Download Ch 5

Semester 2 Review
Inverse functions & Restrictions:
(check INNER domain on composites)
  
sin
sin-1  1,1
  2 , 2 
cos
cos-1  1,1
0,  
tan
Graphs
y = Asin(Bx – C) + D
Amplitude = A
period =
 ,  
tan-1
y = Acsc(Bx – C) + D
first graph the associated sine equation
2
B
phase shift =
  
 , 
 2 2
Zeros become asymptotes.
C
B
Key points every
period
4
y = Atan(Bx – C)


Asymptotes:   Bx  C 
2
2
Zeros: midway between asymptotes
points:
(midway between asy. and zero,  A )
y = Acot(Bx – C)
(midway between zero and asy., A )
(midway between zero and asy., - A )
Revolution x 360 = Degrees
Revolution x 2π = radians
opposite
hypotenuse
adjacent
cos  
hypotenuse
opposite
tan  
adjacent
sin  
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hypotenuse
opposite
hypotenuse
sec =
adjacent
adjacent
cot =
opposite
Asymptotes: 0  Bx  C  
Zeros: midway between asymptotes
points:
(midway between asy. and zero, A )
csc =
0
30
45
sin Θ
cos Θ
tan Θ
Page 1 of 15
60
90
Semester 2 Review
Reciprocal
1 Identities (6)1
sin  
csc 
1
cos  
sec 
1
tan  
cot 
csc =
sin 
1
sec =
cos 
1
cot =
tan 
Quotient Identities (2)
sin 
tan  
cos 
Pythagorean Identities (3)
cos 
cot  
sin 
sin 2   cos 2   1
1  tan 2   sec 2 
1  cot 2   cot 2 
Ch 5
5.1 Find the radian measure of Θ.
Convert each angle in degrees to radians:
4.
60
5.
270
Convert each angle in radians to degrees:

4

7.
8.
4
3
Draw and label each angle in standard position:

3
 

10.
11.
12.
4
4
6.
-300
9.
6
 
7
4
5.2 Find the value of each of the 6 trigonometric functions of Θ in the figure.
1.
2.
c=5
Θ
a=1
Θ
b=4
a=3
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Semester 2 Review
Find each of the following. If a radical appears in the denominator, rationalize.
3.
tan 60
4.
sin 30
5.
cos 45
6.
csc 45
7.
cot 45
8.
sec 60
9. Given sin  
5
2
and cos  
, find the value of the four remaining trig functions.
3
3
10. Given sin  
21
2
and cos  
, find the value of the four remaining functions.
5
5
1
and Θ is an acute angle, find the value of cosΘ using a
2
trigonometric identity.
11. Given that sin  
3
and Θ is an acute angle, find the value of cosΘ using a
5
trigonometric identity.
12. Given that sin  
Find the value of the variable.
21.
a
24
a 750 yd
22.
c
10
a
500 ft
23.
A flagpole 14 meters tall has a shadow 10 meters long. Find the angle of elevation
of the sun to the nearest degree.
5.3
1. Let P(1, -3) be a point on the terminal side of Θ. Find each of the six trig functions of
Θ.
2. Let P(-3, -5) be a point on the terminal side of Θ. Find each of the six trig functions
of Θ.
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Semester 2 Review
Evaluate, if possible, the cosine and cosecant functions at the four following quadrantal
angles:

3
3.   0  0
4.   90 
5.   180  
6.   270 
2
2
7.
8.
If sin   0 and cos  0 , name the quadrant in which Θ lies.
If tan   0 and cos  0 , name the quadrant in which Θ lies.
9.
Given tan   
10.
1
and cos  0 , find sinΘ and secΘ.
3
2
Given tan    and cos  0 , find cosΘ and cscΘ.
3
5.4 Find the values of the trig functions at t on the unit circle.
Find the exact value of each trig function:
 
tan   
3.
4.
cos  60
 3
9
sin
7.
8.
cos 420
4
12.
 5 
sin  

 6 
13.
tan17
6.
cos  45
9.
cos 405
14,
sin
47
4
5.5 Determine the amplitude, period, and phase shift, then graph.
1

y  sin x
y  3cos x
y  3sin x
1.
2.
6.
2
2
7.
y  4 cos  x
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8.
2 

y  4sin  2 x 

3 

9.


y  3sin  2 x  
3

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Semester 2 Review
Write an equation of the form y  A sin( Bx  C ) .
13.
14.
Window
Xmin=-π/4
Xmax=π/2
Xscl=π/8
Ymin=-4
Ymax=4
Yscl=1
5.6 Graph.
17. y  2 tan
21.
x
for -   x  3
2
24.
y  3cot 2 x
5.7 Find the exact values.
2
sin 1
27.
28.
2
36.
 3 
sin 1  sin

2 

39.
cos cos 1  1.2 


18.
y  3 tan 2 x for 

4
x



y  csc  x  
4

25.
y  3sec
3
2
29.
 1
sin 1   
 2
sin 1

3
4
2
x for –π<x<5π
37.
cos cos 1 1.5

38.
sin 1  sin  
40.
5

cos  tan 1 
12 

41.
3

sin  tan 1 
4

5.8 Solve the triangle. Round to two decimal places.
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49.
Semester 2 Review
A = 34.5 , and b = 10.5
50.
A = 62.7 , and a = 8.4.
51.
From a point on level ground 125 feet from the base of a tower, the angle of
elevation is 57.2 . Approximate the height of the tower to the nearest foot.
52.
From a point on level ground 80 feet from the base of the Eiffel Tower, the angle
of elevation is 85.4 . Approximate the height of the Eiffel Tower to the nearest foot.
54.
A guy wire is 13.8 yards long and is attached from the ground to a pole 6.7 yards
above the ground. Find the angle, to the nearest tenth of a degree, that the wire makes
with the ground.
56.
You are standing on level ground 800 feet from Mt. Rushmore, looking at the
sculpture of Abraham Lincoln’s face. The angle of elevation to the bottom of the
sculpture is 32 and the angle of elevation to the top is 35. Find the height of the
sculpture of Lincoln’s face to the nearest tenth of a foot.
58.
A boat leaves the entrance to a harbor and travels 25 miles on a bearing of N 42
E. The captain then turns the boat 90 clockwise and travels 18 miles on a bearing of S
48 E. At this time:
a.
How far is the boat, to the nearest tenth of a mile, from the harbor?
b.
What is the bearing, to the nearest tenth of a degree, of the boat from the
harbor?
61.
A ball on a spring is pulled 6 inches below its rest position and then released. The
period of motion is 4 seconds. Write the equation for the ball’s simple harmonic motion.
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Semester 2 Review
6.1 Verify the identity.
sec x cot x  csc x
1.
2.
csc x tan x  sec x
5.
cos x  cos x sin 2 x  cos3 x
6.
sin x  sin x cos 2 x  sin 3 x
7.
1  sin x
 sec x  tan x
cos x
9.
cos x
1  sin x

 2sec x
1  sin x
cos x
11.
sin x
1  cos x

1  cos x
sin x
14.
6.2 Find the exact values.
17.
cos 15 (Use 60-45=15)
21.
sin
7
,
12
if
7  
 
12 3 4
Verify the identity.
cos    
23.
 cot   tan 
sin  cos 
Find the exact values if sin  
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sec x  csc   x 
sec x csc x
 sin x  cos x
18.
cos 30 (Use 90-60=30)
22.
sin
26.
tan  x     tan x
5
,
12
if
5  
 
12 6 4
12
3
for α in quadrant II and sin   for β in quadrant I.
13
5
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25.
cosα
26.
Semester 2 Review
cosβ
27.
cos(α+β)
28.
sin(α+β)
5
and x lies in quadrant II, find the exact values:
13
sin 2x
34.
cos 2x
35.
tan 2x
6.3 If sin x 
33.
Find the exact values:
2 tan15
39.
1  tan 2 15
40.
cos 2 15  sin 2 15
6.5 Solve the equation.
1.
3sin x  2  5sin x 1
2.
5sin x  3sin x  3
3.
tan 3x  1,
4.
sin
7.
2cos2 x  cos x  1  0, 0  x  2
8.
4sin 2 x  1  0, 0  x  2
9.
tan x sin 2 x  3tan x,
10.
2cos2 x  3sin x  0,
11.
cos 2 x  3sin x  2  0,
12.
cos 2 x  sin x  0,
0  x  2
0  x  2
0  x  2
x 1
 ,
3 2
0  x  2
0  x  2
0  x  2
Solve each equation, correct to four decimal places, for 0 ≤ x < 2π.
21.
tan x = 12.8044
22.
cos x = -0.4317
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Semester 2 Review
23.
sin 2 x  sin x  1  0
24.
cos 2 x  5cos x  3  0
7.1 Solve triangle ABC. Round to the nearest tenth.
27.
A = 46 , C = 63 , and c = 56 in
28.
A = 64 , C = 82 , and c = 14 cm
29.
A = 50 , C = 33.5 , and b = 76
31.
A = 75 , a = 51, and b = 71
30.
32.
A = 43 , a = 81, and b = 62
A = 50 , a = 10, and b = 20
Find the area of a triangle rounded to the nearest square meter.
37.
side lengths 24m and 10m and an included angle of 62.
39.
Two fire-lookout stations are 20 miles apart, with station B directly east of station
A. Both stations spot a fire on a mountain to the north. The bearing from station A to
the fire is N50E. The bearing from station B to the fire is N36W. How far, to the
nearest tenth of a mile, is the fire from station A?
7.2
Solve the triangle. Round lengths to the nearest tenth and angles to the nearest degree.
41.
A = 60, b = 20, and c = 30.
42.
a = 6, b = 9, and c = 4
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Semester 2 Review
45.
Two airplanes leave an airport at the same time on different runways. One flies on
a bearing of N66W at 325 miles per hour. The other plane flies on a bearing of S26W
at 300 miles per hour. How far apart will the planes be after 2 hours?
Find the area of the triangle. Round to the nearest whole unit.
47.
a= 12 yd, b = 16 yd, c = 24 yd
48.
a = 6m, b = 16m, and c = 18m
4.1
The exponential function f(x) = 13.49(0.967)x - 1 describes the number of O-rings
expected to fail, f(x), when the temperature is xF. Find the number of O-rings
expected to fail at the given temperature.
1.
31F
2.
60F
3.
-10F
Graph.
4.
f  x  2
x
5.
g  x  3
x 1
6.
1
h  x   
2
x
10.
The function f(x) = 3.6e0.02x describes world population, f(x), in billions, x years
after 1969. Find the world population in 2020.
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Semester 2 Review
11.
You decide to invest $8000 for 6 years. How much will have if you invest at 7%
per year, compounded monthly? at 6.85% per year, compounded continuously?
13.
You decide to invest $10000 for 5 years at an annual rate of 8%. How much will
have if it is compounded quarterly? compounded continuously?
4.2
Write each equation in its equivalent exponential form:
14.
2 = log5x
15.
3 = logb64
16.
log37 = y
Write each equation in its equivalent logarithmic form:
17.
122 = x
18.
b3 = 8
19.
ey = 9
Evaluate.
20.
log216
21.
log39
22.
log255
23.
log77
24.
log51
25.
log778
26.
6log69
27.
log445
28.
log81
32.
g(x) = log4 (x - 5)
33.
f(x) = ln (3 - x)
log (10x)
39.
log574
Graph.
29.
f(x) = log2x
Find the domain.
31.
f(x) = log4 (x + 3)
34.
g(x) = ln (x - 3)2
4.3
Expand each expression.
37.
log4(7• 5)
38.
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Semester 2 Review
40.
 19 
log 7  
 x 
41.
 e5 
ln  
 11 
43.
log (4x)5
44.
log b x 2 y


42.
ln x
45.
 3x 
log 6 
4 

 36 y 
Write as a single logarithm.
1
log x  4 log  x  1
2
46.
log42 + log432
47.
log (4x - 3) - logx 48.
49.
3 ln (x + 7) - ln x
50.
1
4 log b x  2 log b 6  log b y
2
Evaluate.
51.
log5140
52.
log72506
4.4 Solve.
53.
23x-8 = 16
54.
27x+3 = 9x-1
55.
4x = 15
57.
5x-2 = 42x+3
58.
e2x - 4ex + 3 = 0
56.
40e0.6x - 3 = 237
62.
1
ln  x  2   ln  4 x  3  ln  
x
63.
The risk of having a car accident while under the influence of alcohol can be
modeled by R = 6e12.77x, where x is the blood alcohol concentration and R, given as a
percent, is the risk of a car accident. What blood alcohol level corresponds to a 20% risk
of a car accident?
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64.
Semester 2 Review
How long will it take $25,000 to grow to $500,000 at 9% annual interest
compounded monthly?
 r
A  P 1  
 n
nt
4.5
66.
In 1970, the U.S. population was 203.3 million. By 2003, it had grown to 294
million. Find the exponential growth function that models this data (A = A0ekt). Then find
the year when the population will reach 315 million.
67.
Use the fact that after 5715 years a given amount of carbon-14 will have decayed
to half the original amount to find the exponential decay model for carbon-14. Then
estimate the age of scrolls found in 1947 that contained 76% of their original carbon-14.
69.
Rewrite y = 2.557 (1.017)x in terms of base e.
Ch 8
1.
Bottled water and medical supplies are to be shipped to victims of an earthquake
by plane. Each container of bottled water will serve 10 people and each medical kit will
aid 6 people. Let x represent the number of bottles of water to be shipped and y
represent the number of medical kits.
a. Write the objective function that describes the number of people who can be
helped.
b. Each plane can carry no more than 80,000 pounds. Bottled water weighs 20
pounds per container and each medical kit weighs 10 pounds. Write an inequality
that describes the constraint.
c. Each plane can carry a total volume of supplies that does not exceed 6000 cubic
feet. Each water bottle is 1 cubic foot and each medical kit also has a volume of 1
cubic foot. Write an inequality describing this second constraint.
d. Determine how many bottles of water and how many medical kits should be
sent on each plane to maximize the number of earthquake victims who can be
helped.
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Semester 2 Review
2.
Find the maximum value of the objective function z = 3x + 5y subject to the
constraints x ≥ 0, y ≥ 0, x + y ≥ 1, x + y ≤ 6.
Ch 9
Write the solution set for a system of equations represented by the matrix.
8 
1 2 5 19
1 1 1




4.
5.
9 
0 1 3
0 1 12 15
0 0 1
0 0
4 
1
1 
Use matrices to solve the system.
3 x  y  2 z  31
x  y  2 z  19
6.
7.
x  3 y  2 z  25
8.
2 x  y  2 z  18
x  y  2z  9
x  2y  z  6
w x y  z  4
3 x  y  2 z  31
x  y  2 z  19
2w  x  2 y  z  0
w  2 x  y  2 z  2
9.
x  3 y  2 z  25
3w  2 x  y  3z  4
11.1
Write the first four terms of the sequence whose general term is given.
1.
an = 3n + 4
2.
3.
an = 3an-1 + 2, for a1 = 5 and n≥ 2
4.
Evaluate.
10!
5.
2!8!
8.
7
  2 
k 4
6.
k
 5

 n  1!
n!
7.
an
 1

3n  1
2n
an 
 n  1!
 i
6
i 1
9.
n
2

1
5
3
i1
11.2
12.
Find the first five terms of the arithmetic sequence in which a1 = 77.4 and
an = an-1 - 0.67.
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Semester 2 Review
13.
Find the eighth term of the arithmetic sequence whose first term is 4 and whose
common difference is -7.
14.
According to the U.S. Census Bureau, new one-family houses sold for an average of
$159,000 in 1995. This average sales price has increased by approximately $9700 per
year. Write a formula for the nth term of the arithmetic sequence that describes the
average cost for new one-family houses n years after 1994. How much will new onefamily houses cost, on average, by the year 2010?
15.
Find the sum of the first 100 terms of the arithmetic sequence: 1, 3, 5, 7, …
16.
Find the following sum:
25
  5i  9 
i 1
11.3
18.
Write the first six terms of the geometric sequence with first term 6 and common
ratio 1/3.
19.
Find the eighth term of the geometric sequence whose first term is -4 and whose
common ratio is -2.
21.
Find the sum of the first 18 terms of the geometric sequence:
2, -8, 32, -128, …
22.
Find the following sum:
10
6 2
i
i1
24.
To save for retirement, you decide to deposit $1000 into an IRA at the end of
each year for the next 30 years. If the interest rate is 10% per year compounded
annually, find the value of the IRA after 30 years.
25.
Find the sum of the infinite geometric series:
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3 3 3
3
   
8 16 32 64
.
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