Precalculus
MIDTERM EXAMINATION REVIEW
UNIT TUC
THE UNIT CIRCLE
2)
Convert 85 to radians.
3)
Determine the radian measure of the central angle whose radius is 10 and which has
3p
an arc length of
.
5
4)
A sector of a circle has arc length of
the radius and area.
Convert
7p
to degrees.
18
1)
2p
and a central angle of 45. Determine
7
5)
A sector of a circle has a central angle of 50 and a radius of 9. Determine the
arc length and the area.
6)
A gear with a 14 cm diameter rotates at 180 rev/min.
a)
b)
c)
Convert the angular speed to radians per second.
Determine the linear speed of a point on the circumference of the gear.
If the gear rotates for 3 minutes, how far does the gear travel?
7)
Express each of the following in terms of its reference angle.
a)
cos 153
b)
tan 261
e)
tan
f)
sec
8)
Evaluate each of the following giving exact answers.
a)
cos 120
e)
sec
11p
6
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-23p
7
c)
csc 318
d)
sec 571
g)
cot
22p
15
h)
csc
16p
9
b)
sin 150
c)
tan 180
d)
cot 2
f)
sin
g)
csc
17p
4
h)
cos
19p
3
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Precalculus
9)
Given tan  =
(Quadrant IV), determine the value of the five other
trigonometric functions.
10) Given csc  = -8 and
p
2
trigonometric functions.
<<
3p
, determine the value of the five other
2
11) Evaluate each of the following giving exact answers.
a)
Sin-1 1
b)
Cos-1 -1
c)
Tan-1 -1
d)
Sin-1
e)
Tan-1 0
f)
Sin-1
g)
tan
h)
Sin-1
i)
Tan-1 (cos 0)
j)
Cos-1
k)
cos
l)
Cos-1
m)
csc
n)
sin (Tan-1 4)
o)
Cos-1
UNIT TT
1
2
3
2
TRIANGLE TRIGONOMETRY
1)
The angle of depression from the top a building to a person on the ground is 37.
The person is 94 feet from the base of the building. How tall is the building?
2)
Determine the lengths of the base and altitude of an isosceles triangle whose vertex
angle is 56 and whose legs are 23 feet in length.
3)
If the sides of an isosceles triangle are 10, 10, and 4, determine the measure of
angles.
MPH/CR 12/11
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Precalculus
4)
An object is spotted on the ground from the top of a building that is 75 feet high.
If the angle of depression is measured to be 22, what is the distance from the
object to the base of the building?
5)
A kite is flying 100 feet above the ground. How many feet of string are used if the
angle of elevation is 53?
6)
What is the area of a regular decagon inscribed in a circle with a radius of 12 cm?
7)
A parallelogram has sides 18 and 15 feet in length with an included angle measuring
48. Determine the lengths of the diagonals.
8)
Determine the area of the quadrilateral shown.
9)
Using the information given, determine the number
of solutions for each triangle and whether it should
be solved using the Law of Sines or the Law of
Cosines. Do not solve the triangles!!
8
27
140
13
a)
MID if m = 14, i = 21, and d = 5
b)
TRM if mT = 14, mR = 93, and
m = 17
c)
JAN if m Ð J = 54, a = 12, and
j=8
d)
ICE if mC = 75, i = 26, and
e = 29
e)
SNO if mN = 25, o = 22, and
n = 15
f)
CLD if c = 26, l = 15, and d = 18
g)
SAT if mA = 86, mT = 61, and a = 33
10) A triangular lot has sides measuring 40, 48, and 31 feet. Find the measures of the
angles of the lot.
11) Solve ABC if mA= 35, a = 12, and b = 15.
12) In CAT, mC = 27, a = 12.3, and c = 10.8. Solve the triangle.
MPH/CR 12/11
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Precalculus
UNIT TGIE
1)
TRIGONOMETRIC GRAPHS, IDENTITIES,
AND
EQUATIONS
Complete the following table.
Amplitude
Reflected?
(Yes or No)
Period
1
4
Period
Horizontal
Shift
Vertical
Shift
Axis of
Wave
Max, Min,
Intercept?
a) y = -3 sin 3x + 2
b) y =
1
2
cos
c) y = 2 sin
2p
3
3
2
(x + 4)
(x - p ) - 1
æp
ö
÷
x
p
ç
÷
è2
ø
d) y + 5 = 6 sin ç
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Precalculus
2)
Complete the following table.
Reflected?
(Yes or No)
Period
1
2
Period
Horizontal
Shift
a)
y = tan 4x
b)
y = -2 tan
c)
æ
1ö
y = 4 tan 2p çç x + ÷÷ - 5
7ø
è
d)
æp
ö
y = - tan çç x - 4p ÷÷ + 8
è2
ø
3)
Identify two equations for each of the functions whose graphs are given below.
p
5
Vertical
Shift
Asymptotes
x +1
a)
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5
Precalculus
3)
b)
c)
4)
Write the equation of each function described.
a)
A sine function has a range of -7  y  7. The function is not reflected and not
shifted. The distance between an intercept and the next consecutive maximum
is 5.
b)
A function has a maximum value of 11 at x = 0. The minimum value is -3. The
horizontal distance between two consecutive minima is 8.
c)
The horizontal distance between two consecutive asymptotes of a non-reflected
tangent graph is 4p . The function has been shifted up 4 units and has a
stretch factor of 3.
MPH/CR 12/11
6
Precalculus
5)
Verify the following identities.
a)
sin x (csc x - sin x) = cos2 x
b)
sec x - sin x tan x = cos x
c)
(sec x + tan x)(csc x - 1) = cot x
d)
sec4 x - tan4 x = 1 + 2 tan2 x
f)
(1 - sin2 ) sec  = cos 
e)
6)
Solve for 0  x < 360. Give answers for all exact values in radians. Round to
tenths.
a)
sin x = -0.65
b)
sec x = 2.8
c)
3 tan x = -5
d)
6 cos x + 4 = 1
e)
sin x tan x = sin x
f)
3 cos2 x - cos x - 2 = 0
g)
3 csc2 x = 5 csc x + 2
h)
6 sin x + 3 = 8 cos2 x
i)
sin2 x = sin x
j)
3 cos x + 5 sin x = 0
k)
5 sin 3x + 4 = 3
7)
Determine the angle of inclination to the nearest tenth of a degree for:
8)
a)
3x + 5y = 10
b)
-5x + 8y + 3 = 0
c)
2x + 5y - 7 = 0
d)
through (2, 7) and (4, 1)
e)
parallel to 7x - 9y + 5 = 0
f)
perpendicular to -8x + 5y + 11 = 0
Write the equation of the line in general form for the line described. Round slopes
to the nearest hundredth.
a)
 = 139 thru (-3, 0.5)
c)
 = 110 thru (8, -2)
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b)
 = 41 with y-intercept of 2
7
Precalculus
UNIT TAF
1)
TRIGONOMETRIC ADDITION FORMULAS
Simplify and evaluate each of the following with exact values if possible.
a)
2 cos2 50 - 1
b)
c)
4 sin 15 cos 15
d)
e)
1 – 2 sin2 8x
g)
cos
5p
12
cos
p
4
- sin
5p
12
sin
p
4
f)
sin 48 cos 27 + cos 48 sin 27
h)
sin
7p
10
cos
p
5
- cos
7p
10
sin
2)
Evaluate tan
3)
Evaluate each of the following if sin A =
a)
sin 2B
b)
cos 2A
c)
tan 2B
d)
sin (A + B)
e)
cos (A + B)
f)
tan (B - A)
4)
Verify each of the following identities.
5
if tan x = 3.
a)
sin 2B
= 2 tan B
cos2 B
c)
tan u sin 2u = 2 sin2 u
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p
b)
, cos B =
3
p
3p
, and
<A<
< B < 2.
5
2
2
= cot A
8
Precalculus
UNIT PC
POLAR COORDINATES
1)
Write in polar form.
a)
(6, 6)
b)
(7, -4)
c)
5i
d)
-2 + 2i
e)
-3 - 3 3 i
f)
10
2)
Write in rectangular form.
a)
(9, 30)
b)
c)
6 cis
3p
4
d)
2 cis 
e)
f)
2 cis
4p
3
3)
Express the product in polar form.
a)
(5 cis 25) (6 cis 65)
6 cis 150
b)
4)
Simplify and write answer in rectangular form.
a)
(2 cos 45)4
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b)
c)
(-3 + 3i)4
d)
(2 - 2 3 i)5
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