⎬⎪ .

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