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 П‰. 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 пЂЁxпЂ©пЂ пЂЅsin E5, T1#6, T2#5,6,7 Sketching sinusoidal function x and f пЂЁпЂ xпЂ©пЂ пЂЅcos 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 пЂЁt пЂ©пЂ пЂЅпЂ Asin пЂЁпЃ·t пЂ«BпЂ© 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 = ; ; ; ; ; ; 2ПЂ 6 2ПЂ 4 1ПЂ 3 3ПЂ 2 2ПЂ 6. (1 pt) rochesterLibrary/setTrig01Angles/srw6 1 11.pg Convert 98 ПЂ 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 0в—¦ and 360в—¦ that is coterminal with the 940в—¦ degrees. angle is 9. (1 pt) rochesterLibrary/setTrig01Angles/srw6 1 33.pg The angle between 0в—¦ and 360в—¦ that is coterminal with the degrees. в€’1428в—¦ angle is 10. (1 pt) rochesterLibrary/setTrig01Angles/srw6 1 39.pg The angle between 0 and 2ПЂ in radians that is coterminal with . the angle 49 10 ПЂ 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 О± = 30в—¦ , 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 = ОІ= 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. Generated by c WeBWorK, http://webwork.maa.org, Mathematical Association of America 2 201-305-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 8в—¦ 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 ПЂ to degrees: Convert 20 Convert 420в—¦ to radians: ПЂв€— (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 9ПЂ 6 в€’5ПЂ 4 8ПЂ 3 3ПЂ 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 25в—¦ . How far is the ship from the base of the lighthouse? в€’6ПЂ 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/setTrig03FunctionsRightAngle5ПЂ 6 5ПЂ 4 5ПЂ 3 1ПЂ 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) 3ПЂ 9. rochesterLibrary/setTrig03FunctionsRightAngle- /srw6 2 42.pg (1 pt) rochesterLibrary/setTrig01Angles/ur tr 1 13.pg Find an angle between 0 and 2ПЂ that is coterminal with the given angle. (Note: You can enter ПЂ as вЂ™piвЂ™ in your answers.) (a) 19ПЂ 5 (b) в€’11ПЂ 3 (c) 75ПЂ 2 (d) 13ПЂ 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 23в—¦ . 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 ПЂ as вЂ™piвЂ™ in your answer.) feet Find the distance between a point on the ground directly below the plane and the airport. 1 sin(Оё) = cos(Оё) = tan(Оё) = sec(Оё) = 11. (1 pt) rochesterLibrary/setTrig02FunctionsUnitCircle/p2.pg Find an angle between 0 and 2ПЂ that is coterminal with the given angle. (Note: You can enter ПЂ as вЂ™piвЂ™ in your answers.) (a) 19ПЂ 5 (b) в€’13ПЂ 3 (c) 63ПЂ 2 (d) 15ПЂ 9 15. (1 pt) rochesterLibrary/setTrig02FunctionsUnitCircle/ur tr 1 6.pg If Оё = 1ПЂ 4 , then sin(Оё) equals cos(Оё) equals tan(Оё) equals sec(Оё) 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( 3ПЂ 2 )= 16. (1 pt) rochesterLibrary/setTrig02FunctionsUnitCircle- /ur tr 1 6e.pg If Оё = cos(в€’ ПЂ2 ) = tan(в€’ПЂ) = 5ПЂ 6 , then sin(Оё) equals cos(Оё) equals tan(Оё) equals sec(Оё) equals cot( 3ПЂ 4 )= sec( ПЂ3 ) = 17. (1 pt) rochesterLibrary/setTrig08Equations/p5.pg Solve the following equations in the interval [0,2ПЂ]. Note: Give the answer as a multiple of ПЂ. Do not use decimal numbers. The answer should be a fraction or an integer. Note that ПЂ is already included in the answer so you just have to enter the appropriate multiple. E.g. if the answer is ПЂ/2 you should enter 1/2. If there is more than one answer enter them separated by commas. sin(t) = 12 t= ПЂ sin(t) = в€’ 21 ПЂ t= csc(в€’ 3ПЂ 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 Оё = 5ПЂ 4 , then sin(Оё) = cos(Оё) = tan(Оё) = sec(Оё) = 18. (1 pt) rochesterLibrary/setTrig08Equations/p6.pg Solve the following equations in the interval [0, 2 ПЂ]. Note: Give the answer as a multiple of ПЂ. Do not use decimal numbers. The answer should be a fraction or an integer. Note that ПЂ is already included in the answer so you just have to enter the appropriate multiple. E.g. if the answer is ПЂ/2 you should enter 1/2. If there is more than one answer enter them separated by commas. в€љ cos(t) = в€’ 22 t= ПЂ в€љ cos(t) = 22 t= ПЂ 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 Оё = 2ПЂ 3 , then Generated by c WeBWorK, http://webwork.maa.org, Mathematical Association of America 2 201-305-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 ПЂ]. Note: Give the answer as a multiple of ПЂ. Do not use decimal numbers. The answer should be a fraction or an integer. Note that ПЂ is already included in the answer so you just have to enter the appropriate multiple. E.g. if the answer is ПЂ/2 you should enter 1/2. If there is more than one answer enter them separated by commas. в€љ cos(t) = 23 ПЂ t= cos(t) = 12 ПЂ t= 6. (1 pt) rochesterLibrary/setTrig08Equations/p7.pg Solve the following equation in the interval [0, 2 ПЂ]. Note: Give the answer as a multiple of ПЂ. Do not use decimal numbers. The answer should be a fraction or an integer. Note that ПЂ is already included in the answer so you just have to enter the appropriate multiple. E.g. if the answer is ПЂ/2 you should enter 1/2. If there is more than one answer enter them separated by commas. (cos(t))2 = 21 t= ПЂ 7. (1 pt) rochesterLibrary/setTrig08Equations/p10.pg Solve the given equation in the interval [0,2 ПЂ]. Note: The answer must be written as a multiple of ПЂ. Give exact answers. Do not use decimal numbers. The answer must be an integer or a fraction. Note that ПЂ is already provided with the answer so you just have to find the appropriate multiple. E.g. if the answer is ПЂ2 you should enter 1/2. If there is more than one answer write them separated by commas. 2(sin x)2 в€’ 5 cos x + 1 = 0 x= ПЂ 8. (1 pt) rochesterLibrary/setTrig08Equations/srw7 5 53.pg Find all solutions of the equation 3 sin2 x в€’ 7 sin x + 2 = 0 in the interval [0, 2ПЂ). 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 ПЂ]. Note: Give the answer as a multiple of ПЂ. Do not use decimal numbers. The answer should be a fraction or an integer. Note that ПЂ is already included in the answer so you just have to enter the appropriate multiple. E.g. if the answer is ПЂ/2 you should enter 1/2. If there is more than one answer enter them separated by commas. (sin(t))2 = 43 t= ПЂ 2. (1 pt) rochesterLibrary/setTrig08Equations/p3.pg Solve the following equation in the interval [0, 2 ПЂ]. Note: Give the answer as a multiple of ПЂ. Do not use decimal numbers. The answer should be a fraction or an integer. Note that ПЂ is already included in the answer so you just have to enter the appropriate multiple. E.g. if the answer is ПЂ/2 you should enter 1/2. If there is more than one answer enter them separated by commas. 2(cos(t))2 в€’ cos(t) в€’ 1 = 0 t= ПЂ 3. (1 pt) rochesterLibrary/setTrig08Equations/p4.pg Solve the following equation in the interval [0, 2 ПЂ]. Note: Give the answer as a multiple of ПЂ. Do not use decimal numbers. The answer should be a fraction or an integer. Note that ПЂ is already included in the answer so you just have to enter the appropriate multiple. E.g. if the answer is ПЂ/2 you should enter 1/2. If there is more than one answer enter them separated by commas. 2(sin(t))2 в€’ sin(t) в€’ 1 = 0 t= ПЂ 4. (1 pt) rochesterLibrary/setTrig08Equations/p5.pg Solve the following equations in the interval [0,2ПЂ]. Note: Give the answer as a multiple of ПЂ. Do not use decimal numbers. The answer should be a fraction or an integer. Note that ПЂ is already included in the answer so you just have to enter the appropriate multiple. E.g. if the answer is ПЂ/2 you should enter 1/2. If there is more than one answer enter them separated by commas.в€љ sin(t) = 23 t= ПЂ sin(t) = в€’ 21 t= ПЂ 9. (1 pt) rochesterLibrary/setTrig06Inverses/p14.pg Solve the equation in the interval [0,2 ПЂ]. If there is more than one solution write them separated by commas. 1 (sin(x))2 = 36 x= Generated by c WeBWorK, http://webwork.maa.org, Mathematical Association of America 1 201-305-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 ПЂ]. Note: Give the answer as a multiple of ПЂ. Do not use decimal numbers. The answer should be a fraction or an integer. Note that ПЂ is already included in the answer so you just have to enter the appropriate multiple. E.g. if the answer is ПЂ/2 you should enter 1/2. If there is more than one answer enter them separated by commas. 2(cos(t))2 в€’ cos(t) в€’ 1 = 0 t= ПЂ 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 в€’ 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 ПЂ]. Note: The answer must be written as a multiple of ПЂ. Give exact answers. Do not use decimal numbers. The answer must be an integer or a fraction. Note that ПЂ is already provided with the answer so you just have to find the appropriate multiple. E.g. if the answer is ПЂ2 you should enter 1/2. If there is more than one answer write them separated by commas. 2(sin x)2 в€’ 5 cos x + 1 = 0 x= ПЂ D= ОёD = 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= Оё= 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= Оё= . . D= ОёD = . 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 в€’ 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= ОёE = axis. 1 . degrees counterclockwise from the positive x Generated by c WeBWorK, http://webwork.maa.org, Mathematical Association of America 2 201-305-VA 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 = 140в—¦ , 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 = 50в—¦ and the angle A = 45в—¦ 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 = 110в—¦ and the angle A = 50в—¦ 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 = 110в—¦ and the angle B = 25в—¦ 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 = 103в—¦ and C = 21в—¦ . 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 120в—¦ 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 в€ 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= Generated by c WeBWorK, http://webwork.maa.org, Mathematical Association of America 4 ; 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 ПЂ 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 ПЂ as вЂ™piвЂ™ 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 ПЂ as вЂ™piвЂ™ in your answers.] 3. (1 pt) local/rochesterLibrary/setTrig05Graphs/p2.pg Let y = 13 cos[3(x в€’ ПЂ4 )]. What is the amplitude? What is the period? What is the horizontal shift? [NOTE: If needed, you can enter ПЂ as вЂ™piвЂ™ 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 ПЂ=3.1416... as вЂ™piвЂ™ 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 ПЂ=3.1416... as вЂ™piвЂ™ 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 (в€’12, 0) and (2, 0). Find a sinusoidal 1 function that matches the given graph. If needed, you can enter ПЂ=3.1416... as вЂ™piвЂ™ 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 ПЂ=3.1416... as вЂ™piвЂ™ 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 = в€’2 cos 13 x, its amplitude is ; its period is ; 12. (1 pt) local/rochesterLibrary/setTrig05Graphs/srw5 3 21.pg For y = в€’6 cos(x в€’ ПЂ9 ), 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 (в€’8, в€’1) and (2, в€’1). Find a sinusoidal function that matches the given graph. If needed, you can enter ПЂ=3.1416... as вЂ™piвЂ™ 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 + ПЂ3 ), its amplitude is ; its period is ; ; its horizontal shift is 8. (1 pt) rochesterLibrary/setTrig05Graphs/p23.pg Generated by c WeBWorK, http://webwork.maa.org, Mathematical Association of America 2 201-305-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 в€’C]) + D whose graph is the sine wave shown above. The curve goes through the points (в€’4, 0) and (2, 0). If needed, you can enter ПЂ=3.1416... as вЂ™piвЂ™ in your answer. f (x) = (1 pt) Library/NAU/setGraphSinCos/WPFreq.pg Determine the frequency of the curve determined by y = cos(135ПЂx), 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(84ПЂt) + 708, where t is time in minutes, and v is volume in cubic centimeters. What is the maximum possible volume of air in the athleteвЂ™s lungs? Maximum volume= cubic centimeters What is the minimum possible volume of air in the athleteвЂ™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 в€’C]) + D that models the boutiqueвЂ™s revenue during the year, where x corresponds to the month. If needed, you can enter ПЂ=3.1416... as вЂ™piвЂ™ 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 ПЂ=3.1416... as вЂ™piвЂ™ 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 в€’ПЂ в‰¤ t в‰¤ 2ПЂ. (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 ПЂ radians. degrees ПЂ 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 Оё = в€’80в—¦ (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 25в—¦ 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 (7ПЂx + 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 310в—¦ . Do not give a decimal approximation, and recall in order to enter ПЂ you must type pi. radians 310в—¦ = 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(10ПЂwt) 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: ПЂ y = в€’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) = Generated by c WeBWorK, http://webwork.maa.org, Mathematical Association of America 2 201-305-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) в€’ (2 + 4i) = (2 + 4i)(2 + 4i) = . The complex conjugate of (1 + i) is (1 в€’ i). In general to obtain the complex conjugate reverse the sign of the imaginary part. (Geometrically this corresponds to finding the вЂќmirror imageвЂќ 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 вЂќz barвЂќ. 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 вЂќcomplexвЂќ 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 вЂ“ 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 following numbers in a + bi form: i = (a) в€’3 + i, 2 (b) (в€’4 в€’ 5i) в€’ (в€’5 в€’ 5i) = + i, 2. (1 pt) rochesterLibrary/setComplexNumbers/ur cn 1 5.pg Enter the complex coordinates of the following points: 1 (c) в€’3 = i + i, 12. (1 pt) rochesterLibrary/setComplexNumbers/ur cn 1 19.pg Write the following numbers in the polar form reiОё , 0 в‰¤ Оё < 2ПЂ: 1 (a) 4 r= ,Оё= , (b) 7 + 7i r= ,Оё= , (c) 4 в€’ 4i r= ,Оё= . 5. (1 pt) rochesterLibrary/setComplexNumbers/ur cn 1 8.pg Write the following numbers in a + bi form: (a) (в€’3 + i)2 = + i, 5 в€’ 4i + i, (b) i = 1 (c) 5 в€’ 4i 1 1 = + i. 13. (1 pt) rochesterLibrary/setComplexNumbers/ur cn 1 20.pg Write the following numbers in the polar form reiОё , в€’ПЂ < Оё в‰¤ ПЂ: (a) ПЂi r= в€љ ,Оё= , (b) в€’2 3 в€’ 2i r= , Оёв€љ= , (c) (1 в€’ i)(в€’ 2 + i) r= в€љ ,Оё= , (d) ( 2 в€’ 1i)2 r= , в€љ, Оё = в€’3 + 2i (e) 5 + 3i r= в€љ ,Оё= , в€’ 7(1 + i) (f) в€љ 2+i r= ,Оё= , 6. (1 pt) rochesterLibrary/setComplexNumbers/ur cn 1 9.pg Write the following numbers in a + bi form: (a) (в€’3 + 3i)2 = + i, (b) i(ПЂ в€’ 1i) = + i, в€’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) = в€’5 в€’ i в€’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 в€’ (2 в€’ 2i) (b) (i)2 (в€’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 : iПЂ (a) eв€’ 4 + i, e(1+i4ПЂ) (b) iПЂ e(в€’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 в€’ eв€’5i (a) 2i + i, 9+ iПЂ ( ) 6 (b) 5e + i, iПЂ 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 reiОё , в€’ПЂ < Оё в‰¤ ПЂ. 3в€’i (a) 7 r= ,в€љ Оё= , (b) в€’2ПЂ(6 + i 2) r= ,Оё= , 9. (1 pt) rochesterLibrary/setComplexNumbers/ur cn 1 12.pg Write the following numbers in a + bi form: (a) (в€’5 в€’ 3i)(в€’3 в€’ 2i)(4 в€’ 3i) = + i, (b) ((4 + 4i)2 в€’ 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) iв€’1 = , (h) iв€’2 = , (i) iв€’3 = , (j) iв€’49 = . 11. (1 pt) rochesterLibrary/setComplexNumbers/ur cn 1 14.pg Let z = в€’1 в€’ 4i. Calculate the following: (a) z2 + 2z + 1 = + i, (b) z2 + iz в€’ (в€’4 + i) = + i, (z в€’ 1)2 (c) = + i. z+i 2 (c) (1 + i)7 r= ,Оё= . 17. (1 pt) rochesterLibrary/setComplexNumbers/ur cn 1 24.pg Write each of the given numbers in the polar form reiОё , в€’ПЂ < Оё в‰¤ ПЂ. 2ПЂ 3 2ПЂ + i sin (a) cos 9 9 r= ,Оё= 2 в€’ 2i в€љ (b) в€’ 3+i r= ,Оё= 4i (c) (8+i) 3e ,Оё= r= Generated by c WeBWorK, http://webwork.maa.org, Mathematical Association of America 3 , , . Vanier College 2011.02.25 Student Name.................................. 201-305-VA Applied Mathematics Test 1 1. Find the radian measure of 91в—¦ 2. For 0 < Оё < ПЂ/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(Оё) = 11 then 10 csc(Оё) = sin(Оё) = cos(Оё) = tan(Оё) = cot(Оё) = 3. Find the value of all 6 trigonometric functions of Оё if the point (в€’2, в€’3) is on the terminal side of Оё 4. Solve the following equations for x в€€ [0, 2ПЂ] without using calculator a) sin2 x = 12 sin x b) (2 sin x в€’ c) sec2 x = 4 3 в€љ 3) cos x = 0 5. Solve the following equations using degrees. Make sure you gave all solutions. a) (1 в€’ 3 sin x)(2 в€’ 5 cos x) = 0 b) sin2 x в€’ 3 cos x = 0 c) tan2 x = 0 6. Sketch the following functions. Show two full periods. a) y = в€’2 sin(x в€’ ПЂ2 ) b) y = cos( ПЂ2 (x + 3)) в€’ 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 в€љ a) arcsin(в€’1) = b) arccos(в€’ 2 ) 2 = c) arccos( 21 ) = в€љ d) arcsin(в€’ 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 Оі = 14в—¦ b) a = 12, b = 11, ОІ = 32в—¦ 13. Prove that cos(2О±) = cos2 О± в€’ sin2 О± = 1 в€’ 2 sin2 О± 14. Find without using calculator. Show your work. a) cos 22.5в—¦ b) tan 75в—¦ 15. True or false . Explain a) sin(в€’x) = sin(x) b) cos(в€’x) = cos(x) Trigonometric Identities Sum or difference of two angles sin(a В± b) = sin a cos b В± cos a sin b cos(a В± b) = cos a cos b в€“ sin a sin b tan a В± tan b tan(a В± b) = 1 в€“ tan a tan b Double angle formulas: sin(2a) = 2 sin a cos a cos(2a) = cos2 a в€’ sin2 a = 1 в€’ 2 sin a = 2 cos a в€’ 1 2 tan a tan(2a) = 1 в€’ tan2 a Half angle formulas a =В± 2 a =В± cos 2 a tan =В± 2 sin 1 в€’ cos a 2 1 + cos a 2 1 в€’ cos a 1 + cos a The law of sines sin О± sin ОІ sin Оі = = a b c The law of cosines c2 = a2 + b2 в€’ 2ab cos Оі b2 = a2 + c2 в€’ 2ac cos ОІ a2 = b2 + c2 в€’ 2bc cos О± Vanier Cllege, April 18, 2012 StudentвЂ™s name..................................................................... 201-305-VA Applied Mathematics Test 2 1. (3 points) rectangular form polar form exponential form 3в€’j 3 пїЅ (135в—¦ ) ПЂ 2e 3 j 2. (4 points) Let z1 = 4 в€’ j form. (a) |z1 в€’ 3z2 | (b) z2 + z2 (c) z1 z2 (d) z2 z1 z2 = в€’1 в€’ 2j. Find the following. Give your answer in rectangular 3. (3 points) Let z1 = 1 в€’ 3j forms (a) (z2 )12 (b) в€љ 6 z1 z2 = 1 в€’ j . Give your answer in both (rectangular and polar) 4. ( 6 points) Find all the (complex ) solutions : (a) z 3 = в€’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 в€’ ПЂ4 ) 7. (2 points) On the same coordinate system sketch two functions f (x) = 3 cos (3t) and g(x) = 3 cos (3t) в€’ 2 8. 9. (4 points) Consider the following AC circuit. Use the following data: current I = 5mA, resistance R = 2kв„¦ , reactance XC = 1.5kв„¦ and reactance XL = 1kв„¦ 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).