FE_Review_F11_usethis (1)

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Unformatted text preview: MTH
142
Final
Exam
Formulas
and
Review
Problems
 
 Formulas
 The
following
formulas
will
be
provided
for
students
on
the
final
exam.

Students
will
 not
be
required
to
memorize
these
formulas.

All
other
definitions
and
formulas
 given
in
the
course
must
be
memorized.

Students
are
not
allowed
to
use
their
own
 formulas
sheets
on
the
final
exam.
 
 
 
 Sum
and
Difference
Identities:
 
 
 
 
 
 
 Double
Angle
Identities:
 
 
 
 
 
 
 Half
Angle
or
Power
Reduction
Identities:
 
 
 
 
 
 
 
 Area
of
an
Oblique
Triangle:


 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 Problems
 This
review
is
simply
a
collection
of
problems
from
the
semester
to
–
it
is
not

 comprehensive.

Problems
1
–
30
are
multiple‐choice.

Problems
31
–
35
are
free
 response
(show
all
work).
 
 1.

Convert
 
to
degrees:
 A)

 216 
 
 B)

 108 
 
 C)

 339.3 
 
 D)

 1.88 
 
 E)

None
of
these
 
 
 2.

The
reference
angle 
for
the
angle
 
pictured
below
is
given
by
 
 A)

 


 
 B)

 

 
 
 C)

 
 
 D)

 
 E)

None
of
these.
 
 
 
 
 
 
 
 
 θ 3.

Find
the
exact
value
of
 A)

 
 B)

 
 
 :
 
 
 C)

 D)

 
 
 
 
 
 
 E)

None
of
these
 2 4.

Given

 
,
where
 .

If
 equivalent
to:
 
 A)

 
 B)
 
 C)
 
 D)
 
 E)

None
of
these
 
 5.

Convert
 −165 
to
radian
measure:
 A)

 
 
 
 
is
 
 
 B)

 
is
the
reference
angle,
then
 
 C)

 
 
 D)

 
 
 E)

None
of
these
 
 6.

The
exact
value
of
 A)

 
is:
 
 B)

 
 
 C)

 
 
 D)

undefined

 E)

None
of
these
 
 
 7.

The
reference
angle
for
 A)

 
is:
 

 B)

 
 
 C)

 
 
 D)

 
 
 E)

None
of
these
 3 8.

Match
the
graph
below
with
the
correct
function:
 

















 


























 
 A)

y
=
3
+
csc
x
 
 B)

y
=
x
+
csc
x

 
 C)

y
=
3
+

sec
x
 D)

y
=
x
+
sec
x
 E)

None
of
these
 
 ⎛θ⎞ 9.

 y = −2 sin ⎜ ⎟ 
has
period:
 ⎝ 3⎠ A)

2
 
 B)

6
 
 
 C)

 

 
 D)
 

 
 
 E)

None
of
these
 
 
 10.

The
range
of
 y = tan x 
is:
 ⎛ π π⎞ A)

 ⎜ − , ⎟ 
 ⎝ 2 2⎠ B)

 ⎡ −1,1⎤ 
 ⎣ ⎦ ⎧ π⎫ C)

 ⎨ x | x ≠ odd integer multiples of ⎬
 2⎭ ⎩ D)

 −∞, ∞ 
 
 ( ) E)

None
of
these
 
 4 11.

What
is
the
amplitude
of
 y = 3 − 4 cos( 2 x − 5π ) ?
 A)

3
 
 
 B)

 
 
 
 C)

4
 
 
 D)

 
 
 E)

None
of
these
 
 12.

The
graph
of
 y = sec x 
has
vertical
asymptotes
at:
 A)

 
 B)

 
 C)

 
 D)

there
are
no
vertical
asymptotes
 
 E)

None
of
these
 
 
 13.

The
area
of
the
sector
of
a
circle
intercepted
by
a
central
angle
of
 60 
in
a
circle
 of
radius
10
m
is:
 
 A)

 
sq.m



 B)

 
sq.m



 C)

 
sq.m




 D)

 
sq.m




 E)

None
of
these
 
 
 14.

For
the
right
triangle
shown
below,
solve
forside
 x :
 A)
 
 
 B)

 
 C)

14
 
 
 
 D)

 
 
 
 E)

None
of
these
 
 5 15.

 
is:
 A)

 
 
 B)

 
 
 C)

 
 
 D)

 

 
 E)

None
of
these
 
 
 16.

An
equivalent
expression
for
 cot arctan x 
is:
 ( ) A)

 x 
 
 1 B)

 
 
 x 1 C)

 2 

 
 x D)

 x 

 
 E)

None
of
these
 
 
 17.

The
domain
of
 y = cos −1 ( x ) is
 A)

 
 B)

 
 C)

 
 D)
 
 E)

None
of
these
 
 
 18.

The
range
of
 y = sin −1 ( x ) is
 ⎡ π π⎤ A)

 ⎢ − , ⎥ 
 ⎣ 2 2⎦ B)

 −∞, ∞ 
 ( ) C)

 D)
 
 
 E)

None
of
these
 6 19.

Simplify
 :
 A)

 

 B)

 
 
 C)

 
 
 D)

 
 
 E)

None
of
these
 
 cos x 20.

Simplify
 :
 1 + tan 2 x A)

 cos x + cos x cot 2 x 

 B)

 cos x sin 2 x 
 
 C)

 sec x 
 
 3 D)

 cos x 


 
 E)
None
of
these
 
 21.

Use
a
sum
identity
to
find
the
exact
value
of
 sin 105 :
 A)

 

 B)

 

 C)

 

 D)

 

 E)

None
of
these
 










 22.

Find
 :
 A)

 

 B)

 
 
 C)

 
 
 
 D)

 
 
 E)

None
of
these
 7 ⎛ π⎞ 23.

Use
the
difference
identity
to
rewrite
 cos ⎜ x − ⎟ :
 4⎠ ⎝ A)

 B)

 
 
 2 cos x + sin x 
 
 2 ( ) C)

 2 cos x sin x 
 D)

0




 E)

None
of
these
 
 
 24.

Find
all
solutions
for
the
equation
 sin x + 3 = − sin x 

in
the
interval
 A)

 

 
 B)

 C)

 D)
 :
 

 
 


 
 
 

 



 E)

None
of
these
 
 25.

Find
all
solutions
for
the
equation
 cos( 2 x ) = A)

 B)

 2 

in
the
interval
 2 :
 

 
 

 
 C)

 
 



 D)
 
 

 


 E)

None
of
these
 
 
 
 
 
 
 8 26.

If
 A)

 
and
 ,
find
 :
 

 B)

 







 
 C)

 
 
 D)
 
 
 E)
None
of
these
 
 
 27.

The
radius
of
a
circle
is
8
m.

Find
the
length
of
an
arc
of
the
circle
intercepted
 by
a
central
angle
of
 A)

 
m
 
 B)

 m
 
 C)

 m
 
 D)

 
m
 :
 
 E)

None
of
these
 
 28.

Given
an
oblique
triangle
with
 C = 72 , A = 15 , 
and
 b = 342.6 ,
find
side
 a .
 A)

1258.92
 
 
 B)

88.79

 
 C)

6323.1

 
 D)

326.28

 
 E)

None
of
these
 
 
 29.

Given
an
oblique
triangle
with
 a = 80, b = 51, 
and
 c = 113 ,
find
angle
 C .
 A)

 

 
 
 B)

 

 
 C)

 

 
 D)

 

 
 E)

None
of
these
 
 
 9 30.

What
is
the
angle
of
elevation
(in
degrees)
to
the
top
of
a
building
200
feet
tall
 from
a
point
12
feet
from
the
base
of
the
building?
 
 A)

 3.4 

 
 
 B)

 93.4 
 
 C)

 86.6 

 
 D)

 12 

 
 E)

None
of
these
 
 _________________________________________________________________________________________________
 
 
 Show
all
work
for
Problems
31
–
35.
 
 31.

Verify
the
identity:
 sin 2 θ = sec θ − cosθ 
 





 
 
 
 
 cosθ 
 32.

Solve
the
equation
 2 cos 2 θ − cosθ = 1 
for
 θ 
in
 .


 
 33.

To
find
the
height
of
a
building,
a
surveyor
takes
two
measurements
as
 indicated
in
the
figure
below.

Use
the
information
given
below
to
find
the
height
of
 the
building.

Round
your
answer
to
2
decimal
places
 














 
 
 
 
 
 
 
 
 
 
 
 34.

A
plane
is
flying
in
a
race
in
which
the
course
is
in
the
shape
of
a
triangle
with
 sides
as
given
in
the
figure
below.

Find
the
angle
(in
degrees)
between
the
220
 miles
and
the
310
miles
sides.

Round
your
answer
to
2
decimal
places.

Circle
your
 final
answer.
 



































 
 10 
 ⎛ 3π ⎞ 35.


Verify
the
identity:

 cos ⎜ + θ ⎟ = sin θ .
 ⎝2 ⎠ 

 
 
 
 
 
 
 








































































 
 
 Answers
 
 
 
 Multiple‐Choice
 1.
B
 6.
C
 11.
C
 16.
B
 21.
D
 26.
A
 2.
C
 7.
A
 12.
B
 17.
C
 22.
B
 27.
A
 3.
B
 8.
A
 13.
D
 18.
A
 23.
B
 28.
B
 4.
C
 9.
E
 14.
A
 19.
B
 24.
E
 29.
A
 5.
D
 10.
D
 15.
B
 20.
D
 25.
C
 30.
C
 
 
 
 
 Free
–Response
 
 
 
31.


 
 
 
 
 
 











32. 
 sin 2 θ = sec θ − cosθ cosθ Work on RHS: 2 cos 2 θ − cosθ = 1 1 2 cos 2 θ − cosθ − 1 = 0 = − cosθ cosθ ( 2 cosθ + 1)(cosθ − 1) = 0 1 cosθ cosθ 






















 2 cosθ + 1 = 0 or cosθ − 1 = 0 
 = − cosθ cosθ 1 cosθ = − or cosθ = 1 1 − cos 2 θ 2 = cosθ 2π 4π θ = 0, , sin 2 θ 33 = cosθ 
 
 
 
 
 
 
 11 
 
 33.
 
 
 Find the angles in the oblique triangle first, then use Law of Sines: sin 10o sin 15o = 200 x where x is the hypotenuse of the smaller right triangle. 
 Solving for x : x= o 200 sin 15 = 298.0958955 sin 10o Use right triangle trig on the smaller right triangle: h sin 25o = (h is the height of the building) x h = x sin 25o = ( 298.0958955) sin 25o = 125.98 ft 
 
 34.

 Use Law of Cosines 1522 = 2202 + 3102 − 2( 220)(310) cos C 1522 − 2202 − 3102 = cos C −2( 220)(310) 
 .89 = cos C 
 35.

 C = cos −1 (0.89) = 27.13o 
 ⎛ 3π ⎞ cos ⎜ + θ ⎟ = sin θ ⎝2 ⎠ Expand the left side using the sum identity: ⎛ 3π ⎞ 3π 3π cos ⎜ + θ ⎟ = cos cosθ − sin sin θ 2 2 ⎝2 ⎠ 

 
 = 0 ⋅ cosθ − ( −1) ⋅ sin θ = 0 + sin θ = sin θ 12 ...
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This note was uploaded on 04/05/2012 for the course MTH 142 taught by Professor John during the Fall '04 term at Moraine Valley Community College.

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