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Converting Degrees and Radians – Class WorkConvert the following degree measures to radians and radian measures to degrees. Sketch each angle.2π335°225°π5150°14π9310°10π7Converting Degrees and Radians – HomeworkConvert the following degree measures to radians and radian measures to degrees. Sketch each angle.5π375°200°π6175°17π9350°9π7Co-terminal Angles – ClassworkName one positive angle and one negative angle that is co-terminal with the given angle. 2π335°225°π5150°14π9310°10π7Co-terminal Angles – HomeworkName one positive angle and one negative angle that is co-terminal with the given angle. 5π375°200°π6175°17π9350°9π7Arc Length and Sector Area - ClassworkRound all lengths to the nearest tenth. For problems 33 - 36 below, θ is the radian measure of a central angle that intercepts an arc of length s in a circle with a radius r. If s=10 and r=5, find θ. If θ=π3 and r=6, find s. If s=5.4 and r=1.8, find θ. If s=15 and θ=3π4, find r. If r=9 and θ=3π, find s. If θ=6π and s=9, find r.Find the area of a sector with radius 5 inches and central angle θ=π12.Find the area of a sector with radius 6 cm and central angle θ=150°.Find the radius of a sector with area 45 sq in and central angle θ=5π12. The central angle of a circle has a measure of 7 radians and it intercepts an arc whose length is 9 meters. What is the length in meters of the radius of the circle? The minute hand of a clock makes what angle as it moves from 6:15 to 6:45? If the length of the intercepted arc is 15 inches, what is the length of the minute hand?The wheels of a car have a diameter of 36 inches. The wheels of a scooter have a diameter of 10 inches. If each wheel makes one complete rotation, do the car and the scooter travel the same distance? If no, which travels farther, and by how much? A wedge of a round cake is cut to be one-sixth of the cake. If the diameter of the cake is 10 inches, what is the length of the intercepted arc of the top of the cake?Billy Bob got 1/3 of a 6-inch pie and Sally Sue got ? of an 8-inch pie. Who got more pie and by what percent? Go back to the dartboard problem on slide 30. What is the probability that a dart thrown at random at the board lands in the black space? Arc Length and Sector Area - HomeworkFor problems 43 - 46 below, θ is the radian measure of a central angle that intercepts an arc of length s in a circle with a radius r. If s=8 and r=9, find θ. If θ=5π3 and r=6, find s. If s=.001 and r=.00025, find θ. If s=20 and θ=9π4, find r. If r=1.5 and θ=π, find s. If θ=4π and s=18, find r.Find the area of a sector with radius 11 inches and central angle θ=π9.Find the area of a sector with radius 9 cm and central angle θ=-140°.Find the radius of a sector with area 12 sq in and central angle θ=3π4. If a circle has a radius of 6 inches and a central angle intercepts an arc of 11 inches, what is the radian measure of the central angle? The minute hand of a clock makes what angle as it moves from 8:05 to 8:57? If the length of the intercepted arc is 18 inches, what is the length of the minute hand?The wheel of a unicycle has a radius of 24 inches. The wheels of a tricycle have a radius of 16 inches. If each wheel makes one complete rotation, do the car and the scooter travel the same distance? If no, which travels farther, and by how much? A wedge of pie is cut to be one-seventh of the pie. If the length of the intercepted arc of the top of the pie is 4.3 inches, what is the diameter of the pie?Billy Bob got 38 of an 18-inch pizza pie and Sally Sue got 49 of a 16-inch pie. Who got more pizza and by what percent? Unit Circle – Class WorkGiven the terminal point 37,-2107 find tanθ and θ.Given the terminal point -513,-1213 findcotθ and θ.Given cos θ = 23 and the terminal point in the fourth quadrant, find sin θ.Given cot θ = 45 and the terminal point in the third quadrant, find sec θ.For problems 53 - 56, for each given function value, find the values of the other five trig functions.sinθ=-14 and the terminal point is in the fourth quadrant.tanθ=-2 and the terminal point is in the second quadrant. cscθ=85 and the terminal point is in the second quadrant.secθ=3 and the terminal point is in the fourth quadrant.State the quadrant in which θ lies:sinθ>0, cosθ>0sinθ<0, tanθ>0cscθ<0, secθ>0sinθ>0, cotθ>0Find the exact value of the given expression.cos4π3sin7π4sec2π3tan-5π6cot15π4csc-9π2Find the exact value of the sine, cosine and tangent of the given angle. 4π3–π211π4210°-315°Unit Circle – HomeworkGiven the terminal point 725,-2425 find cotθ and θ.Given the terminal point -429,79 find tanθ and θ.Given sin θ= 78 and the terminal point in the second quadrant, find sec θ.Given csc θ = 5-4 and the terminal point in the third quadrant find cot θ.For problems 68 - 71, for each given function value, find the values of the other five trig functions.sinθ=941 and the terminal point is in the second quadrant.cotθ=-3 and the terminal point is in the second quadrant. cosθ=-35 and the terminal point is in the third quadrant.sinθ=0.7 and the terminal point is in the second quadrant.State the quadrant in which θ lies:sinθ>0, cosθ<0sinθ<0, tanθ<0cscθ>0, secθ>0sinθ<0, cotθ<0Find the exact value of the given expression.cos5π3sin3π4 sec4π3 tan-7π6 cot13π4 csc-11π2Find the exact value of the sine, cosine and tangent of the given angle. 8π35π4-7π6690°-240°Graphing ClassworkUse the functions below to answer questions 108 – 111.y=2cosxy=-2sin2xy=-3sinx2+1y=cosx-π3y=sinx+π4y=2cos2x-π3y=-4sin0.5x+π+1Find the amplitude of each function.Find the period of each function.Find the phase shift of each function.Find the vertical shift of each function.Sketch one cycle of each function on graph paper. Is the graph of y=cosx is the same as the graph of y=sinx-π2? Justify your answer.For each graph below, name the amplitude, period and vertical shift. Write an equation to represent each graph. 365760026670228600-3810114.115.0184150116.117.320040087630State the amplitude, period, phase shift, and vertical shift for each function. Draw the graph by hand and then check it with a graphing calculator.y=2cosx+π3+1y=-3cos4x-π-2y=sin23x+π6+3y=-1cos3x-2π-1y=23cos4x-2π+2The musical note A above middle C on a piano makes a sound that can be modeled by the sine wave ?y=sin(880πx), where x represents time in seconds, and y represents the sound pressure. What is the period of this function? 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 boat’s position y at time t, if the boat is at its low point at t = 0. Graphing – HomeworkUse the functions below to answer questions 124 – 127.y=-3cosx y=-2sin2xy=-sinx6y=cosx+2π3y=-2sinx+π4y=4cosx-π3-2y=-2sinx+3π+5Find the amplitude of each function.Find the period of each function.Find the phase shift of each function.Find the vertical shift of each function. 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 then check it with a graphing calculator.y=-4cos12x-π3+2y=-2cos4x-3π-3y=2sin14x+π2+1y=-1cos6x-2π-1y=32cos4x-3π-2The musical notes C# (C sharp) and E can be modeled by the sine waves y=sin(1100πx), and y=sin1320πx respectively , where x represents time in seconds, and y represents the sound pressure. What are the periods of these functions?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 raft’s position y at time t, if the raft is at its low point at t = 0. Trigonometric Identities – Class WorkSimplify the expression137. cscxtanx138. cotxsecxsinx139. sinxcscx-sinx140. 1+cot2x1-cos2x141. 1-tan2x÷sec2x142. sinx-cosx2143. cot2x1-sin2x144. cosxsecx+tanx145. sinxtanx+cosxVerify the Identity146. 1-sinx1+sinx=cos2x147. tanxcotx secx=cosx148. 1-cos2x1+tan2x=tan2x149. 1secx+tanx+1secx-tanx=2secxTrigonometric Identities – HomeworkSimplify the expression150. tanx+cotx 2151. 1-sinxcosx+cosx1-sinx152. cosx-cosysinx+siny+sinx-sinycosx+cosy153. 1sinx-1cscx154. 1+sec2x1+tan2x155. sin2xtan2x+cos2xcot2x156. tan2x1+tan2x157. cosxsecx+sinxcscx158. 1+sec2x1+tan2x+cos2xcot2xVerify the Identity159. cos2x-sin2x=1-2sin2x160. tanxcosxcscx=1 161. 1+cotxcscx=sinx+cosx162. cosxcscxcotx=1Unit ReviewMultiple ChoiceHow many degrees is 4π9?160°110°80°62°Which angle is 11π3?c. d. Which of the following angles is/are co-terminal with 170° (choose all correct answers)?340° 190° -190° 530°Which is larger and by how much: an angle of 258°, or an angle of 10π7 radians?258° by 67°258° by 67 radian10π7 radians by 17°10π7 radians by 67°The central angle of a circle has a measure of 5π4 radians and it intercepts an arc whose length is 5 meters. What is the approximate length in meters of the radius of the circle?19.6 m2.0 m1.3 m12.6 mθ is the radian measure of a central angle that intercepts an arc of length s in a circle with a radius r. If θ=2π3 and r = 9, what is the value of s?18.84.30.2356.5A 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? 1,119,744 inches36 inches24 inches576 inches Given the terminal point of 22,-22 find tanθ.π4-π4-11Knowing secx=-54 and the terminal point is in the second quadrant find cotθ.-4535-43-34If cscx=-1312 and the terminal point is in the third quadrant, which of the following is NOT true?cosx=-513tanx=125secx=-135sinx=1213What is the phase shift of y=53cos6x-2π+3?12ππ3132πName the amplitude and vertical shift of y=-0.5cos3x+π-3.Amplitude: -0.5, Vertical Shift: -3Amplitude: 0.5, Vertical Shift: -3Amplitude: -π3, Vertical Shift: 3Amplitude: π3, Vertical Shift: -3Which graph represents y=-2cos3x-π3+1?c. d. The difference between the maximum of y=2cos2x+π3+1 and y=-3cos4x-π-2 is1238secx+tanxsecx-tanx=1+2secxtanx1-secxtanx 11-sec2xsinxFind the exact value of sin5π612-323222On the interval 0,2π, if sin2x=0, what is x?0π23π2all of the aboveIf the angle is placed in standard position, its terminal side lies in quadrant II and sinθ=45 What is the value of cos(θ+3π). (This problem is from the NJ Model Curriculum assessment for Algebra II Unit 3.)-0.8c. 0.75-0.75d. 0.8A 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. b. c.d.Extended ResponseSketch the graph of y=-4sin2x-π3-1The 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. Which trig function would be the best fit for this model (assuming 9AM is t=0)? Write the equation that models this situation. Determine the amplitude, period, and midline. Predict the water level at midnight. The average daily production, M (in hundreds of gallons), on a dairy farm is modeled byM=19.6sin2πd365+12.6+45where d is the day, d=1 is January first.a. What is the period of the function?What is the average daily production on the last day of the year (d=365)?Using the graph of M(d), what months during the year is production over 5500 gallons a day?14630408102604. 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.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? 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.cosx+1=sinxcos2x+2cosx +1=sin2xcos2x+2cosx +1=1-cos2x2cosx=0cosx=0π2,3π2Answer KeyFor sketches of #1 – 16, see end of key120°7π365π436°5π6280°31π18257.14°300°5π1210π930°35π36340°35π18231.4°8π3, -4π3395°, -325°585°, -135°11π5, -9π5510°, -210°32π9, -4π9670°, -50°24π7, -4π711π3, -π3435°, -285°560°, -160°13π6, -11π6535°, -185°35π9, -π9710°, -30°23π7, -5π72 radians6.33 radians6.484.80.483.3 in247.1 in28.3 in97 m-180 °, 4.8 mThe car travels 81.7 inches farther5.2 in. Sally got 33% more (12.6 vs. 9.4 in2) about 37% 89 radians31.44 radians2.84.71.4 21.1 in299 cm2 3.2 in 116radians26π15 radians, 3.3 inThe unicycle goes 50.3 inches farther9.6 inBilly Bob got 8% more (56.5 vs. 52.5) tan θ=-2103, θ=-64.6°cot θ=512, θ=247.3°-53-414cosθ=154, tanθ=-1515, cscθ=-4, secθ=41515, cotθ=-15cos θ=-55, sinθ=255, secθ=-5, cscθ=52, cotθ=-12sinθ=58, cosθ=-398, tan θ=-53939, secθ=-83939, cotθ=-395sinθ=-223, cosθ=13, tanθ=-22, cscθ=-324, cotθ=-24Quadrant IQuadrant IIIQuadrant IVQuadrant I-12-22-233-1-1sin4π3=-32, cos4π3=-12, tan4π3=3sin-π2=-1, cos-π2=0, tan-π2=undefinedsin11π4=22, cos11π4=-22, tan11π4=-1sin210°=-12, cos210°=-32, tan210°=33sin-315°=22, cos-315°=22, tan-315°=1cotθ=-724, and θ=-73.7°tanθ=-728, and θ = 128.9°-8151534cosθ=-4041, tanθ=-940, cscθ=419,secθ=-4140,cotθ=-409sinθ=1010, cosθ=-31010, tanθ=-13, cscθ=10,secθ=-`103sinθ=-45, tanθ=43, cscθ=-54,secθ=-53,cotθ=34cosθ=-0.7, tanθ=-1, cscθ=107,secθ=-107,cot-1Quadrant IIQuadrant IVQuadrant IQuadrant IV1222-2-3311sin8π3=32, cos8π3=-12, tan8π3=-3sin5π4=-22, cos5π4=-22, tan5π4=1sin-7π6=12, cos-7π6=-32, tan-7π6=-33sin690°=-12, cos690°=32, tan690°=-33sin-240°=32, cos-240°=-12, tan-240°=-3a. 2, b. 2, c. 3, d. 1, e. 1, f. 2, g. 4a. 2π, b. π, c. 4π, d. 2π, e. 2π, f. π, g. 4πa. 0, b. 0, c. 0, d. π3, e. -π4, f. π6, g. -2πa. 0, b. 0, c. 1, d. 0, e. 0, f. 0, g. 1 a.No they are not the same, they are reflections of each other. For example, when x = 0, cos x = 1, and sin (x - π2) = -1.A: 2, P: π, VS: -1, y = 2 cos (2x) -1 (one possible answer)A: 1, P: π, VS: 3, y = sin (2x) + 3A: 4, P: π, VS: 0, y = 4 sin (2x)A: 0.5, P: 2 π, VS: 0, y = -0.5 cos xA: 2, P: 2π, PS:-π3, VS: 1A: 3, P: π2, PS: π4, VS: -2A: 1, P: 3π, PS: -π6, VS: 3A: 1, P: 2π3, PS: 2π3, VS: -1A: 23, P: π2, PS: π2, VS: 21440y=-5cos2π11xa. 3, b. 2, c. 1, d. 1, e. 2, f. 4, g. 2a. 2 π, b. π, c. 12 π, d. 2 π, e. 2 π, f. 2 π, g. 2 πa. 0, b. 0, c. 0, d. -2π3, e. -π4, f. π3, g. -3 πa. 0, b. 0, c. 0, d. 0, e. 0, f. -2, g. 5a. A: 4, P: 4π, PS: π3, VS: 2A: 2, P: π2, PS: 3π4, VS: -3A; 2, P: 8π, PS: -π2, VS: 1A: 1, P: π3, PS: π3, VS: -1A: 32, P: π2, PS: 3π4, VS: -21550, 1660 y=3.5cosπ9xsec x1cos2 x1cos2 x1 – 2cos x sin x csc2 x1 – sinxsecx1-sin2x=cos2xsinxcosx?cosxsinxsecx=1secx=cosxsin2x?sec2x=sin2x?1cos2x=sin2xcos2x=tan2xsecx-tanx(secx+tanx)(secx-tanx)+secx+tanx(secx+tanx)(secx-tanx)=2secxsec2x-tan2x=2secx1Sec2x + Csc2x2Secx0CosxCotxCos2x + 11Sin2x121-sin2x-sin2x=1-2sin2xsinxcosxcosx1sinx=11+cosxsinxsinx=sinx+cosx 162. cosx1sinxsinxcosx=cosxsinxsinxcosx=1MC1. CMC2. DMC3. C, DMC4. AMC5. CMC6. AMC7. CMC8. CMC9. CMC10. DMC11. BMC12. BMC13. CMC14. BMC15. CMC16. AMC17. DMC18. DMC19. CER1. ER2A. cosineER2B. y=-3.5cosπ6t+8.5ER2C. amplitude = 3.5, period = 12, midline = 8.5 ftER2D. 8.5 feetER3A. 365 daysER3B. 4,500 GallonsER4A. 22.6 inches, 33.9 inchesER4B. 305.4 square inchesER5. 3 radiansER6. 2cos2x+2cosx=0cos2x+cosx=0π2, π,3π21. 2. 3. 4. 5. 6. 7. 8. 9. 10.11.12.13.14.15.16. ................
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