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Algebra / Geometry III: Unit 8- Periodic Trig SUCCESS CRITERIA: 1. Be able to find an additional positive and negative co-terminal angle in degrees and radians. 2. Be able to use the unit circle to find the exact value of special angles. 3. Be able to graph trig functions and their transformations finding key features. 4. Be able to transform trig expressions and solve equations using trig identities. INSTRUCTOR: Craig Sherman Hidden Lake High School Westminster Public Schools PMI-NJ Center for Teaching & Learning ~1~ NJCTL.org EMPOWER Recorded TARGET SCALE THEME MA.11.F.05.04 Extend the Domain of Trigonometric Functions Using the Unit Circle MA.11.F.06.04 Periodic Phenomena MA.11.F.07.04 Trigonometric Identities MA.11.G.01.04 Apply Trigonometry VOCABULARY o o o o radian measure terminal side co-terminal side unit circle o o o o cotangent secant cosecant amplitude o o o o frequency phase shift period trig idenity PROFICIENCY SCALE: SCORE REQUIREMENTS 4.0 In addition to exhibiting Score 3.0 performance, in-depth inferences and applications that go BEYOND what was taught in class. Score 4.0 does not equate to more work but rather a higher level of performance. 3.5 3.0 In addition to Score 3.0 performance, in-depth inferences and applications with partial success. The learner exhibits no major errors or omissions regarding any of the information and processes (simple or complex) that were explicitly taught. o Be able to find an additional positive and negative co-terminal angle in degrees and radians AND o Be able to use the unit circle to find the exact value of special angles, AND o Graph polynomial equations and identify its key features, AND o Be able to graph trig functions and their transformations finding key features, AND o Be able to transform trig expressions and solve equations using trig identities. 2.0 Can do one or more of the following skills / concepts: There are no major errors or omissions regarding the simpler details and processes as the learnerβ¦ o Convert angle degrees into radians, OR o Convert radian measures to degrees, OR o Graph periodic trig functions, OR o Identify key features of periodic functions, OR o Graph periodic transformations, OR o Identify key features of periodic transformations, OR o Simplify trig expressions using the trig identities, OR o Use trig identities to solve trig proofs. 1.0 Know and use the vocabulary Identify the Basic Elements With help, a partial understanding of some of the simpler details and process PMI-NJ Center for Teaching & Learning ~2~ NJCTL.org Converting Degrees and Radians INSTRUCTION 1: KHAN ACADEMY INSTRUCTION 2: SOPHIA Class Work Convert the following degree measures to radians and radian measures to degrees. Sketch each angle. 2π 5. 150° 1. 3 14π 9 2. 35° 6. 3. 225° 7. 310° 4. Ο 5 8. 10Ο 7 Homework Convert the following degree measures to radians and radian measures to degrees. Sketch each angle. Ο 5π 12. 6 9. 3 10. 75° 13. 175° 11. 200° 14. 17π 9 15. 350° 16. 9Ο 7 Co-terminal Angles INSTRUCTION 1: SOPHIA Classwork Name one positive angle and one negative angle that is co-terminal with the given angle. 2π 21. 150° 17. 3 14π 9 18. 35° 22. 19. 225° 23. 310° Ο 20. 5 24. 10Ο 7 Homework Name one positive angle and one negative angle that is co-terminal with the given angle. 5π 29. 175° 25. 3 17π 9 26. 75° 30. 27. 200° 31. 350° Ο 28. 6 PMI-NJ Center for Teaching & Learning 32. ~3~ 9Ο 7 NJCTL.org INSTRUCTION 1: KHAN ACADEMY INSTRUCTION 2: SOPHIA UNIT CIRCLE TRIG VALUES ( _______ ,________ ) ( _______ ,________ ) ( _______ ,________ ) ( _______ ,________ ) ( _______ ,________ ) ( _______ ,________ ) ( _______ ,________ ) ( _______ ,________ ) ( _______ ,________ ) ( _______ ,________ ) ( _______ ,________ ) ( _______ ,________ ) ( _______ ,________ ) ( _______ ,________ ) ( _______ ,________ ) ( _______ ,________ ) ( _______ ,________ ) ( _______ ,________ ) ( _______ ,________ ) ( _______ ,________ ) PMI-NJ Center for Teaching & Learning ( _______ ,________ ) ~4~ NJCTL.org Class Work 3 β2β10 ) 7 7 33. Given the terminal point ( , β5 β12 ) 13 34. Given the terminal point ( 13 , find tanΞΈ and π. find cot π and π. 2 35. Given cos π = 3 and the terminal point in the fourth quadrant, find sin π. 4 36. Given cot π = 5 and the terminal point in the third quadrant, find sec π. For problems 53 - 56, for each given function value, find the values of the other five trig functions. 1 37. sin π = β 4 and the terminal point is in the fourth quadrant. 38. tan π = β2 and the terminal point is in the second quadrant. 8 5 39. csc π = and the terminal point is in the second quadrant. 40. sec π = 3 and the terminal point is in the fourth quadrant. State the quadrant in which π lies: 70. sin π > 0, cos π > 0 71. sin π < 0, tan π > 0 72. csc π < 0, sec π > 0 73. sin π > 0, cot π > 0 Find the exact value of the given expression. 74. cos 4Ο 3 76. sec 2Ο 3 78. cot 15Ο 4 75. sin 7Ο 4 77. tan -5Ο 6 79. csc -9Ο 2 Find the exact value of the sine, cosine and tangent of the given angle. 80. 4π 3 π 81. β 2 82. 11π 4 83. 210° 84. -315° PMI-NJ Center for Teaching & Learning ~5~ NJCTL.org Homework 7 β24 , ) 25 25 find cotΞΈ and π. β4β2 7 , 9) 9 find tanΞΈ and π. 85. Given the terminal point ( 86. Given the terminal point ( 7 87. Given sin π= 8 and the terminal point in the second quadrant, find sec π. 5 88. Given csc π = β4 and the terminal point in the third quadrant find cot π. For problems 68 - 71, for each given function value, find the values of the other five trig functions. 9 89. sin π = 41 and the terminal point is in the second quadrant. 90. cot π = β3 and the terminal point is in the second quadrant. 3 5 91. cos π = β and the terminal point is in the third quadrant. 92. sin π = 0.7 and the terminal point is in the second quadrant. State the quadrant in which π lies: 93. sin π > 0, cos π < 0 94. sin π < 0, tan π < 0 95. csc π > 0, sec π > 0 96. sin π < 0, cot π < 0 Find the exact value of the given expression. 4Ο 3 97. cos 5Ο 3 99. sec 98. sin 3Ο 4 100. tan β7Ο 6 101. cot 13Ο 4 102. csc β11π 2 Find the exact value of the sine, cosine and tangent of the given angle. 7π 6 103. 8π 3 105. β 104. 5π 4 106. 690° 107. -240° PMI-NJ Center for Teaching & Learning ~6~ NJCTL.org Graphing INSTRUCTION 1 : KHAN ACADEMY INSTRUCTION 2 SOPHIA Function Transformations ± a function ( ± b X ± c ) ± d ACTIONS DIRECTION Reflection ( - ) Vertical (outside) Stretch/Shrink ( a & b) Horizontal (inside) Phase Shift ( ± c & d) INSTRUCTION 1 : KHAN ACADEMY PMI-NJ Center for Teaching & Learning INSTRUCTION 2 SOPHIA ~7~ NJCTL.org Classwork Use the functions below to answer questions 108 β 111. a. π¦ = 2 cos π₯ b. π¦ = β2 sin 2π₯ π₯ c. π¦ = β3 sin 2 + 1 π 4 e. π¦ = sin (π₯ + ) π d. π¦ = cos (π₯ β 3 ) f. π 3 π¦ = 2 cos (2π₯ β ) g. π¦ = β4 sin(0.5π₯ + π) + 1 108. Find the amplitude of each function. 110. Find the phase shift of each function. 109. Find the period of each function. 111. Find the vertical shift of each function. 112. Sketch one cycle of each function on graph paper. 113. Is the graph of π¦ = cos π₯ is the same as the graph of π¦ = sin (π₯ β )? Justify your answer π 2 For each graph below, name the amplitude, period and vertical shift. Write an equation to represent each graph. 114. 115. 116. 117. State the amplitude, period, phase shift, and vertical shift for each function. Draw the graph by hand and check with a graphing calculator. π 2 π 121. π¦ = β1 cos(3π₯ β 2π) β 1 118. π¦ = 2 cos (π₯ + ) + 1 120. π¦ = sin ( (π₯ + )) + 3 3 119. 3 6 122. π¦ = β3 cos(4π₯ β π) β 2 2 π¦ = 3 cos(4π₯ β 2π) + 2 123. The musical note A above middle C on a piano makes a sound that can be modeled by the sine wave π¦ = sin(880ππ₯), where x represents time in seconds, and y represents the sound pressure. What is the period of this function? 124. A row boat in the ocean oscillates up and down with the waves. The boat moves a total of 10 feet from its low point to its high point and then returns to its low point every 11 seconds. Write an equation to represent the boatβs position y at time t, if the boat is at its low point at t = 0. PMI-NJ Center for Teaching & Learning ~8~ NJCTL.org Homework Use the functions below to answer questions 125 β 128. a. π¦ = β3 cos π₯ b. π¦ = β2 sin 2π₯ π₯ c. π¦ = β sin 6 d. π¦ = cos (π₯ + π e. π¦ = β2 sin (π₯ + 4 ) f. 2π ) 3 π π¦ = 4 cos (π₯ β 3 ) β 2 g. π¦ = β2 sin(π₯ + 3π) + 5 125. Find the amplitude of each function. 127. Find the phase shift of each function. 126. Find the period of each function. 128. Find the vertical shift of each function. 129. Sketch one cycle of each function on graph paper. State the amplitude, period, phase shift, and vertical shift for each function. Draw the graph by hand and check with a graphing calculator. 1 2 π 3 130. π¦ = β4 cos ( (π₯ β )) + 2 131. π¦ = β2 cos(4π₯ β 3π) β 3 1 4 π 2 132. π¦ = 2 sin ( (π₯ + )) + 1 133. π¦ = β1 cos(6π₯ β 2π) β 1 3 2 134. π¦ = cos(4π₯ β 3π) β 2 135. The musical notes C# (C sharp) and E can be modeled by the sine waves π¦ = sin(1100ππ₯), and π¦ = sin(1320ππ₯) respectively , where x represents time in seconds, and y represents the sound pressure. What are the periods of these functions? 136. A swimmer on a raft in the ocean oscillates up and down with the waves. The raft moves a total of 7 feet from its low point to its high point and then returns to its low point every 8 seconds. Write an equation to represent the raftβs position y at time t, if the raft is at its low point at t = 0. PMI-NJ Center for Teaching & Learning ~9~ NJCTL.org Trigonometric Identities INSTRUCTION 1 : KHAN ACADEMY Class Work Simplify the expression 137. csc π₯ tan π₯ INSTRUCTION 2 SOPHIA 139. sin x (csc x β sin x) 138. cot π₯ sec π₯ sin π₯ 140. (1 + cot 2 x)(1 β cos 2 x) 141. 1 β tan2 x ÷ sec 2 π₯ 142. (sin x β cos x)2 143. cos π₯ cot2 x 1βsin2x 145. sin π₯ tan π₯ + cos π₯ 144. sec π₯+tan π₯ Verify the Identity 146. (1 β sin π₯)(1 + sin π₯) = cos 2 x 147. tan π₯ cot π₯ sec π₯ = cos π₯ 148. (1 β cos2 x)(1 + tan2 x) = tan2 x 149. 1 sec x+tan x + 1 sec xβtan x = 2 sec x Homework Simplify the expression 150. (tan x + cot x )2 1 151. 1 153. sin π₯ β csc π₯ sin2 x cos2 x 155. tan2 x + cot2 x 157. cos x sec x + sin x csc x Verify the Identity 159. πππ 2 π₯ β π ππ2 π₯ = 1 β 2π ππ2 π₯ 161. 1+cot x csc x = sin x + cos x PMI-NJ Center for Teaching & Learning 1βsin x cos x + 1βsin x cos x 152. 154. 1+sec2 x 1+tan2 x 156. π‘ππ2 π₯ 1+π‘ππ2 π₯ 158. 1+sec2 x cos2 x + 2 1+tan2 x cot x cos xβcos y sin x+sin y sin xβsin y + cos x+cos y 160. tan π₯ cos π₯ csc π₯ = 1 162. cos x csc x cot x ~10~ =1 NJCTL.org PERIODIC TRIG UNIT REVIEW Multiple Choice 1. How many degrees is a. b. c. d. 4Ο 9 ? 160° 110° 80° 62° 2. Which angle is 11π 3 ? a. c. b. d. 3. Which of the following angles is/are co-terminal with 170° (choose all correct answers)? a. 340° c. -190° b. 190° d. 530° 4. Which is larger and by how much: an angle of 258°, or an angle of 6 a. 258° by ° c. b. 258° by radian d. 7 6 7 5. The central angle of a circle has a measure of 5π 4 10π radians? 7 10π 7 10π 7 1 radians by ° 7 6 radians by ° 7 radians and it intercepts an arc whose length is 5 meters. What is the approximate length in meters of the radius of the circle? a. 19.6 m c. 1.3 m b. 2.0 m d. 12.6 m 6. π is the radian measure of a central angle that intercepts an arc of length π in a circle with a radius π. If π= 2π 3 and r = 9, what is the value of s? a. 18.8 c. 0.23 b. 4.3 d. 56.5 7. A windshield wiper of a car makes an angle of 170°. If the area covered by the blade is 864 square inches, how long is the blade? a. 1,119,744 inches c. 24 inches b. 36 inches d. 576 inches PMI-NJ Center for Teaching & Learning ~11~ NJCTL.org 8. Given the terminal point of ( a. β2 ββ2 , 2 2 ) find tan π. Ο 4 b. β Ο 4 c. -1 d. 1 5 9. Knowing sec π₯ = β and the terminal point is in the second quadrant find cot π. 4 a. b. β4 c. 5 3 d. 5 10. If csc π₯ = β a. cos π₯ = b. tan π₯ = 13 12 β5 β4 3 β3 4 and the terminal point is in the third quadrant, which of the following is NOT true? c. 13 12 sec π₯ = β d. sin π₯ = 5 13 5 12 13 5 11. What is the phase shift of π¦ = cos(6π₯ β 2π) + 3? 3 a. b. 1 c. 2Ο Ο 1 3 d. 2π 3 12. Name the amplitude and vertical shift of π¦ = β0.5 cos(3π₯ + π) β 3. a. Amplitude: -0.5, Vertical Shift: -3 b. Amplitude: 0.5, Vertical Shift: -3 π c. Amplitude: β , Vertical Shift: 3 d. Amplitude: π 3 3 , Vertical Shift: -3 π 13. Which graph represents π¦ = β2 cos (3π₯ β ) + 1? 3 a. c. b. d. π 14. The difference between the maximum of π¦ = 2 cos (2 (π₯ + )) + 1 and π¦ = β3 cos(4π₯ β π) β 2 is 3 a. 1 b. 2 15. (sec π₯ + tan π₯)(sec π₯ β tan π₯) = a. 1 + 2 sec π₯ tan π₯ b. 1 β sec π₯ tan π₯ PMI-NJ Center for Teaching & Learning c. 3 d. 8 c. 1 d. 1 β sec 2 π₯ sin π₯ ~12~ NJCTL.org 16. Find the exact value of sin a. b. 5π 6 1 c. 2 ββ3 d. 2 β3 2 β2 2 17. On the interval [0, 2Ο), if sin 2π₯ = 0, what is π₯? a. 0 Ο b. c. 2 3Ο 2 d. all of the above 18. If the angle ο± is placed in standard position, its terminal side lies in quadrant II and sin π = What is the value of 5 cos(π + 3π). (This problem is from the NJ Model Curriculum assessment for Algebra II Unit 3.) a. β0.8 c. 0.75 b. β0.75 d. 0.8 4 19. A mass is attached to a spring, as shown in the figure above. If the mass is pulled down and released, the mass will move up and down for a period of time. The height of the mass above the floor, in inches, can be modeled by the function, f(t), t seconds after the mass is set in motion. The mass is 4 feet above the floor before it is pulled down. It is pulled 3 inches below the starting point and makes one full oscillation in 0.2 second. If the spring is at its lowest point at t = 0, which of the following functions models h ? (This problem is from the NJ Model Curriculum assessment for Algebra II Unit 3.) a. ο¦ 2ο° οΆ h ο¨t ο© ο½ 48 ο 3cos ο§ tο· ο¨ 5 οΈ b. ο¦ 2ο° h ο¨t ο© ο½ 48 ο« 3cos ο§ ο¨ 5 c. h ο¨t ο© ο½ 48 ο 3cos ο¨10ο° t ο© d. h ο¨t ο© ο½ 48 ο« 3cos ο¨10ο° t ο© οΆ tο· οΈ PMI-NJ Center for Teaching & Learning ~13~ NJCTL.org Extended Response 1. Sketch the graph of π¦ = β4 sin (2π₯ β π ) 3 β1 2. The water in the bay at Long Beach Island, NJ at a particular pier measures 5 feet deep at 9PM, which is low tide. High tide is reached at 3AM when the gauge reads 12 feet. a. Which trig function would be the best fit for this model (assuming 9AM is t=0)? b. Write the equation that models this situation. c. Determine the amplitude, period, and midline. d. 3. Predict the water level at midnight. The average daily production, M (in hundreds of gallons), on a dairy farm is modeled by 2ππ π = 19.6 sin ( + 12.6) + 45 365 where d is the day, d=1 is January first. a. What is the period of the function? b. What is the average daily production on the last day of the year (d=365)? c. Using the graph of M(d), what months during the year is production over 5500 gallons a day? 4. A door has a stained glass window at the top made of panes that are arranged in a semicircular shape as shown below. The radius of the semicircular shape is 1.5 feet. Its outside edge is trimmed with metal cord. The red sectors are trimmed with gold cord and the yellow sectors are trimmed with silver cord, as shown in the diagram below. a. If all of the sectors are of equal size, how many inches of silver cord will be needed, and how many inches of gold cord will be needed? b. What is the total area in square inches of all of the red sectors? 5. A monster truck has tires that are 66 inches in diameter. If a truck rolls a distance of 100 feet, what is the angle, in radians, that each tire has turned in rolling that distance? 6. Cal C. was asked to solve the following equation over the interval [0, 2π). During his calculations he might have made an error. Identify the error and correct his work so that he gets the right answer. cos π₯ + 1 = sin π₯ cos x + 2 cos x + 1 = π ππ2 π₯ cos 2 x + 2 cos x + 1 = 1 β πππ 2 π₯ 2 cos π₯ = 0 cos π₯ = 0 Ο 3Ο , 2 2 2 PMI-NJ Center for Teaching & Learning ~14~ NJCTL.org PMI-NJ Center for Teaching & Learning ~15~ NJCTL.org