1. science
2. mathematics
3. algebra

Sample 201-305-VA Applied Math Assessments
EVALUATION OF ASSESSMENT TOOLS USED TO MEASURE ACHIEVEMENT OF IET COURSE COMPETENCIES
Please attach copies of all assessment tools used in this section of the course
Instructions: Scroll over Headings to learn more about the requested information
Teacher Name: Anna Krasowska
Course Number: 201-305-VA
Section Number: all
Ponderation:
Semester: A2012
Competency code and statement:
Elements of the Competency
(Objectives)
1. Solve trigonometric
problems..
Performance Criteria
(Standards)
Assessment Tools
Relevance of Assessment Tool
Identification of different types of triangles:
acute, obtuse, scalene, isosceles, equilateral,
and right
T1#10
Sketching different types
of triangles
Use of formulas to solve for the side or angle
of a right triangle including Pythagoras
theorem.
Also sin, cos, and tan
E#1, E#2, T1#2, T1#3
Using trigonometric
functions for right
triangle
T1#11
Finding length of one side
of a right triangle.
Use of formulas to solve for the side or angle
of a triangle using sine law.
E#7, T1#12
Using sine and cosine laws to
find the lengths and angles in
a triangle
The unit circle
E#3, T1#4
Unit circle is used to find
solutions of easy
trigonometric equations.
T1#9
Accurate conversion of units: degrees to
radians and vice-versa, and angular velocity ω.
Graphing of trigonometric functions and,
translation of functions.
T1#1
T2#5, T2#6, T2#7
Understanding of inverse
trig functions through
the unit circle
Conversion degree to
radians
.Graphing trigonometric
functions and
performing horizontal
and vertical shifts.
Proper use of method for addition of functions
E#3,E#4,T1#4,T1#5
Algebraic manipulations in conformity with
rules.
Solving trigonometric
equations
T1#14
Using trigonometric
identities
Graphing of trigonometric functions f xsin
E5, T1#6, T2#5,6,7
Sketching sinusoidal function
x and f xcos x , translation of functions
E6, T1#7
Finding equation of sinusoidal
function given the graph
Calculate and interpret the values of sine and
time-dependant functions.
f t Asin t B
2. Apply operations on
vectors.
Graphic representation of vectors in the
Cartesian plane
E#16
Representation of vectors in
3-space
Translation of vectors in the plane.
E#16
Identifying translated vectors.
Addition of vectors.
E#12, E#13
Vectors must be resolved
before addition
3. Apply operations on
complex numbers
Scalar product of vectors.
E#15
Using scalar product to
find the angle between
given vectors
Algebraic manipulations in conformity with
rules.
E#12,E#13
Vector addition using
components
Proper graphic representation of complex
numbers.
A8#2,3
Introduction to Real and
Imaginary axes.
Proper use of polar and rectangular
coordinates.
T2#1
Conversion between
rectangular and polar forms
E#9, E#10, T2#2, T2#3
4. Analyze the elements of an
industrial electronics
Computations must be done
in the required form
Proper methods for the adding and multiplying
of complex numbers.
E#9, E#10, T2#2, T2#3
Basic operations on complex
numbers in rectangular and
polar form
Accurate interpretation of information
T2#9
Understanding of
impedance, resistance
and reactance in terms of
complex numbers
addition
problem.
Proper determination of operations to be
performed
T2#9
Finding impedance and phase
angle
Accurate interpretation of units of
measurement
T2#9
Use of units : amperes ,
ohms.
Competency code and statement:
Elements of the Competency
(Objectives)
1.
Performance Criteria
(Standards)
1.1
1.2
1.3
1.4
1.5
1.6
2.
2.1
2.2
2.3
2.4
2.5
2.6
Assessment Tools
Relevance of Assessment Tool
2.7
2.8
3.
3.1
3.2
3.3
4.
4.1
4.2
4.3
4.4
5
5.1
5.2
5.3
5.4
201-305-VA
Assignment Set 01 due 01/26/2012 at 10:00pm EST
amathanna
1. (1 pt) rochesterLibrary/setTrig03FunctionsRightAngle/srw6 2 1.pg
Click on the graph to view a larger graph
For the given angle x in the triangle given in the graph
sin x =
cos x =
tan x =
cot x =
sec x =
csc x =
sin x =
cos x =
tan x =
cot x =
sec x =
csc x =
;
;
;
;
;
;
;
;
;
;
;
;
4. (1 pt) rochesterLibrary/setTrig03FunctionsRightAngle/srw6 2 9.pg
Click on the graph to view a larger graph
In the triangle given in the graph
2. (1 pt) rochesterLibrary/setTrig03FunctionsRightAngle/srw6 2 3.pg
Click on the graph to view a larger graph
For the given angle x in the triangle given in the graph
the length of the side x =
.
5. (1 pt) rochesterLibrary/setTrig01Angles/p1.pg
For each of the following angles, find the degree measure of the
angle with the given radian measure:
sin x =
cos x =
tan x =
cot x =
sec x =
csc x =
;
;
;
;
;
;
2π
6
2π
4
1π
3
3π
2
2π
6. (1 pt) rochesterLibrary/setTrig01Angles/srw6 1 11.pg
Convert 98 π in radians to degrees:
.
3. (1 pt) rochesterLibrary/setTrig03FunctionsRightAngle/srw6 2 5.pg
Click on the graph to view a larger graph
For the given angle x in the triangle given in the graph
1
Your answer is
7. (1 pt) rochesterLibrary/setTrig01Angles/srw6 1 13.pg
Convert -0.3 in radians to degrees:
.
million miles.
14. (1 pt) rochesterLibrary/setTrig03FunctionsRightAngle/p8.pg
Refer to the right triangle in the figure. Click on the picture to
see it more clearly.
8. (1 pt) rochesterLibrary/setTrig01Angles/srw6 1 31.pg
The angle between 0◦ and 360◦ that is coterminal with the 940◦
degrees.
angle is
9. (1 pt) rochesterLibrary/setTrig01Angles/srw6 1 33.pg
The angle between 0◦ and 360◦ that is coterminal with the
degrees.
−1428◦ angle is
10. (1 pt) rochesterLibrary/setTrig01Angles/srw6 1 39.pg
The angle between 0 and 2π in radians that is coterminal with
.
the angle 49
10 π in radians is
11. (1 pt) rochesterLibrary/setTrig01Angles/srw6 1 41.pg
In a circle of radius 7, the length of the arc that subtends a central angle of 295 degrees is
.
If , BC = 9 and the angle α = 30◦ , find any missing angles or
sides. Give your answer to at least 3 decimal digits.
12. (1 pt) rochesterLibrary/setTrig01Angles/srw6 1 45.pg
In a circle of radius 3 miles, the length of the arc that subtends
a central angle of 3 radians is
miles.
AB =
AC =
β=
13. (1 pt) rochesterLibrary/setTrig01Angles/srw6 1 53.pg
Find the distance that the earth travels in one day in its path
around the sun. Assume that a year has 365 days and that the
path of the earth around the sun is a circle of radius 93 million
miles.
c
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201-305-VA
Assignment Set 02 due 02/03/2012 at 10:30pm EST
amathanna
1. (1 pt) rochesterLibrary/setTrig03FunctionsRightAngle/p2.pg
The angle of elevation to the top of a building is found to be 8◦
from the ground at a distance of 4500 feet from the base of the
building. Find the height of the building.
(Show the student hint after 5 attempts: )
Hint: Did you convert degrees to radians?
5. (1 pt) rochesterLibrary/setTrig01Angles/p2.pg
6
π to degrees:
Convert 20
Convert 420◦ to radians:
π∗
(Show the student hint after 5 attempts: )
6. (1 pt) rochesterLibrary/setTrig01Angles/p3.pg
For each of the followings angles, find the degree measure of
the angle with the given radian measure:
Hint: Did you convert degrees to radians?
2.
(1 pt) rochesterLibrary/setTrig03FunctionsRightAngle/p6.pg
9π
6
−5π
4
8π
3
3π
2
The captain of a ship at sea sights a lighthouse which is 120
feet tall.
The captain measures the the angle of elevation to the top of
the lighthouse to be 25◦ .
How far is the ship from the base of the lighthouse?
−6π
7. (1 pt) rochesterLibrary/setTrig01Angles/srw6 1 5.pg
The radian measure of an angle of 245 degrees is
.
(Show the student hint after 5 attempts: )
8. (1 pt) rochesterLibrary/setTrig01Angles/ur tr 1 1.pg
For each of the following angles, find the degree measure of the
angle with the given radian measure:
Hint: Did you convert degrees to radians?
3.
(1
pt)
rochesterLibrary/setTrig03FunctionsRightAngle5π
6
5π
4
5π
3
1π
2
/srw6 2 35.pg
The angle of elevation to the top of the Empire State Building
in New York is found to be 11 degrees from the ground at a
distance of 1 mile from the base of the building. Using this
information, find the height of the Empire State Building.
Your answer is
feet.
4.
(1
pt)
3π
9.
rochesterLibrary/setTrig03FunctionsRightAngle-
/srw6 2 42.pg
(1 pt) rochesterLibrary/setTrig01Angles/ur tr 1 13.pg
Find an angle between 0 and 2π that is coterminal with the
given angle. (Note: You can enter π as ’pi’ in your answers.)
(a) 19π
5
(b) −11π
3
(c) 75π
2
(d) 13π
7
A plane is flying at an elevation of 21000 feet.
It is within sight of the airport and the pilot finds that the
angle of depression to the airport is 23◦ .
Find the distance between the plane and the airport.
10. (1 pt) rochesterLibrary/setTrig01Angles/ur tr 3 4.pg
A circular arc of length 11 feet subtends a central angle of 30
degrees. Find the radius of the circle in feet. (Note: You can
enter π as ’pi’ in your answer.)
feet
Find the distance between a point on the ground directly below the plane and the airport.
1
sin(θ) =
cos(θ) =
tan(θ) =
sec(θ) =
11. (1 pt) rochesterLibrary/setTrig02FunctionsUnitCircle/p2.pg
Find an angle between 0 and 2π that is coterminal with the
given angle. (Note: You can enter π as ’pi’ in your answers.)
(a) 19π
5
(b) −13π
3
(c) 63π
2
(d) 15π
9
15. (1 pt) rochesterLibrary/setTrig02FunctionsUnitCircle/ur tr 1 6.pg
If θ = 1π
4 , then
sin(θ) equals
cos(θ) equals
tan(θ) equals
sec(θ) equals
12. (1 pt) rochesterLibrary/setTrig02FunctionsUnitCircle/p5.pg
Evaluate the following expressions.
Note: Your answer must be in EXACT form: it cannot contain
decimals. It must be either an integer or a fraction. If the answer
involves a square root write it as sqrt . For instance, the square
root of 2 should be written as sqrt(2).
sin( 3π
2 )=
16.
(1
pt)
rochesterLibrary/setTrig02FunctionsUnitCircle-
/ur tr 1 6e.pg
If θ =
cos(− π2 ) =
tan(−π) =
5π
6 ,
then
sin(θ) equals
cos(θ) equals
tan(θ) equals
sec(θ) equals
cot( 3π
4 )=
sec( π3 ) =
17. (1 pt) rochesterLibrary/setTrig08Equations/p5.pg
Solve the following equations in the interval [0,2π].
Note: Give the answer as a multiple of π. Do not use decimal
numbers. The answer should be a fraction or an integer. Note
that π is already included in the answer so you just have to enter
the appropriate multiple. E.g. if the answer is π/2 you should
enter 1/2. If there is more than one answer enter them separated
by commas.
sin(t) = 12
t=
π
sin(t) = − 21
π
t=
csc(− 3π
4 )=
13. (1 pt) rochesterLibrary/setTrig02FunctionsUnitCircle/p6.pg
Evaluate the following expressions.
Note: Your answer must be in EXACT form: it cannot contain
decimals. It must be either an integer or a fraction. If the answer
involves a square root write it as sqrt . For instance, the square
root of 2 should be written as sqrt(2).
If θ = 5π
4 , then
sin(θ) =
cos(θ) =
tan(θ) =
sec(θ) =
18. (1 pt) rochesterLibrary/setTrig08Equations/p6.pg
Solve the following equations in the interval [0, 2 π].
Note: Give the answer as a multiple of π. Do not use decimal
numbers. The answer should be a fraction or an integer. Note
that π is already included in the answer so you just have to enter
the appropriate multiple. E.g. if the answer is π/2 you should
enter 1/2. If there is more than one answer enter them separated
by commas. √
cos(t) = − 22
t=
π
√
cos(t) = 22
t=
π
14. (1 pt) rochesterLibrary/setTrig02FunctionsUnitCircle/p7.pg
Evaluate the following expressions.
Note: Your answer must be in EXACT form: it cannot contain
decimal numbers. Give the answer either as an integer or a fraction. If the answer involves a square root write it as sqrt . For
instance, the square root of 2 should be written as sqrt(2).
If θ = 2π
3 , then
c
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201-305-VA
Assignment Set 03 due 02/11/2012 at 10:00pm EST
amathanna
5. (1 pt) rochesterLibrary/setTrig08Equations/p6.pg
Solve the following equations in the interval [0, 2 π].
Note: Give the answer as a multiple of π. Do not use decimal
numbers. The answer should be a fraction or an integer. Note
that π is already included in the answer so you just have to enter
the appropriate multiple. E.g. if the answer is π/2 you should
enter 1/2. If there is more than one answer enter them separated
by commas. √
cos(t) = 23
π
t=
cos(t) = 12
π
t=
6. (1 pt) rochesterLibrary/setTrig08Equations/p7.pg
Solve the following equation in the interval [0, 2 π].
Note: Give the answer as a multiple of π. Do not use decimal
numbers. The answer should be a fraction or an integer. Note
that π is already included in the answer so you just have to enter
the appropriate multiple. E.g. if the answer is π/2 you should
enter 1/2. If there is more than one answer enter them separated
by commas.
(cos(t))2 = 21
t=
π
7. (1 pt) rochesterLibrary/setTrig08Equations/p10.pg
Solve the given equation in the interval [0,2 π].
Note: The answer must be written as a multiple of π. Give exact answers. Do not use decimal numbers. The answer must be
an integer or a fraction. Note that π is already provided with the
answer so you just have to find the appropriate multiple. E.g. if
the answer is π2 you should enter 1/2. If there is more than one
answer write them separated by commas.
2(sin x)2 − 5 cos x + 1 = 0
x=
π
8. (1 pt) rochesterLibrary/setTrig08Equations/srw7 5 53.pg
Find all solutions of the equation 3 sin2 x − 7 sin x + 2 = 0 in the
interval [0, 2π).
and x2 =
with x1 < x2 .
The answer is x1 =
1. (1 pt) rochesterLibrary/setTrig08Equations/p1.pg
Solve the following equation in the interval [0, 2 π].
Note: Give the answer as a multiple of π. Do not use decimal
numbers. The answer should be a fraction or an integer. Note
that π is already included in the answer so you just have to enter
the appropriate multiple. E.g. if the answer is π/2 you should
enter 1/2. If there is more than one answer enter them separated
by commas.
(sin(t))2 = 43
t=
π
2. (1 pt) rochesterLibrary/setTrig08Equations/p3.pg
Solve the following equation in the interval [0, 2 π].
Note: Give the answer as a multiple of π. Do not use decimal
numbers. The answer should be a fraction or an integer. Note
that π is already included in the answer so you just have to enter
the appropriate multiple. E.g. if the answer is π/2 you should
enter 1/2. If there is more than one answer enter them separated
by commas.
2(cos(t))2 − cos(t) − 1 = 0
t=
π
3. (1 pt) rochesterLibrary/setTrig08Equations/p4.pg
Solve the following equation in the interval [0, 2 π].
Note: Give the answer as a multiple of π. Do not use decimal
numbers. The answer should be a fraction or an integer. Note
that π is already included in the answer so you just have to enter
the appropriate multiple. E.g. if the answer is π/2 you should
enter 1/2. If there is more than one answer enter them separated
by commas.
2(sin(t))2 − sin(t) − 1 = 0
t=
π
4. (1 pt) rochesterLibrary/setTrig08Equations/p5.pg
Solve the following equations in the interval [0,2π].
Note: Give the answer as a multiple of π. Do not use decimal
numbers. The answer should be a fraction or an integer. Note
that π is already included in the answer so you just have to enter
the appropriate multiple. E.g. if the answer is π/2 you should
enter 1/2. If there is more than one answer enter them separated
by commas.√
sin(t) = 23
t=
π
sin(t) = − 21
t=
π
9. (1 pt) rochesterLibrary/setTrig06Inverses/p14.pg
Solve the equation in the interval [0,2 π]. If there is more than
one solution write them separated by commas.
1
(sin(x))2 = 36
x=
c
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201-305-VA
Assignment Set 04 due 02/17/2012 at 02:30pm EST
amathanna
6. (1 pt) dcdsLibrary/Physics/vectors/vcomp1.pg
The vector A has a magnitude of A=5.5 and a direction of 280
degrees from the positive x axis. What are the x and y components of the vector?
1. (1 pt) rochesterLibrary/setTrig08Equations/p3.pg
Solve the following equation in the interval [0, 2 π].
Note: Give the answer as a multiple of π. Do not use decimal
numbers. The answer should be a fraction or an integer. Note
that π is already included in the answer so you just have to enter
the appropriate multiple. E.g. if the answer is π/2 you should
enter 1/2. If there is more than one answer enter them separated
by commas.
2(cos(t))2 − cos(t) − 1 = 0
t=
π
Ax =
Ay =
7. (1 pt) dcdsLibrary/Physics/vectors/vadd1.pg
The vector A has a magnitude of 10 and a direction of 115.5
degrees. The vector B has a magnitude of 4.5 and a direction of
147.5 degrees. The vector C has a magnitude of 5.5 and a direction of 30.5 degrees. All angles are measured counterclockwise
from the positive x axis. The vector D follows the following
relation: D = A + B − C What are the magnitude and direction
of the vector D?
2. (1 pt) rochesterLibrary/setTrig08Equations/p10.pg
Solve the given equation in the interval [0,2 π].
Note: The answer must be written as a multiple of π. Give exact answers. Do not use decimal numbers. The answer must be
an integer or a fraction. Note that π is already provided with the
answer so you just have to find the appropriate multiple. E.g. if
the answer is π2 you should enter 1/2. If there is more than one
answer write them separated by commas.
2(sin x)2 − 5 cos x + 1 = 0
x=
π
D=
θD =
3. (1 pt) dcdsLibrary/Physics/vectors/vcomp2.pg
The vector B has an x component of 15 and a y component of
11.5. What are the magnitude and direction of this vector?
B=
θ=
5.
.
degrees from the positive x axis.
8. (1 pt) dcdsLibrary/Physics/vectors/vadd2.pg
The vector A has a magnitude of 19 and a direction of 45 N of
E. The vector B has a magnitude of 8 and a direction of 65.5
S of W. The vector C has a magnitude of 14.5 and a direction
of 69.5 E of S. The vector D follows the following relation:
D = A + B + C What are the magnitude and direction of the
vector D?
.
degrees from the positive x axis.
4. (1 pt) dcdsLibrary/Physics/vectors/vcomp3.pg
The vector H has an x component of -2 and a y component of
-12. What are the magnitude and direction of this vector?
H=
θ=
.
.
D=
θD =
.
degrees from the positive x axis.
.
degrees from the positive x axis.
9. (1 pt) dcdsLibrary/Physics/vectors/vadd3.pg
The vector A has a magnitude of 7.5 and a direction of 183. The
vector B has a magnitude of 20 and a direction of 66.5. The
vector C has a magnitude of 17.5 and a direction of 195. The
vector D has a magnitude of 6.5 and a direction of 44. All angles
are measured counterclockwise from the positive x axis. The
vector E follows the following relation: E = A + 4B − C + D
What are the magnitude and direction of the vector E?
(1 pt) rochesterLibrary/setVectors2DotProduct/UR VC 1 9.pg
A child walks due east on the deck of a ship at 1 miles per
hour.
The ship is moving north at a speed of 14 miles per hour.
Find the speed and direction of the child relative to the surface of the water.
Speed =
mph
The angle of the direction from the north =
(radians)
E=
θE =
axis.
1
.
degrees counterclockwise from the positive x
c
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Assignment Set 05 due 03/08/2012 at 10:30pm EST
amathanna
3. (1 pt) rochesterLibrary/setTrig09Laws/p3.pg
Consider the triangle below. Click on the picture to see it more
clearly.
1. (1 pt) rochesterLibrary/setTrig09Laws/p1.pg
Consider the triangle below. Click on the picture to see it more
clearly.
If a = 7, b = 8 and the angle C = 140◦ , find the remaining
side c and the other two angles A and B. Give your answer to at
least 2 decimal places.
c=
A=
B=
If a = 6, the angle C = 50◦ and the angle A = 45◦ find the other
angle B and the remaining sides b and c. Give your answer to at
least 3 decimal places.
B=
b=
c=
degrees
degrees
degrees
2. (1 pt) rochesterLibrary/setTrig09Laws/p2.pg
Consider the triangle below. Click on the picture to see it more
clearly.
4. (1 pt) rochesterLibrary/setTrig09Laws/p4.pg
Consider the triangle below. Click on the picture to see it more
clearly.
If b = 8, the angle C = 110◦ and the angle A = 50◦ find the other
angle B and the remaining sides a and c. Give your answer to at
least 3 decimal places.
If c = 9, the angle C = 110◦ and the angle B = 25◦ find the other
angle A and the remaining sides a and b. Give your answer to at
least 3 decimal places.
B=
a=
c=
A=
a=
b=
degrees
1
5. (1 pt) rochesterLibrary/setTrig09Laws/p5.pg
Consider the triangle below. Click on the picture to see it more
clearly.
If a = 1, b = 3 and c = 3, find the angles A, B and C. Give your
answer in degrees to at least 3 decimal places.
Click on the graph to view a larger graph
(a) How far is the satellite from station A? Your answer is
miles;
(b) How high is the satellite above the ground? Your answer is
miles;
A=
B=
C=
6. (1 pt) rochesterLibrary/setTrig09Laws/p6.pg
To find the distance AB across a river, a distance BC = 220 is
laid off on one side of the river. It is found that B = 103◦ and
C = 21◦ . Find AB.
See the picture below. Click on the picture to see it more clearly.
AB =
7. (1 pt) rochesterLibrary/setTrig09Laws/p8.pg
Two ships leave a harbor at the same time, traveling on courses
that have an angle of 120◦ between them. If the first ship travels
at 30 miles per hour and the second ship travels at 28 miles per
hour, how far apart are the two ships after 2.6 hours?
distance =
9. (1 pt) rochesterLibrary/setTrig09Laws/srw6 4 27.pg
A communication tower (the side CB) is located at the top (the
point C) of a steep hill. The angle of inclination of the hill is
58 degrees. A guy wire is to be attached to the top (the point
B) of the tower and to the ground (the point A), 95 m downhill
from the base of the tower (the side AC). The angle ∠BAC in the
figure is 12 degrees. See the graph
8. (1 pt) rochesterLibrary/setTrig09Laws/srw6 4 25.pg
The path of a satellite orbiting the earth causes it to pass directly over two tracking stations A and B, which are 52 miles
apart. When the satellite is on one side of the two stations, the
angles of elevation at A and B are measured to be 87 degrees
and 84 degrees, respectively, see the graph
2
x=
;
11. (1 pt) rochesterLibrary/setTrig09Laws/srw6 5 5.pg
Click on the graph to view a larger graph
Use the Law of Cosines to find the indicated angle x given in the
graph
x=
degrees;
12. (1 pt) rochesterLibrary/setTrig09Laws/srw6 5 31.pg
A pilot flies in a straight path for 1 h 30 min. She then makes a
course correction, heading 10 degrees to the right of her original course, and flies 2 h in the new direction. If she maintains
a constant speed of 615 mi/h, how far is she from her starting
position?
Your answer is
mi;
13. (1 pt) rochesterLibrary/setTrig09Laws/srw6 5 19.pg
Click on the graph to view a larger graph
Find the indicated side x of the triangle ABC given in the graph
Click on the graph to view a larger graph
Find the length of cable (the side AB) required for the guy wire.
Your answer is
m;
10. (1 pt) rochesterLibrary/setTrig09Laws/srw6 5 1.pg
Click on the graph to view a larger graph
Use the Law of Cosines to find the indicated side x given in the
graph
x=
;
14. (1 pt) rochesterLibrary/setTrig09Laws/srw6 5 23.pg
Click on the graph to view a larger graph
Find the indicated angle x of the triangle ABC given in the graph
3
x=
degrees;
15. (1 pt) rochesterLibrary/setTrig09Laws/srw6 5 25.pg
Click on the graph to view a larger graph
Find the indicated side x of the triangle ABC given in the graph
x=
c
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201-305-VA
Assignment Set 06 due 03/14/2012 at 10:00pm EDT
1.
amathanna
(1 pt) local/rochesterLibrary/setTrig05Graphs/mec 4 6.pg
5.
(1 pt) rochesterLibrary/setTrig05Graphs/p5.pg
π
4 )].
Let y = 3 cos[6(x +
What is the amplitude?
What is the period?
What is the horizontal shift?
[NOTE: If needed, you can enter π as ’pi’ in your answers.]
2.
(1 pt) local/rochesterLibrary/setTrig05Graphs/mec 4 7.pg
Let y = 10 sin(5x + 2).
What is the amplitude?
What is the period?
What is the horizontal shift?
[NOTE: If needed, you can enter π as ’pi’ in your answers.]
3.
(1 pt) local/rochesterLibrary/setTrig05Graphs/p2.pg
Let y = 13 cos[3(x − π4 )].
What is the amplitude?
What is the period?
What is the horizontal shift?
[NOTE: If needed, you can enter π as ’pi’ in your answers.]
4.
To get a better look at the graph, you can click on it.
The curve above is the graph of a sinusoidal function. It
goes through the point (0, 2) and (4, 2). Find a sinusoidal function that matches the given graph. If needed, you can enter
π=3.1416... as ’pi’ in your answer, otherwise use at least 3 decimal digits.
f (x) =
(1 pt) rochesterLibrary/setTrig05Graphs/p3.pg
6.
To get a better look at the graph, you can click on it.
The curve above is the graph of a sinusoidal function. It goes
through the point (8, 0). Find a sinusoidal function that matches
the given graph. If needed, you can enter π=3.1416... as ’pi’ in
your answer, otherwise use at least 3 decimal digits.
f (x) =
(1 pt) rochesterLibrary/setTrig05Graphs/p8.pg
To get a better look at the graph, you can click on it.
The curve above is the graph of a sinusoidal function. It
goes through the points (−12, 0) and (2, 0). Find a sinusoidal
1
function that matches the given graph. If needed, you can enter π=3.1416... as ’pi’ in your answer, otherwise use at least 3
decimal digits.
f (x) =
To get a better look at the graph, you can click on it.
The curve above is the graph of a sinusoidal function. It goes
through the point (1, 2). Find a sinusoidal function that matches
the given graph. If needed, you can enter π=3.1416... as ’pi’ in
your answer, otherwise use at least 3 decimal digits.
f (x) =
7. (1 pt) rochesterLibrary/setTrig05Graphs/p9.pg
9. (1 pt) rochesterLibrary/setTrig05Graphs/srw5 3 11.pg
For y = cos 2x,
its amplitude is
;
its period is
;
10. (1 pt) rochesterLibrary/setTrig05Graphs/srw5 3 13.pg
For y = 10 sin 9x,
its amplitude is
;
its period is
;
11. (1 pt) rochesterLibrary/setTrig05Graphs/srw5 3 17.pg
For y = −2 cos 13 x,
its amplitude is
;
its period is
;
12. (1 pt) local/rochesterLibrary/setTrig05Graphs/srw5 3 21.pg
For y = −6 cos(x − π9 ),
its amplitude is
;
its period is
;
;
its horizontal shift is
To get a better look at the graph, you can click on it.
The curve above is the graph of a sinusoidal function. It
goes through the points (−8, −1) and (2, −1). Find a sinusoidal
function that matches the given graph. If needed, you can enter π=3.1416... as ’pi’ in your answer, otherwise use at least 3
decimal digits.
f (x) =
13. (1 pt) local/rochesterLibrary/setTrig05Graphs/srw5 3 33.pg
For y = sin(5x + π3 ),
its amplitude is
;
its period is
;
;
its horizontal shift is
8. (1 pt) rochesterLibrary/setTrig05Graphs/p23.pg
c
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201-305-VA
Assignment Set 07 due 03/28/2012 at 05:29pm EDT
1.
amathanna
To get a better look at the graph, you can click on it.
Find a function of the form f (x) = A sin(B [x −C]) + D
whose graph is the sine wave shown above. The curve goes
through the points (−4, 0) and (2, 0).
If needed, you can enter π=3.1416... as ’pi’ in your answer.
f (x) =
(1 pt) Library/NAU/setGraphSinCos/WPFreq.pg
Determine the frequency of the curve determined by y =
cos(135πx), where x is time in seconds.
Frequency
2.
(1 pt) Library/NAU/setGraphSinCos/TrigApp1.pg
5.
(1 pt) Library/./ASU-topics/setTrigGraphs/p5.pg
The volume of air contained in the lungs of a certain athlete
is modeled by the equation v = 500 sin(84πt) + 708, where t is
time in minutes, and v is volume in cubic centimeters.
What is the maximum possible volume of air in the athlete’s
lungs?
Maximum volume=
cubic centimeters
What is the minimum possible volume of air in the athlete’s
lungs?
Minimum volume=
cubic centimeters
How many breaths does the athlete take per minute?
breaths per minute
3.
(1 pt) Library/NAU/setGraphSinCos/TrigApp2.pg
Over the past several years, the owner of a boutique on Aspen Avenue has observed a pattern in the amount of revenue for
the store. The revenue reaches a maximum of about $ 54000
in January and a minimum of about $ 28000 in July. Suppose
the months are numbered 1 through 12, and write a function of
the form f (x) = A sin(B [x −C]) + D that models the boutique’s
revenue during the year, where x corresponds to the month.
If needed, you can enter π=3.1416... as ’pi’ in your answer.
f (x) =
4.
To get a better look at the graph, you can click on it.
The curve above is the graph of a sinusoidal function. It
goes through the point (0, 1) and (2, 1). Find a sinusoidal function that matches the given graph. If needed, you can enter
π=3.1416... as ’pi’ in your answer, otherwise use at least 3 decimal digits.
f (x) =
(1 pt) Library/NAU/setGraphSinCos/WriteTrigEqn3.pg
6. (1 pt) Library/LoyolaChicago/Precalc/Chap6Review/Q28.pg
Estimate the amplitude, midline, and period of the sinusoidal
function graphed below:
1
(e) Based on your answers above, without a calculator sketch
the graph of the function above over the interval −π ≤ t ≤ 2π.
(Click on the graph to get a larger version.)
(a) The amplitude is
.
.
(b) The midline is y =
(c) The period is
.
12. (1 pt) Library/LoyolaChicago/Precalc/Chap6Review/Q08.pg
Determine the exact degree measure for the angle π radians.
degrees
π radians =
7. (1 pt) Library/LoyolaChicago/Precalc/Chap6Review/Q02.pg
Find approximations to at least two decimal places for the coordinates of point Z in the figure below. The angle θ = −80◦
(denoted Q in the figure) and radius r = 9 are labeled in the
figure.
13. (1 pt) Library/LoyolaChicago/Precalc/Chap6Review/Q17.pg
Find the arc length corresponding to the given angle (in degrees)
on a circle of radius 5.5.
An angle of 25◦ has an arc length of
units.
14. (1 pt) Library/LoyolaChicago/Precalc/Chap6Review/Q18.pg
3,0,1,2; 2,3,1,0
Without a calculator, match each of the equations below to
one of the graphs by placing the corresponding letter of the
equation under the appropriate graph.
A. y = sin (t + 2)
B. y = sin (t) + 2
C. y = 2 sin (t)
D. y = sin (2t)
Z=
(retain at least two decimal places)
8. (1 pt) Library/LoyolaChicago/Precalc/Chap6Review/Q20.pg
Find period, amplitude, and midline of the following function:
y = 4 sin (7πx + 2) + 7
1.
(a) The period of the graph is
(b) The midline of the graph is y =
(c) The amplitude of the graph is
2.
3.
4.
(click on an image to enlarge each individual graph)
15. (1 pt) umichLibrary/sv calc/Chap1Sec5/Q39.pg
A mass is oscillating on the end of a spring. The distance, y, of
the mass from its equilibrium point is given by the formula
9. (1 pt) Library/LoyolaChicago/Precalc/Chap6Review/Q04.pg
Determine the exact radian measure for the angle 310◦ . Do not
give a decimal approximation, and recall in order to enter π you
must type pi.
radians
310◦ =
10. (1 pt) Library/LoyolaChicago/Precalc/Chap6Review/Q13.pg
What angle (in degrees) corresponds to 19.5 rotations around
the unit circle?
19.5 rotations is an angle of
degrees.
y = 2z cos(10πwt)
where y is in centimeters, t is time in seconds, and z and w are
positive constants.
(a) What is the furthest distance of the mass from its equilibrium point?
cm
(b) How many oscillations are completed in 1 second?
11. (1 pt) Library/LoyolaChicago/Precalc/Chap6Review/Q24.pg
State the period, amplitude, phase shift, and horizontal shift of
the following function:
π
y = −6 sin 2t +
3
(a) The period of the graph is
(give an exact answer)
(b) The amplitude of the graph is
(give an exact answer)
(c) The phase shift of the graph is
(give an exact answer)
(d) The horizontal shift of the graph is
(give an exact
answer)
16. (1 pt) umichLibrary/sv calc/Chap1Sec5/Q43.pg
A population of animals oscillates sinusoidally between a
low of 300 on January 1 and a high of 700 on July 1. Graph
the population against time and use your graph to find a formula
for the population P as a function of time t, in months since the
start of the year.
P(t) =
c
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201-305-VA
Assignment Set 08 due 04/14/2012 at 06:48pm EDT
amathanna
1. (1 pt) rochesterLibrary/setComplexNumbers/ur cn 1 1.pg
For some practice working with complex numbers:
Calculate
(2 + 4i) + (2 + 4i) =
,
,
(2 + 4i) − (2 + 4i) =
(2 + 4i)(2 + 4i) =
.
The complex conjugate of (1 + i) is (1 − i). In general to obtain
the complex conjugate reverse the sign of the imaginary part.
(Geometrically this corresponds to finding the ”mirror image”
point in the complex plane by reflecting through the x-axis. The
complex conjugate of a complex number z is written with a bar
over it: z and read as ”z bar”.
Notice that if z = a + ib, then
(z) (z) = |z|2 = a2 + b2
which is also the square of the distance of the point z from the
origin. (Plot z as a point in the ”complex” plane in order to see
this.)
If z = 2 + 4i then z =
and |z| =
.
You can use this to simplify complex fractions. Multiply the
numerator and denominator by the complex conjugate of the denominator to make the denominator real.
2 + 4i
=
+i
.
2 + 4i
Two convenient functions to know about pick out the real and
imaginary parts of a complex number.
Re(a + ib) = a (the real part (coordinate) of the complex
number), and
Im(a + ib) = b (the imaginary part (coordinate) of the complex
number. Re and Im are linear functions – now that you know
about linear behavior you may start noticing it often.
+
+
+
A:
B:
C:
i,
i,
i.
3. (1 pt) rochesterLibrary/setComplexNumbers/ur cn 1 6.pg
Enter the complex coordinates of the following points:
A:
B:
C:
,
,
.
4.
(1 pt) rochesterLibrary/setComplexNumbers/ur cn 1 7.pg
Write the
numbers in a + bi form:
following
i
=
(a) −3
+
i,
2
(b) (−4 − 5i) − (−5 − 5i) =
+
i,
2. (1 pt) rochesterLibrary/setComplexNumbers/ur cn 1 5.pg
Enter the complex coordinates of the following points:
1
(c)
−3
=
i
+
i,
12. (1 pt) rochesterLibrary/setComplexNumbers/ur cn 1 19.pg
Write the following numbers in the polar form reiθ , 0 ≤ θ < 2π:
1
(a)
4
r=
,θ=
,
(b) 7 + 7i
r=
,θ=
,
(c) 4 − 4i
r=
,θ=
.
5. (1 pt) rochesterLibrary/setComplexNumbers/ur cn 1 8.pg
Write the following numbers in a + bi form:
(a) (−3 + i)2 =
+
i,
5 − 4i
+
i,
(b) i
=
1
(c)
5 − 4i
1
1
=
+
i.
13. (1 pt) rochesterLibrary/setComplexNumbers/ur cn 1 20.pg
Write the following numbers in the polar form reiθ , −π < θ ≤ π:
(a) πi
r= √
,θ=
,
(b) −2 3 − 2i
r=
, θ√=
,
(c) (1 − i)(− 2 + i)
r= √
,θ=
,
(d) ( 2 − 1i)2
r=
,
√, θ =
−3 + 2i
(e)
5 + 3i
r= √
,θ=
,
− 7(1 + i)
(f) √
2+i
r=
,θ=
,
6. (1 pt) rochesterLibrary/setComplexNumbers/ur cn 1 9.pg
Write the following numbers in a + bi form:
(a) (−3 + 3i)2 =
+
i,
(b) i(π − 1i) =
+
i,
−4 + 3i
+
i.
(c)
=
i
7. (1 pt) rochesterLibrary/setComplexNumbers/ur cn 1 10.pg
Write the following numbers in a + bi form:
3 + 5i
+
i,
(a)
=
−5 − i
−2
2
(b)
+
=
+
i,
5i
2i
(c) (4i)3 =
+
i.
8. (1 pt) rochesterLibrary/setComplexNumbers/ur cn 1 11.pg
Write the following numbers in a + bi form:
2
2+i
+
i,
(a)
=
5i − (2 − 2i)
(b) (i)2 (−5 + i)2 =
+
i.
14. (1 pt) rochesterLibrary/setComplexNumbers/ur cn 1 21.pg
Write each of the given numbers in the form a + bi :
iπ
(a) e− 4
+
i,
e(1+i4π)
(b)
iπ
e(−1+ 2 )
+
i,
i
(c) ee
+
i.
15. (1 pt) rochesterLibrary/setComplexNumbers/ur cn 1 22.pg
Write each of the given numbers in the form a + bi :
e5i − e−5i
(a)
2i
+
i,
9+ iπ
(
)
6
(b) 5e
+
i,
iπ
2e( 3 )
(c) e
+
i.
16. (1 pt) rochesterLibrary/setComplexNumbers/ur cn 1 23.pg
Write each of the given numbers in the polar form reiθ , −π <
θ ≤ π.
3−i
(a)
7
r=
,√
θ=
,
(b) −2π(6 + i 2)
r=
,θ=
,
9. (1 pt) rochesterLibrary/setComplexNumbers/ur cn 1 12.pg
Write the following numbers in a + bi form:
(a) (−5 − 3i)(−3 − 2i)(4 − 3i) =
+
i,
(b) ((4 + 4i)2 − 4)i =
+
i.
10. (1 pt) rochesterLibrary/setComplexNumbers/ur cn 1 13.pg
Calculate the following:
(a) i2 =
,
,
(b) i3 =
(c) i4 =
,
(d) i5 =
,
,
(e) i72 =
(f) i0 =
,
(g) i−1 =
,
(h) i−2 =
,
(i) i−3 =
,
(j) i−49 =
.
11. (1 pt) rochesterLibrary/setComplexNumbers/ur cn 1 14.pg
Let z = −1 − 4i. Calculate the following:
(a) z2 + 2z + 1 =
+
i,
(b) z2 + iz − (−4 + i) =
+
i,
(z − 1)2
(c)
=
+
i.
z+i
2
(c) (1 + i)7
r=
,θ=
.
17. (1 pt) rochesterLibrary/setComplexNumbers/ur cn 1 24.pg
Write each of the given numbers in the polar form reiθ , −π <
θ ≤ π.
2π 3
2π
+ i sin
(a) cos
9
9
r=
,θ=
2 − 2i
√
(b)
− 3+i
r=
,θ=
4i
(c) (8+i)
3e
,θ=
r=
c
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,
,
.
Vanier College 2011.02.25
Student Name..................................
201-305-VA Applied
Mathematics Test 1
1. Find the radian measure of 91◦
2. For 0 < θ < π/2, find the values of the trigonometric functions based on the given one. Note:
The answer must be given as a fraction. NO DECIMALS. If sec(θ) = 11
then
10
csc(θ) =
sin(θ) =
cos(θ) =
tan(θ) =
cot(θ) =
3. Find the value of all 6 trigonometric functions of θ if the point (−2, −3) is on the terminal side
of θ
4. Solve the following equations for x ∈ [0, 2π] without using calculator
a) sin2 x = 12 sin x
b) (2 sin x −
c) sec2 x =
4
3
√
3) cos x = 0
5. Solve the following equations using degrees. Make sure you gave all solutions.
a) (1 − 3 sin x)(2 − 5 cos x) = 0
b) sin2 x − 3 cos x = 0
c) tan2 x = 0
6. Sketch the following functions. Show two full periods.
a) y = −2 sin(x − π2 )
b) y = cos( π2 (x + 3)) − 1
7. Find equations of the following graphs.
8. Sketch in the same coordinate system
y = cos x and y = sec x
9. Find the following without using calculator
√
a) arcsin(−1) =
b) arccos(−
2
)
2
=
c) arccos( 21 ) =
√
d) arcsin(−
3
)
2
=
10. Sketch any example of
a) acute scalene triangle
b) obtuse isosceles triangle
11. A boy 160 cm tall, stands 360 cm from a lamp post at night. His shadow from the light is 90
cm long. How high is the lamp post?
12. Solve the following triangles
a) a = 3, b = 3.5 γ = 14◦
b) a = 12, b = 11, β = 32◦
13. Prove that cos(2α) = cos2 α − sin2 α = 1 − 2 sin2 α
14. Find without using calculator. Show your work.
a) cos 22.5◦
b) tan 75◦
15. True or false . Explain
a) sin(−x) = sin(x)
b) cos(−x) = cos(x)
Trigonometric Identities
Sum or difference of two angles
sin(a ± b) = sin a cos b ± cos a sin b
cos(a ± b) = cos a cos b ∓ sin a sin b
tan a ± tan b
tan(a ± b) =
1 ∓ tan a tan b
Double angle formulas:
sin(2a) = 2 sin a cos a
cos(2a) = cos2 a − sin2 a = 1 − 2 sin a = 2 cos a − 1
2 tan a
tan(2a) =
1 − tan2 a
Half angle formulas
s
1 − cos a
a
sin
=±
2
2
s
1 + cos a
a
=±
cos
2
2
s
1 − cos a
a
tan
=±
2
1 + cos a
The law of sines
sin α
sin β
sin γ
=
=
a
b
c
The law of cosines
c2 = a2 + b2 − 2ab cos γ
b2 = a2 + c2 − 2ac cos β
a2 = b2 + c2 − 2bc cos α
Vanier Cllege, April 18, 2012
Student’s name.....................................................................
201-305-VA Applied
Mathematics Test 2
1. (3 points)
rectangular form
polar form exponential form
3−j
3 � (135◦ )
π
2e 3 j
2. (4 points) Let z1 = 4 − j
form.
(a) |z1 − 3z2 |
(b) z2 + z2
(c) z1 z2
(d)
z2
z1
z2 = −1 − 2j. Find the following. Give your answer in rectangular
3. (3 points) Let z1 = 1 − 3j
forms
(a) (z2 )12
(b)
√
6
z1
z2 = 1 − j . Give your answer in both (rectangular and polar)
4. ( 6 points) Find all the (complex ) solutions :
(a) z 3 = −1
(b) z 3 = 1 + j
(c) 2z 2 + z + 4 = 0
5. (2 points) On the same coordinate system sketch two functions f (x) = cos(x) and g(x) = sec(x)
6. (2 points) On the same coordinate system sketch two functions
f (x) = 3 sin ( 12 t) and g(x) = 3 sin ( 12 t − π4 )
7. (2 points) On the same coordinate system sketch two functions
f (x) = 3 cos (3t) and g(x) = 3 cos (3t) − 2
8.
9. (4 points) Consider the following AC circuit.
Use the following data: current I = 5mA, resistance R = 2kΩ , reactance XC = 1.5kΩ and reactance XL = 1kΩ to find the following:
(a) The voltage across the resistor
(b) The voltage across the capacitor
(c) The magnitude of the impedance across the combination of the resistor and the capacitor
(d) The phase angle between the current and the voltage for this combination ( resistor and
capacitor).