Sample 201-305-VA Applied Math Assessmentsof complex numbers. E#9, E#10, T2#2, T2#3 Basic operations...

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)

Performance Criteria

(Standards)

Assessment Tools

Relevance of Assessment Tool

1. Solve trigonometric

problems..

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

T1#11

Using trigonometric

functions for right

triangle

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

T1#9

Unit circle is used to find

solutions of easy

trigonometric equations.

Understanding of inverse

trig functions through

the unit circle

Accurate conversion of units: degrees to

radians and vice-versa, and angular velocity ω. T1#1

Conversion degree to

radians

Graphing of trigonometric functions and,

translation of functions.

T2#5, T2#6, T2#7 .Graphing trigonometric

functions and

performing horizontal

and vertical shifts.

Proper use of method for addition of functions

Algebraic manipulations in conformity with

rules. E#3,E#4,T1#4,T1#5

T1#14

Solving trigonometric

equations

Using trigonometric

identities

Calculate and interpret the values of sine and

time-dependant functions.

Graphing of trigonometric functions f xsin

x and f xcos x , translation of functions

f t Asin t B

E5, T1#6, T2#5,6,7

E6, T1#7

Sketching sinusoidal function

Finding equation of sinusoidal

function given the graph

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

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

3. Apply operations on

complex numbers

Proper graphic representation of complex

numbers.

A8#2,3

Introduction to Real and

Imaginary axes.

Proper use of polar and rectangular

coordinates.

T2#1

E#9, E#10, T2#2, T2#3

Conversion between

rectangular and polar forms

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

4. Analyze the elements of an

industrial electronics

Accurate interpretation of information

T2#9 Understanding of

impedance, resistance

and reactance in terms of

problem.

complex numbers

addition

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)

Performance Criteria

(Standards)

Assessment Tools

Relevance of Assessment Tool

1. 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

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

amathanna201-305-VAAssignment Set 01 due 01/26/2012 at 10:00pm EST

1. (1 pt) rochesterLibrary/setTrig03FunctionsRightAngle/srw6 2 1.pgClick on the graph to view a larger graphFor the given angle x in the triangle given in the graph

sinx = ;cosx = ;tanx = ;cotx = ;secx = ;cscx = ;





4. (1 pt) rochesterLibrary/setTrig03FunctionsRightAngle/srw6 2 9.pgClick on the graph to view a larger graphIn the triangle given in the graph

the length of the side x = .

5. (1 pt) rochesterLibrary/setTrig01Angles/p1.pgFor each of the following angles, find the degree measure of theangle with the given radian measure:

2π

62π

41π

33π

22π

6. (1 pt) rochesterLibrary/setTrig01Angles/srw6 1 11.pgConvert 8

9 π in radians to degrees: .

1

7. (1 pt) rochesterLibrary/setTrig01Angles/srw6 1 13.pgConvert -0.3 in radians to degrees: .

8. (1 pt) rochesterLibrary/setTrig01Angles/srw6 1 31.pgThe angle between 0 and 360 that is coterminal with the 940

angle is degrees.

9. (1 pt) rochesterLibrary/setTrig01Angles/srw6 1 33.pgThe angle between 0 and 360 that is coterminal with the−1428 angle is degrees.

10. (1 pt) rochesterLibrary/setTrig01Angles/srw6 1 39.pgThe angle between 0 and 2π in radians that is coterminal withthe angle 49

10 π in radians is .

11. (1 pt) rochesterLibrary/setTrig01Angles/srw6 1 41.pgIn a circle of radius 7, the length of the arc that subtends a cen-tral angle of 295 degrees is .

12. (1 pt) rochesterLibrary/setTrig01Angles/srw6 1 45.pgIn a circle of radius 3 miles, the length of the arc that subtendsa central angle of 3 radians is miles.

13. (1 pt) rochesterLibrary/setTrig01Angles/srw6 1 53.pgFind the distance that the earth travels in one day in its patharound the sun. Assume that a year has 365 days and that thepath of the earth around the sun is a circle of radius 93 millionmiles.

Your answer is million miles.

14. (1 pt) rochesterLibrary/setTrig03FunctionsRightAngle/p8.pgRefer to the right triangle in the figure. Click on the picture tosee it more clearly.

If , BC = 9 and the angle α = 30, find any missing angles orsides. Give your answer to at least 3 decimal digits.

AB =AC =β=

Generated by c©WeBWorK, http://webwork.maa.org, Mathematical Association of America

2


1. (1 pt) rochesterLibrary/setTrig03FunctionsRightAngle/p2.pgThe 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 thebuilding. Find the height of the building.

(Show the student hint after 5 attempts: )

Hint: Did you convert degrees to radians?

2. (1 pt) rochesterLibrary/setTrig03FunctionsRightAngle/p6.pg

The captain of a ship at sea sights a lighthouse which is 120feet tall.

The captain measures the the angle of elevation to the top ofthe lighthouse to be 25.

How far is the ship from the base of the lighthouse?



3. (1 pt) rochesterLibrary/setTrig03FunctionsRightAngle-/srw6 2 35.pgThe angle of elevation to the top of the Empire State Buildingin New York is found to be 11 degrees from the ground at adistance of 1 mile from the base of the building. Using thisinformation, find the height of the Empire State Building.Your answer is feet.

4. (1 pt) rochesterLibrary/setTrig03FunctionsRightAngle-/srw6 2 42.pg

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.

Find the distance between a point on the ground directly be-low the plane and the airport.



5. (1 pt) rochesterLibrary/setTrig01Angles/p2.pgConvert 6

20 π to degrees:

Convert 420 to radians:π∗

6. (1 pt) rochesterLibrary/setTrig01Angles/p3.pgFor each of the followings angles, find the degree measure ofthe angle with the given radian measure:

9π

6−5π

48π

33π

2−6π

7. (1 pt) rochesterLibrary/setTrig01Angles/srw6 1 5.pgThe radian measure of an angle of 245 degrees is .

8. (1 pt) rochesterLibrary/setTrig01Angles/ur tr 1 1.pgFor each of the following angles, find the degree measure of theangle with the given radian measure:

5π

65π

45π

31π

23π

9. (1 pt) rochesterLibrary/setTrig01Angles/ur tr 1 13.pg

Find an angle between 0 and 2π that is coterminal with thegiven angle. (Note: You can enter π as ’pi’ in your answers.)

(a) 19π

5(b) −11π

3(c) 75π

2(d) 13π

7

10. (1 pt) rochesterLibrary/setTrig01Angles/ur tr 3 4.pgA circular arc of length 11 feet subtends a central angle of 30degrees. Find the radius of the circle in feet. (Note: You canenter π as ’pi’ in your answer.)

feet

1

11. (1 pt) rochesterLibrary/setTrig02FunctionsUnitCircle/p2.pg

Find an angle between 0 and 2π that is coterminal with thegiven angle. (Note: You can enter π as ’pi’ in your answers.)

(a) 19π

5(b) −13π

3(c) 63π

2(d) 15π

9

12. (1 pt) rochesterLibrary/setTrig02FunctionsUnitCircle/p5.pgEvaluate the following expressions.Note: Your answer must be in EXACT form: it cannot containdecimals. It must be either an integer or a fraction. If the answerinvolves a square root write it as sqrt . For instance, the squareroot of 2 should be written as sqrt(2).

sin( 3π

2 ) =

cos(−π

2 ) =tan(−π) =

cot( 3π

4 ) =

sec(π

3 ) =

csc(− 3π

4 ) =

13. (1 pt) rochesterLibrary/setTrig02FunctionsUnitCircle/p6.pgEvaluate the following expressions.Note: Your answer must be in EXACT form: it cannot containdecimals. It must be either an integer or a fraction. If the answerinvolves a square root write it as sqrt . For instance, the squareroot of 2 should be written as sqrt(2).

If θ = 5π

4 , then

sin(θ) =cos(θ) =tan(θ) =sec(θ) =

14. (1 pt) rochesterLibrary/setTrig02FunctionsUnitCircle/p7.pgEvaluate the following expressions.Note: Your answer must be in EXACT form: it cannot containdecimal numbers. Give the answer either as an integer or a frac-tion. If the answer involves a square root write it as sqrt . Forinstance, the square root of 2 should be written as sqrt(2).

If θ = 2π

3 , then

sin(θ) =cos(θ) =tan(θ) =sec(θ) =

15. (1 pt) rochesterLibrary/setTrig02FunctionsUnitCircle/ur tr 1 6.pgIf θ = 1π

4 , then

sin(θ) equalscos(θ) equalstan(θ) equalssec(θ) equals

16. (1 pt) rochesterLibrary/setTrig02FunctionsUnitCircle-/ur tr 1 6e.pgIf θ = 5π

6 , then

sin(θ) equalscos(θ) equalstan(θ) equalssec(θ) equals

17. (1 pt) rochesterLibrary/setTrig08Equations/p5.pgSolve the following equations in the interval [0,2π].Note: Give the answer as a multiple of π. Do not use decimalnumbers. The answer should be a fraction or an integer. Notethat π is already included in the answer so you just have to enterthe appropriate multiple. E.g. if the answer is π/2 you shouldenter 1/2. If there is more than one answer enter them separatedby commas.

sin(t) = 12

t= π

sin(t) =− 12

t= π

18. (1 pt) rochesterLibrary/setTrig08Equations/p6.pgSolve the following equations in the interval [0, 2 π].Note: Give the answer as a multiple of π. Do not use decimalnumbers. The answer should be a fraction or an integer. Notethat π is already included in the answer so you just have to enterthe appropriate multiple. E.g. if the answer is π/2 you shouldenter 1/2. If there is more than one answer enter them separatedby commas.

cos(t) =−√

22

t = π

cos(t) =√

22

t = π


2


1. (1 pt) rochesterLibrary/setTrig08Equations/p1.pgSolve the following equation in the interval [0, 2 π].Note: Give the answer as a multiple of π. Do not use decimalnumbers. The answer should be a fraction or an integer. Notethat π is already included in the answer so you just have to enterthe appropriate multiple. E.g. if the answer is π/2 you shouldenter 1/2. If there is more than one answer enter them separatedby commas.

(sin(t))2 = 34

t = π


2(cos(t))2− cos(t)−1 = 0t = π


2(sin(t))2− sin(t)−1 = 0t = π

4. (1 pt) rochesterLibrary/setTrig08Equations/p5.pgSolve the following equations in the interval [0,2π].Note: Give the answer as a multiple of π. Do not use decimalnumbers. The answer should be a fraction or an integer. Notethat π is already included in the answer so you just have to enterthe appropriate multiple. E.g. if the answer is π/2 you shouldenter 1/2. If there is more than one answer enter them separatedby commas.

sin(t) =√

32

t= π

sin(t) =− 12

t= π

5. (1 pt) rochesterLibrary/setTrig08Equations/p6.pgSolve the following equations in the interval [0, 2 π].Note: Give the answer as a multiple of π. Do not use decimalnumbers. The answer should be a fraction or an integer. Notethat π is already included in the answer so you just have to enterthe appropriate multiple. E.g. if the answer is π/2 you shouldenter 1/2. If there is more than one answer enter them separatedby commas.

cos(t) =√

32

t = π

cos(t) = 12

t = π


(cos(t))2 = 12

t = π

7. (1 pt) rochesterLibrary/setTrig08Equations/p10.pgSolve the given equation in the interval [0,2 π].Note: The answer must be written as a multiple of π. Give ex-act answers. Do not use decimal numbers. The answer must bean integer or a fraction. Note that π is already provided with theanswer so you just have to find the appropriate multiple. E.g. ifthe answer is π

2 you should enter 1/2. If there is more than oneanswer write them separated by commas.

2(sinx)2−5cosx+1 = 0x= π

8. (1 pt) rochesterLibrary/setTrig08Equations/srw7 5 53.pgFind all solutions of the equation 3sin2 x−7sinx+2 = 0 in theinterval [0,2π).The answer is x1 = and x2 = with x1 < x2.

9. (1 pt) rochesterLibrary/setTrig06Inverses/p14.pgSolve the equation in the interval [0,2 π]. If there is more thanone solution write them separated by commas.

(sin(x))2 = 136

x =


1



2(cos(t))2− cos(t)−1 = 0t = π

2. (1 pt) rochesterLibrary/setTrig08Equations/p10.pgSolve the given equation in the interval [0,2 π].Note: The answer must be written as a multiple of π. Give ex-act answers. Do not use decimal numbers. The answer must bean integer or a fraction. Note that π is already provided with theanswer so you just have to find the appropriate multiple. E.g. ifthe answer is π

2 you should enter 1/2. If there is more than oneanswer write them separated by commas.

2(sinx)2−5cosx+1 = 0x= π

3. (1 pt) dcdsLibrary/Physics/vectors/vcomp2.pgThe vector B has an x component of 15 and a y component of11.5. What are the magnitude and direction of this vector?

B= .θ = degrees from the positive x axis.

4. (1 pt) dcdsLibrary/Physics/vectors/vcomp3.pgThe 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 = degrees from the positive x axis.

5. (1 pt) rochesterLibrary/setVectors2DotProduct/UR VC 1 9.pg

A child walks due east on the deck of a ship at 1 miles perhour.The ship is moving north at a speed of 14 miles per hour.

Find the speed and direction of the child relative to the sur-face of the water.

Speed = mphThe angle of the direction from the north =

(radians)

6. (1 pt) dcdsLibrary/Physics/vectors/vcomp1.pgThe vector A has a magnitude of A=5.5 and a direction of 280degrees from the positive x axis. What are the x and y compo-nents of the vector?

Ax = .Ay = .

7. (1 pt) dcdsLibrary/Physics/vectors/vadd1.pgThe vector A has a magnitude of 10 and a direction of 115.5degrees. The vector B has a magnitude of 4.5 and a direction of147.5 degrees. The vector C has a magnitude of 5.5 and a direc-tion of 30.5 degrees. All angles are measured counterclockwisefrom the positive x axis. The vector D follows the followingrelation: D = A + B−C What are the magnitude and directionof the vector D?

D= .θD = degrees from the positive x axis.

8. (1 pt) dcdsLibrary/Physics/vectors/vadd2.pgThe vector A has a magnitude of 19 and a direction of 45 N ofE. The vector B has a magnitude of 8 and a direction of 65.5S of W. The vector C has a magnitude of 14.5 and a directionof 69.5 E of S. The vector D follows the following relation:D = A + B + C What are the magnitude and direction of thevector D?

D= .θD = degrees from the positive x axis.

9. (1 pt) dcdsLibrary/Physics/vectors/vadd3.pgThe vector A has a magnitude of 7.5 and a direction of 183. Thevector B has a magnitude of 20 and a direction of 66.5. Thevector C has a magnitude of 17.5 and a direction of 195. Thevector D has a magnitude of 6.5 and a direction of 44. All anglesare measured counterclockwise from the positive x axis. Thevector E follows the following relation: E = A + 4B−C + DWhat are the magnitude and direction of the vector E?

E= .θE = degrees counterclockwise from the positive xaxis.

1


2


1. (1 pt) rochesterLibrary/setTrig09Laws/p1.pgConsider the triangle below. Click on the picture to see it moreclearly.

If a = 7, b = 8 and the angle C = 140, find the remainingside c and the other two angles A and B. Give your answer to atleast 2 decimal places.

c =A = degreesB = degrees


If b = 8, the angle C = 110 and the angle A = 50 find the otherangle B and the remaining sides a and c. Give your answer to atleast 3 decimal places.

B = degreesa =c =


If a = 6, the angle C = 50 and the angle A = 45 find the otherangle B and the remaining sides b and c. Give your answer to atleast 3 decimal places.

B = degreesb =c =


If c = 9, the angle C = 110 and the angle B = 25 find the otherangle A and the remaining sides a and b. Give your answer to atleast 3 decimal places.

A =a =b =

1


If a = 1, b = 3 and c = 3, find the angles A, B and C. Give youranswer in degrees to at least 3 decimal places.

A =B =C =

6. (1 pt) rochesterLibrary/setTrig09Laws/p6.pgTo find the distance AB across a river, a distance BC = 220 islaid off on one side of the river. It is found that B = 103 andC = 21. Find AB.See the picture below. Click on the picture to see it more clearly.

AB =

7. (1 pt) rochesterLibrary/setTrig09Laws/p8.pgTwo ships leave a harbor at the same time, traveling on coursesthat have an angle of 120 between them. If the first ship travelsat 30 miles per hour and the second ship travels at 28 miles perhour, how far apart are the two ships after 2.6 hours?

distance =

8. (1 pt) rochesterLibrary/setTrig09Laws/srw6 4 25.pgThe path of a satellite orbiting the earth causes it to pass di-rectly over two tracking stations A and B, which are 52 milesapart. When the satellite is on one side of the two stations, theangles of elevation at A and B are measured to be 87 degreesand 84 degrees, respectively, see the graph

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;

9. (1 pt) rochesterLibrary/setTrig09Laws/srw6 4 27.pgA communication tower (the side CB) is located at the top (thepoint C) of a steep hill. The angle of inclination of the hill is58 degrees. A guy wire is to be attached to the top (the pointB) of the tower and to the ground (the point A), 95 m downhillfrom the base of the tower (the side AC). The angle ∠BAC in thefigure is 12 degrees. See the graph

2

Click on the graph to view a larger graphFind the length of cable (the side AB) required for the guy wire.Your answer is m;

10. (1 pt) rochesterLibrary/setTrig09Laws/srw6 5 1.pgClick on the graph to view a larger graphUse the Law of Cosines to find the indicated side x given in thegraph

x = ;

11. (1 pt) rochesterLibrary/setTrig09Laws/srw6 5 5.pgClick on the graph to view a larger graphUse the Law of Cosines to find the indicated angle x given in thegraph

x = degrees;

12. (1 pt) rochesterLibrary/setTrig09Laws/srw6 5 31.pgA pilot flies in a straight path for 1 h 30 min. She then makes acourse correction, heading 10 degrees to the right of her origi-nal course, and flies 2 h in the new direction. If she maintainsa constant speed of 615 mi/h, how far is she from her startingposition?Your answer is mi;

13. (1 pt) rochesterLibrary/setTrig09Laws/srw6 5 19.pgClick on the graph to view a larger graphFind the indicated side x of the triangle ABC given in the graph

x = ;

14. (1 pt) rochesterLibrary/setTrig09Laws/srw6 5 23.pgClick on the graph to view a larger graphFind the indicated angle x of the triangle ABC given in the graph

3

x = degrees;

15. (1 pt) rochesterLibrary/setTrig09Laws/srw6 5 25.pgClick on the graph to view a larger graphFind the indicated side x of the triangle ABC given in the graph

x = ;


4

amathanna201-305-VAAssignment Set 06 due 03/14/2012 at 10:00pm EDT

1. (1 pt) local/rochesterLibrary/setTrig05Graphs/mec 4 6.pg

Let y = 3cos[6(x+ π

4 )].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 = 10sin(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 = 13cos[3(x− π

4 )].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. (1 pt) rochesterLibrary/setTrig05Graphs/p3.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 point (8,0). Find a sinusoidal function that matchesthe given graph. If needed, you can enter π=3.1416... as ’pi’ inyour 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 (0,2) and (4,2). Find a sinusoidal func-tion that matches the given graph. If needed, you can enterπ=3.1416... as ’pi’ in your answer, otherwise use at least 3 dec-imal digits.

f (x) =



goes through the points (−12,0) and (2,0). Find a sinusoidal1

function that matches the given graph. If needed, you can en-ter π=3.1416... as ’pi’ in your answer, otherwise use at least 3decimal digits.

f (x) =



goes through the points (−8,−1) and (2,−1). Find a sinusoidalfunction that matches the given graph. If needed, you can en-ter π=3.1416... as ’pi’ in your answer, otherwise use at least 3decimal 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 matchesthe given graph. If needed, you can enter π=3.1416... as ’pi’ inyour answer, otherwise use at least 3 decimal digits.

f (x) =

9. (1 pt) rochesterLibrary/setTrig05Graphs/srw5 3 11.pgFor y = cos2x,its amplitude is ;its period is ;

10. (1 pt) rochesterLibrary/setTrig05Graphs/srw5 3 13.pgFor y = 10sin9x,its amplitude is ;its period is ;

11. (1 pt) rochesterLibrary/setTrig05Graphs/srw5 3 17.pgFor y =−2cos 1

3 x,its amplitude is ;its period is ;

12. (1 pt) local/rochesterLibrary/setTrig05Graphs/srw5 3 21.pgFor y =−6cos(x− π

9 ),its amplitude is ;its period is ;its horizontal shift is ;

13. (1 pt) local/rochesterLibrary/setTrig05Graphs/srw5 3 33.pgFor y = sin(5x+ π

3 ),its amplitude is ;its period is ;its horizontal shift is ;


2


1. (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

The volume of air contained in the lungs of a certain athleteis modeled by the equation v = 500sin(84πt)+ 708, where t istime in minutes, and v is volume in cubic centimeters.

What is the maximum possible volume of air in the athlete’slungs?Maximum volume= cubic centimeters

What is the minimum possible volume of air in the athlete’slungs?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 As-pen Avenue has observed a pattern in the amount of revenue forthe store. The revenue reaches a maximum of about $ 54000in January and a minimum of about $ 28000 in July. Supposethe months are numbered 1 through 12, and write a function ofthe form f (x) = Asin(B [x−C])+D that models the boutique’srevenue 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. (1 pt) Library/NAU/setGraphSinCos/WriteTrigEqn3.pg

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 goesthrough the points (−4,0) and (2,0).If needed, you can enter π=3.1416... as ’pi’ in your answer.

f (x) =

5. (1 pt) Library/./ASU-topics/setTrigGraphs/p5.pg


goes through the point (0,1) and (2,1). Find a sinusoidal func-tion that matches the given graph. If needed, you can enterπ=3.1416... as ’pi’ in your answer, otherwise use at least 3 dec-imal digits.

f (x) =

6. (1 pt) Library/LoyolaChicago/Precalc/Chap6Review/Q28.pgEstimate the amplitude, midline, and period of the sinusoidalfunction graphed below:

1

(Click on the graph to get a larger version.)(a) The amplitude is .(b) The midline is y = .(c) The period is .

7. (1 pt) Library/LoyolaChicago/Precalc/Chap6Review/Q02.pgFind approximations to at least two decimal places for the co-ordinates of point Z in the figure below. The angle θ = −80

(denoted Q in the figure) and radius r = 9 are labeled in thefigure.

Z =(retain at least two decimal places)

8. (1 pt) Library/LoyolaChicago/Precalc/Chap6Review/Q20.pgFind period, amplitude, and midline of the following function:

y = 4sin(7πx+2)+7

(a) The period of the graph is(b) The midline of the graph is y =(c) The amplitude of the graph is

9. (1 pt) Library/LoyolaChicago/Precalc/Chap6Review/Q04.pgDetermine the exact radian measure for the angle 310. Do notgive a decimal approximation, and recall in order to enter π youmust type pi.

310 = radians10. (1 pt) Library/LoyolaChicago/Precalc/Chap6Review/Q13.pg

What angle (in degrees) corresponds to 19.5 rotations aroundthe unit circle?

19.5 rotations is an angle of degrees.

11. (1 pt) Library/LoyolaChicago/Precalc/Chap6Review/Q24.pgState the period, amplitude, phase shift, and horizontal shift ofthe following function:

y =−6sin(

2t +π

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 exactanswer)

(e) Based on your answers above, without a calculator sketchthe graph of the function above over the interval −π≤ t ≤ 2π.

12. (1 pt) Library/LoyolaChicago/Precalc/Chap6Review/Q08.pgDetermine the exact degree measure for the angle π radians.

π radians = degrees

13. (1 pt) Library/LoyolaChicago/Precalc/Chap6Review/Q17.pgFind 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.pg3,0,1,2; 2,3,1,0

Without a calculator, match each of the equations below toone of the graphs by placing the corresponding letter of theequation under the appropriate graph.

A. y = sin(t +2)B. y = sin(t)+2C. y = 2sin(t)D. y = sin(2t)

1. 2. 3. 4.

(click on an image to enlarge each individual graph)

15. (1 pt) umichLibrary/sv calc/Chap1Sec5/Q39.pgA mass is oscillating on the end of a spring. The distance, y, ofthe mass from its equilibrium point is given by the formula

y = 2zcos(10πwt)

where y is in centimeters, t is time in seconds, and z and w arepositive constants.

(a) What is the furthest distance of the mass from its equilib-rium point? cm

(b) How many oscillations are completed in 1 second?

16. (1 pt) umichLibrary/sv calc/Chap1Sec5/Q43.pg

A population of animals oscillates sinusoidally between alow of 300 on January 1 and a high of 700 on July 1. Graphthe population against time and use your graph to find a formulafor the population P as a function of time t, in months since thestart of the year.P(t) =


2


1. (1 pt) rochesterLibrary/setComplexNumbers/ur cn 1 1.pgFor 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 obtainthe 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. Thecomplex conjugate of a complex number z is written with a barover 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 theorigin. (Plot z as a point in the ”complex” plane in order to seethis.)

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 de-nominator to make the denominator real.

2+4i2+4i

= +i .

Two convenient functions to know about pick out the real andimaginary parts of a complex number.

Re(a + ib) = a (the real part (coordinate) of the complexnumber), andIm(a + ib) = b (the imaginary part (coordinate) of the complexnumber. Re and Im are linear functions – now that you knowabout linear behavior you may start noticing it often.

2. (1 pt) rochesterLibrary/setComplexNumbers/ur cn 1 5.pgEnter the complex coordinates of the following points:

A: + i,B: + i,C: + i.

3. (1 pt) rochesterLibrary/setComplexNumbers/ur cn 1 6.pgEnter the complex coordinates of the following points:

A: ,B: ,C: .

4. (1 pt) rochesterLibrary/setComplexNumbers/ur cn 1 7.pg

Write the following numbers in a + bi form:

(a) −3(

i2

)= + i,

(b) (−4−5i)− (−5−5i) = + i,1

(c)−3i

= + i,


Write the following numbers in a + bi form:(a) (−3+ i)2 = + i,

(b)5−4i

i1

= + i,

(c)5−4i

11

= + i.


Write the following numbers in a + bi form:(a) (−3+3i)2 = + i,(b) i(π − 1i) = + i,

(c)−4+3i

i= + i.

7. (1 pt) rochesterLibrary/setComplexNumbers/ur cn 1 10.pgWrite the following numbers in a + bi form:

(a)3+5i−5− i

= + i,

(b)−25i

+22i

= + i,

(c) (4i)3 = + i.


(a)(

2+ i5i − (2−2i)

)2

= + i,

(b) (i)2(−5+ i)2 = + i.


(a) (−5−3i)(−3−2i)(4−3i) = + i,(b) ((4+4i)2−4)i = + i.

10. (1 pt) rochesterLibrary/setComplexNumbers/ur cn 1 13.pgCalculate 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.pgLet z = −1−4i. Calculate the following:

(a) z2 + 2z+1 = + i,(b) z2 + iz − (−4 + i) = + i,

(c)(z − 1)2

z + i= + i.

12. (1 pt) rochesterLibrary/setComplexNumbers/ur cn 1 19.pgWrite the following numbers in the polar form reiθ, 0≤ θ < 2π:

(a)14

r = , θ = ,(b) 7 + 7ir = , θ = ,(c) 4−4ir = , θ = .13. (1 pt) rochesterLibrary/setComplexNumbers/ur cn 1 20.pg

Write the following numbers in the polar form reiθ,−π < θ≤ π:(a) πir = , θ = ,(b) −2

√3 − 2i

r = , θ = ,(c) (1 − i)(−

√2 + i)

r = , θ = ,(d) (

√2 −1i)2

r = , θ = ,

(e)−3 +

√2i

5 + 3ir = , θ = ,

(f)−√

7(1 + i)√2 + i

r = , θ = ,

14. (1 pt) rochesterLibrary/setComplexNumbers/ur cn 1 21.pgWrite each of the given numbers in the form a+bi :

(a) e−iπ4

+ i,

(b)e(1+i4π)

e(−1+ iπ2 )

+ i,(c) eei

+ i.15. (1 pt) rochesterLibrary/setComplexNumbers/ur cn 1 22.pg

Write each of the given numbers in the form a+bi :

(a)e5i− e−5i

2i+ i,

(b) 5e(9+ iπ6 )

+ i,

(c) e

(2e(

iπ3 )

)+ i.

16. (1 pt) rochesterLibrary/setComplexNumbers/ur cn 1 23.pgWrite each of the given numbers in the polar form reiθ, −π <θ≤ π.

(a)3− i

7r = , θ = ,(b) −2π(6+ i

√2)

r = , θ = ,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.

(a)(

cos2π

9+ isin

2π

9

)3

r = , θ = ,

(b)2−2i−√

3+ ir = , θ = ,

(c)4i

3e(8+i)r = , θ = .


3

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

10then

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 sideof θ

4. Solve the following equations for x ∈ [0, 2π] without using calculator

a) sin2 x = 12

sinx

b) (2 sinx−√

3) cosx = 0

c) sec2 x = 43

5. Solve the following equations using degrees. Make sure you gave all solutions.

a) (1− 3 sinx)(2− 5 cosx) = 0

b) sin2 x− 3 cosx = 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 = cosx and y = secx

9. Find the following without using calculator

a) arcsin(−1) = b) arccos(−√22

) = c) arccos(12) = d) arcsin(−

√32

) =

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 90cm 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 bcos(a± b) = cos a cos b∓ sin a sin b

tan(a± b) =tan a± tan b

1∓ tan a tan b

Double angle formulas:

sin(2a) = 2 sin a cos acos(2a) = cos2 a− sin2 a = 1− 2 sin a = 2 cos a− 1

tan(2a) =2 tan a

1− tan2 a

Half angle formulas

sin(a

2

)= ±

√1− cos a

2

cos(a

2

)= ±

√1 + cos a

2

tan(a

2

)= ±

√1− cos a

1 + cos a

The law of sines

sinα

a=

sin β

b=

sin γ

c

The law of cosines

c2 = a2 + b2 − 2ab cos γb2 = a2 + c2 − 2ac cos βa2 = b2 + c2 − 2bc cosα

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)

2eπ3 j

2. (4 points) Let z1 = 4− j z2 = −1− 2j. Find the following. Give your answer in rectangular

form.

(a) |z1 − 3z2|

(b) z2 + z2

(c) z1z2

(d)z2z1

3. (3 points) Let z1 = 1 − 3j z2 = 1 − j . Give your answer in both (rectangular and polar)

forms

(a) (z2)12

(b) 6√z1

4. ( 6 points) Find all the (complex ) solutions :

(a) z3 = −1

(b) z3 = 1 + j

(c) 2z2 + 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 (12t) and g(x) = 3 sin (

12t−

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 , reac-

tance 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).

Sample 201-305-VA Applied Math Assessmentsof complex numbers. E#9, E#10, T2#2, T2#3 Basic operations...

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Transcript of Sample 201-305-VA Applied Math Assessmentsof complex numbers. E#9, E#10, T2#2, T2#3 Basic operations...