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All Textbook Solutions for Precalculus
For Exercises 64-68, use the fundamental trigonometric identities as needed. Give that cosx0.6691 , Approximate the given function values. Round to 4 decimal places. a. sinx b. sin2x c. tanx d. cos2x e. secx f. cot2xFor Exercises 64-68, use the fundamental trigonometric identities as needed. Given thatcos12=2+64 , five the exact function values. a. sin512 b. sin12 c. sec12For Exercises 64-68, use the fundamental trigonometric identities as needed. Given that tan36=525 , give the exact function values. a. sec36 b. csc54 c. cot5468PE69PEAn airplane traveling 400 mph at a cruising altitude of 6.6 mi begins its descent. If the angle of descent is 2 from the horizontal, determine the new altitude after 15 min. Round to the nearest tenth of a mile.A scientist standing at the top of a mountain 2mi above sea level measures the angle of depression to the ocean horizon to be 1.82 . Use this information to approximate the radius of the Earth to the nearest mile.72PEFind the exact lengths x,y , and z .Use the figure to explain why tan=cot90 .75PEAn athlete is in a boat at point A, 14mi from the nearest point D on a straight shoreline. She can row at a speed of 3mph and run at a speed of 6mph . Her planned workout is to row to point D and then run to point C farther down the shoreline. However, the current pushes her at an angle of 24 from her original path so that she comes ashore at point B , 2mi from her final destination at point C . How many minutes will her trip take? Round to the nearest minute.77PEIn the figure, CD=15,DE=8,tan=43 and sin=35 , Find the lengths of a. AC b. AD c. DB d. BE e. AB79PE80PEUse a cofunction relationship to show that the product tan1tan2tan3tan87tan88tan89 is equal to 1 .For Exercises 82-85, use a calculator to approximate the values of the left- and right-hand sides of each statement for A=30 and B=45 . Based on the approximation from your calculator, determine if the statement appears to be true or false. a. sinA+B=sinA+sinB b. sinA+B=sinAcosB+cosAsinBFor Exercises 82-85, use a calculator to approximate the values of the left- and right-hand sides of each statement for A=30 and B=45 . Based on the approximation from your calculator, determine if the statement appears to be true or false. a. tanAB=tanAtanB b. tanAB=tanAtanB1+tanAtanB84PEFor Exercises 82-85, use a calculator to approximate the values of the left- and right-hand sides of each statement for A=30 and B=45 . Based on the approximation from your calculator, determine if the statement appears to be true or false. a. tanB2=1cosBsinB b. tanB2=sinB1+cosBLet p5,7 be a point on the terminal side of angle drawn in standard position. Find the values of the six trigonometric functions of .Find the reference angle . a. =150 b. =157.5 c. =5 d. =133Evaluate the functions. a. cos56 b. cot120 c. csc74Evaluate the functions. a. csc11 b. cos600Given cos=38 and sin0 , find and tan .Given sin=513 for in Quadrant IV, find cos , and tan .The distance from the origin to a point Px,y is given by .2PEIf is an angle in standard position, the angle for is the acute angle formed by the terminal side of and the horizontal axis.The values tan and sec are undefined for odd multiples of .The values cot and csc are undefined for multiples of .For what values of is sin greater than 1 ?Let px,y be a point on the terminal side of an angle drawn in standard position and let r be the distance from p to the origin. Fill in the boxes to form the ratios defining the six trigonometric functions. a. sin= b. cos= c. tan= d. csc= e. sec= f. cot=Fill in the cells in the table with the appropriate sign for each trigonometric function for in Quadrants I, II, III, and IV. The signs for the sine function are done for you.For Exercises 9-14, given the stated conditions, identify the quadrant in which lies. sin0 and tan0For Exercises 9-14, given the stated conditions, identify the quadrant in which lies. csc0 and cot011PEFor Exercises 9-14, given the stated conditions, identify the quadrant in which lies. cot0 and sin0For Exercises 9-14, given the stated conditions, identify the quadrant in which lies. cos0 and cot0For Exercises 9-14, given the stated conditions, identify the quadrant in which lies. cot0 and sec015PEFor Exercises 15-20, a point is given on the terminal side of an angle drawn in standard position. Find the values of the six trigonometric functions of (See Example 1) 8,15For Exercises 15-20, a point is given on the terminal side of an angle drawn in standard position. Find the values of the six trigonometric functions of (See Example 1) 3,5For Exercises 15-20, a point is given on the terminal side of an angle drawn in standard position. Find the values of the six trigonometric functions of (See Example 1) 2,3For Exercises 15-20, a point is given on the terminal side of an angle drawn in standard position. Find the values of the six trigonometric functions of (See Example 1) 32,2For Exercises 15-20, a point is given on the terminal side of an angle drawn in standard position. Find the values of the six trigonometric functions of (See Example 1) 5,83Complete the table for the given angles.22PEFor Exercises 23-30, find the reference angle for the given angle. (See Example 2) a. 135 b. 330 c. 660 d. 690For Exercises 23-30, find the reference angle for the given angle. (See Example 2) a. 120 b. 225 c. 1035 d. 510For Exercises 23-30, find the reference angle for the given angle. (See Example 2) a. 23 b. 56 c. 134 d. 103For Exercises 23-30, find the reference angle for the given angle. (See Example 2) a. 54 b. 116 c. 173 d. 196For Exercises 23-30, find the reference angle for the given angle. (See Example 2) a. 2017 b. 9920 c. 110 d. 42228PEFor Exercises 23-30, find the reference angle for the given angle. (See Example 2) a. 1.8 b. 1.8 c. 5.1 d. 5.1For Exercises 23-30, find the reference angle for the given angle. (See Example 2) a. 0.6 b. 0.6 c. 100 d. 100For Exercises 31-54, use reference angles to find the exact value. (See Examples 3 and 4) sin120For Exercises 31-54, use reference angles to find the exact value. (See Examples 3 and 4) cos22533PEFor Exercises 31-54, use reference angles to find the exact value. (See Examples 3 and 4) sin56For Exercises 31-54, use reference angles to find the exact value. (See Examples 3 and 4) sec330For Exercises 31-54, use reference angles to find the exact value. (See Examples 3 and 4) csc225For Exercises 31-54, use reference angles to find the exact value. (See Examples 3 and 4) sec133For Exercises 31-54, use reference angles to find the exact value. (See Examples 3 and 4) csc53For Exercises 31-54, use reference angles to find the exact value. (See Examples 3 and 4) cot240For Exercises 31-54, use reference angles to find the exact value. (See Examples 3 and 4) tan150For Exercises 31-54, use reference angles to find the exact value. (See Examples 3 and 4) tan54For Exercises 31-54, use reference angles to find the exact value. (See Examples 3 and 4) cot34For Exercises 31-54, use reference angles to find the exact value. (See Examples 3 and 4) cos630For Exercises 31-54, use reference angles to find the exact value. (See Examples 3 and 4) sin63045PEFor Exercises 31-54, use reference angles to find the exact value. (See Examples 3 and 4) cos114For Exercises 31-54, use reference angles to find the exact value. (See Examples 3 and 4) sec1170For Exercises 31-54, use reference angles to find the exact value. (See Examples 3 and 4) csc750For Exercises 31-54, use reference angles to find the exact value. (See Examples 3 and 4) csc550PEFor Exercises 31-54, use reference angles to find the exact value. (See Examples 3 and 4) tan2400For Exercises 31-54, use reference angles to find the exact value. (See Examples 3 and 4) cot90053PEFor Exercises 31-54, use reference angles to find the exact value. (See Examples 3 and 4) tan184For Exercises 55-58, find two angles between 0 and 360 for the given condition. sin=12For Exercises 55-58, find two angles between 0 and 360 for the given condition. cos=22For Exercises 55-58, find two angles between 0 and 360 for the given condition. cot=358PEFor Exercises 59-62, find two angles between 0 and 2 for the given condition. sec=260PEFor Exercises 59-62, find two angles between 0 and 2 for the given condition. tan=33For Exercises 59-62, find two angles between 0 and 2 for the given condition. cot=1In Exercises 63-68, find the values of the trigonometric functions from the given information. (See Example 5) Given tan=2021 and cos0, find sin and cos .In Exercises 63-68, find the values of the trigonometric functions from the given information. (See Example 5) Given cot=1160 and sin0, find cos and sin .In Exercises 63-68, find the values of the trigonometric functions from the given information. (See Example 5) Given sin=310 and tan0, find cos and cot .In Exercises 63-68, find the values of the trigonometric functions from the given information. (See Example 5) Given cos=58 and csc0, find sin and tan .In Exercises 63-68, find the values of the trigonometric functions from the given information. (See Example 5) Given sec=587 and cot0, find csc and cos .In Exercises 63-68, find the values of the trigonometric functions from the given information (See Example 5) Given csc=265 and cos0, find sin and cot .69PEFor Exercises 69-74, use fundamental trigonometric identities to find the values of the functions. (See Example 6) Given sin=817 for in Quadrant III, find cos and cot .For Exercises 69-74, use fundamental trigonometric identities to find the values of the functions. (See Example 6) Given tan=4 for in Quadrant II, find sec and cot .For Exercises 69-74, use fundamental trigonometric identities to find the values of the functions. (See Example 6) Given sec=5 for in Quadrant IV, find csc and cos .73PEFor Exercises 69-74, use fundamental trigonometric identities to find the values of the functions. (See Example 6) Given csc=73 for in Quadrant II, find cot and cos .75PEFor Exercises 75-76, find the sign of the expression for in each quadrant.For Exercises 77-84, suppose that is an acute angle. Identify each statement as true or false. If the statement is false, rewrite the statement to give the correct answer for the right side. cos180=cosFor Exercises 77-84, suppose that is an acute angle. Identify each statement as true or false. If the statement is false, rewrite the statement to give the correct answer for the right side. tan180=tanFor Exercises 77-84, suppose that is an acute angle. Identify each statement as true or false. If the statement is false, rewrite the statement to give the correct answer for the right side. tan180+=tanFor Exercises 77-84, suppose that is an acute angle. Identify each statement as true or false. If the statement is false, rewrite the statement to give the correct answer for the right side. sin180+=sinFor Exercises 77-84, suppose that is an acute angle. Identify each statement as true or false. If the statement is false, rewrite the statement to give the correct answer for the right side. csc=csc82PE83PEFor Exercises 77-84, suppose that is an acute angle. Identify each statement as true or false. If the statement is false, rewrite the statement to give the correct answer for the right side. sin+=sinFor Exercises 85-90, find the value of each expression. sin30cos150sec60csc120For Exercises 85-90, find the value of each expression. cos45sin240tan135cot6087PE88PEFor Exercises 85-90, find the value of each expression. 2tan1161tan211690PE91PEFor Exercises 91-94, verify the statement for the given values. cosBA=cosBcosA+sinBsinA;A=330,B=120For Exercises 91-94, verify the statement for the given values. tanA+B=tanA+tanB1tanAtanB;A=210,B=120For Exercises 91-94, verify the statement for the given values. cotA+B=cotAcotB1cotA+cotB;A=300,B=150For Exercises 95-96, give the exact values if possible. Otherwise, use a calculator and approximate the result to 4 decimal places. a. sin30 b. sin30 c. sin30For Exercises 95-96, give the exact values if possible. Otherwise, use a calculator and approximate the result to 4 decimal places. a. cos0.25 b. cos0.25 c. cos25Explain why neither sinnorcos can be greater than 1 . Refer to the figure for your explanation.Explain why tan is undefined at =2 but cot is defined at =2 .99PEThe circle shown is centered at the origin with a radius of 1 . The segment BD is tangent to the circle at D . Match the length of each segment with the appropriate trigonometric function. ACi.tanBDii.cosOBiii.sinOCiv.secCircle A, with radius a, and circle B , with radius b , are tangent to each other and to PQ (see figure). PR passes through the center of each circle. Let x be the distance from point P to a point S where PR intersects circle A on the left. Let denote RPQ . a. Show that sin=ax+a and sin=bx+2a+b . b. Use the results from part (a) to show that sin=bab+a .Graph the function and identify the key points on one full period. a. y=2cosx b. y=13sinxGiven fx=2sin4x, a. Identify the amplitude and period. b. Graph the function and identify the key points on one full period.Given y=cos3x2, a. Identify the amplitude and period. b. Graph the function and identify the key points on one full period.Given y=2cos3x+3, a. Identify the amplitude, period, phase shift, and vertical shift. b. Graph the function and identify the key points on one full period.Given y=2sin6x2+1, a. Identify the amplitude, period, phase shift, and vertical shift. b. Graph the function and identify the key points on one full period.A mechanical metronome uses an inverted pendulum that makes a regular, rhythmic click as it swings to the left and right. With each swing, the pendulum moves 3in . to the left and right of the center position. The pendulum is initially pulled to the right 3in . and then released. It returns to its starting position in 0.8 sec. Assuming that this pattern continues indefinitely and behaves like a cosine wave, write a function of the form xt=AcosBtC+D . The value xt is the horizontal position (in inches) relative to the center line of the pendulum.The value of sinx (increases/decreases) on 0,2 and (increases/decreases) on 2, .The value of cosx (increases/decreases) on 0,2 and (increases/decreases) on 2, .The graph of y=sinx and y=cosx differ by a horizontal shift of units.Given y=AsinBxC+D or y=AcosBxC+D , for B0 the amplitude is , the period is , the phase shift is , and the vertical shift is .The sine function is an (even/odd) function because sinx= . The cosine function is an (even/odd) function because cosx= .Given B0 , how would the equation y=AsinBxC+D be rewritten to obtain a positive coefficient on x ?Given B0 , how would the equation y=AcosBxC+D be rewritten to obtain a positive coefficient on x ?Given y=sinBx and y=cosBx , for B1 , is the period less than or greater than 2 ? If 0B1, is the period less than or greater than 2 ?From memory, sketch y=sinx on the interval 0,2 .10PEFor y=cosx , a. The domain is . b. The range is . c. The amplitude is . d. The period is . e. The cosine function is symmetric to the -axis. f. On the interval 0,2 , the x -intercepts are . g. On the interval 0,2 , the maximum points are and , and the minimum point is .For y=sinx , a. The domain is . b. The range is . c. The amplitude is . d. The period is . e. The cosine function is symmetric to the . f. On the interval 0,2 , the x-intercepts are . g. On the interval 0,2 , the maximum points is and the minimum point is .a. Over what interval (s) taken between 0 and 2 is the graph of y=sinx increasing? b. Over what interval (s) taken between 0 and 2 is the graph of y=sinx decreasing?a. Over what interval (s) taken between 0 and 2 is the graph of y=cosx increasing? b. Over what interval (s) taken between 0 and 2 is the graph of y=cosx decreasing?15PEFor Exercises 15-16, identify the amplitude of the function. a. y=2cosx b. y=12cosx c. y=2cosxBy how many units does the graph of y=14cosx deviate from the x-axis ?By how many units does the graph of y=5sinx deviate from the x-axis ?For Exercises 19-24, graph the function and identify the key points on one full period. (See Examples 1) y=5cosxFor Exercises 19-24, graph the function and identify the key points on one full period. (See Examples 1) y=4sinx21PEFor Exercises 19-24, graph the function and identify the key points on one full period. (See Examples 1) y=14cosxFor Exercises 19-24, graph the function and identify the key points on one full period. (See Examples 1) y=2cosxFor Exercises 19-24, graph the function and identify the key points on one full period. (See Examples 1) y=3sinxFor Exercises 25-26, identify the period. a. sin2x b. sin2x c. sin23x26PEFor Exercises 27-32 a. Identify the amplitude and period. b. Graph the function and identify the key points on one full period. (See Example2) y=2cos3xFor Exercises 27-32 a. Identify the amplitude and period. b. Graph the function and identify the key points on one full period. (See Example2) y=6sin4xFor Exercises 27-32 a. Identify the amplitude and period. b. Graph the function and identify the key points on one full period. (See Example2) y=4sin3xFor Exercises 27-32 a. Identify the amplitude and period. b. Graph the function and identify the key points on one full period. (See Example2) y=5cos6xFor Exercises 27-32 a. Identify the amplitude and period. b. Graph the function and identify the key points on one full period. (See Example2) y=sin13xFor Exercises 27-32 a. Identify the amplitude and period. b. Graph the function and identify the key points on one full period. (See Example2) y=cos12xWrite a function of the form fx=AcosBx for the given graph.Write a function of the form gx=AsinBx for the given graph.35PEA respiratory cycle is defined as the beginning of one breath to the beginning of the next breath. The rate of air intake r (in L/sec) during a respiratory cycle for a physically fit male can be approximated by rt=0.9sin3.5t , where t is the number of seconds into the cycle. A positive value for r represents inhalation and a negative value represents exhalation. a. How long is the respiratory cycle? b. What is the maximum rate of air intake? c. Graph one cycle of the function. On what interval does inhalation occur? On what interval does exhalation occur?37PEFor Exercises 37-38, identify the phase shift and indicate whether the shift is to the left or to the right. a. sinx+8 b. sin2x8 c. sin4x8For Exercises 39-44, a. Identify the amplitude, period, and phase shift. b. Graph the function and identify the key points on one full period. (See Example 3) y=2cosx+For Exercises 39-44, a. Identify the amplitude, period, and phase shift. b. Graph the function and identify the key points on one full period. (See Example 3) y=4sinx+2For Exercises 39-44, a. Identify the amplitude, period, and phase shift. b. Graph the function and identify the key points on one full period. (See Example 3) y=sin2x3For Exercises 39-44, a. Identify the amplitude, period, and phase shift. b. Graph the function and identify the key points on one full period. (See Example 3) y=cos3x443PEFor Exercises 39-44, a. Identify the amplitude, period, and phase shift. b. Graph the function and identify the key points on one full period. (See Example 3) y=5sin13x+6Write a function of the form fx=AcosBxC for the given graph.46PEGiven y=2sin2x67 , a. Is the period less than or greater than 2 ? b. Is the phase shift to the left or right? c. Is the vertical shift upward or downward?48PEFor Exercises 49-52, rewrite the equation so that the coefficient on x is positive. y=cos2x+64For Exercises 49-52, rewrite the equation so that the coefficient on x is positive. y=4cos3x+5For Exercises 49-52, rewrite the equation so that the coefficient on x is positive. y=sin2x+64For Exercises 49-52, rewrite the equation so that the coefficient on x is positive. y=4sin3x+5Given y=2sin6x+23 , is the phase shift to the right or left?54PEFor Exercises 55-68, a. Identify the amplitude, period, phase shift, and vertical shift. b. Graph the function and identify the key points on one full period. (See Example 4-5) hx=3sin4x+5For Exercises 55-68, a. Identify the amplitude, period, phase shift, and vertical shift. b. Graph the function and identify the key points on one full period. (See Example 4-5) gx=2sin3x4For Exercises 55-68, a. Identify the amplitude, period, phase shift, and vertical shift. b. Graph the function and identify the key points on one full period. (See Example 4-5) fx=4cos3x21For Exercises 55-68, a. Identify the amplitude, period, phase shift, and vertical shift. b. Graph the function and identify the key points on one full period. (See Example 4-5) kx=5cos2x2+1For Exercises 55-68, a. Identify the amplitude, period, phase shift, and vertical shift. b. Graph the function and identify the key points on one full period. (See Example 4-5) y=12sin13xFor Exercises 55-68, a. Identify the amplitude, period, phase shift, and vertical shift. b. Graph the function and identify the key points on one full period. (See Example 4-5) y=23sin12x61PEFor Exercises 55-68, a. Identify the amplitude, period, phase shift, and vertical shift. b. Graph the function and identify the key points on one full period. (See Example 4-5) mx=2.4cos4xFor Exercises 55-68, a. Identify the amplitude, period, phase shift, and vertical shift. b. Graph the function and identify the key points on one full period. (See Example 4-5) y=2sin2x+5For Exercises 55-68, a. Identify the amplitude, period, phase shift, and vertical shift. b. Graph the function and identify the key points on one full period. (See Example 4-5) y=3sin4x7For Exercises 55-68, a. Identify the amplitude, period, phase shift, and vertical shift. b. Graph the function and identify the key points on one full period. (See Example 4-5) px=cos2x+2For Exercises 55-68, a. Identify the amplitude, period, phase shift, and vertical shift. b. Graph the function and identify the key points on one full period. (See Example 4-5) qx=cos3x267PEFor Exercises 55-68, a. Identify the amplitude, period, phase shift, and vertical shift. b. Graph the function and identify the key points on one full period. (See Example 4-5) y=sin3x2+4The temperature TinF for Kansas City, Missouri, over a several day period in April can be approximated by Tt=5.9cos0.262t1.245+48.2 , where t is the number of hours since midnight on day 1 . a. What is the period of the function? Round to the nearest hour. b. What is the significance of the term 48.2 in this model? c. What is the significance of the factor 5.9 in this model? d. What was the minimum temperature for the day? When did it occur? e. What was the maximum temperature for the day? When did it occur?The duration of daylight and darkness varies during the year due to the angle of the Sun in the sky. The model dt=2.65sin0.51t1.32+12 approximates the amount of daylight dt (in hours) for Sacramento, California, as a function of the time t (in months) after January 1 for a recent year; that is, t=0 is January 1 , t=0 is February 1 , and so on. The model y=nt represents the amount of darkness as a function of t a. Describe the relationship between the graphs of the functions and the line y=12 . b. Use the result of part (a) and a transformation of y=dt to write an equation representing n as a function of t . c. What do the points of intersection of the two graphs represent? d. What do the relative minima and relative maxima of the graphs represent? e. What does Tt=dt+nt represent?The probability of precipitation in Modesto, California, varies from a peak of 0.3434 in January to a low of 0.044 in July. Assume that the percentage of precipitation varies monthly and behaves like a cosine curve. a. Write a function of the form Pt=AcosBtC+D to model the precipitation probability. The value Pt is the probability of precipitation (as a decimal), for month t with January ast=1 . b. Graph the function from part (a) on the interval 0,13 and plot the points 1,0.34,7,0.04 , and 13,0.34 to check the accuracy of your model.The monthly high temperature for Atlantic City. New Jersey, peaks at an average high of 86 in July and goes down to an average high of 64 in January. Assume that this pattern for monthly high temperatures continues indefinitely and behaves like a cosine wave. a. Write a function of the form Ht=AcosBtC+D to model the average high temperature. The value Ht is the average high temperature for month t , with January as t=0. b. Graph the function from part (a) on the interval 0,13 and plot the points 0,64,6,86 and 12,64 to check the accuracy of your model.An adult human at rest inhales and exhales approximately 500mL of air (called the tidal volume) in approximately 5sec . However, at the end of each exhalation, the lungs still contain a volume of air, called the functional residual capacity (FRC), which is approximately 2000mL . (See Example 6) a. What volume of air is in the lungs after inhalation? b. What volume of air is in the lungs after exhalation? c. What is the period of a complete respiratory cycle? d. Write a function Vt=AcosBt+D to represent the volume of air in the lungs t seconds after the end of an inhalation. e. What is the average amount of air in the lungs during one breathing cycle? f. During hyperventilation, breathing is more rapid with deep inhalations and exhalations. What parts of the equation from part (d) change?The times for high and low tides are given in the table for a recent day in Jacksonville Beach, Florida. The times are rounded to the nearest hour and the tide levels are measured relative to mean sea level (MSL). a. Write a model ht=AcosBtC to represent the tide level ht (in feet) in terms of the amount of time t elapsed since midnight. b. Use the model from part (a) to estimate the tide level at 3:00P.M .The data in the table represent the monthly power bills (in dollars) for a homeowner in southern California. a. Enter the data in a graphing utility and use the sinusoidal regression tool (SinReg) to find a model of the formAt=asinbt+c+d , where At represents the amount of the bill for month t . (t=1 represents January, t=2 represents February, and so on). b. Graph the data and the resulting function.The data in the table represent the duration of daylight dt (in hours) for Houston, Texas, for the first day of the month, t months after January 1 for a recent year. a. Enter the data in a graphing utility and use the sinusoidal regression tool (SinReg) to find a model of the form dt=asinbt+c+d . b. Graph the data and the resulting function.For Exercises 77-78, write the range of the function in interval notation. a. y=8cos2x+4 b. y=3cosx+35For Exercises 77-78, write the range of the function in interval notation. a. y=6sin3x22 b. y=2sin3x+2+12Given fx=cosx and hx=3x+2 , find hfx and hfx , a. Find the amplitude. b. Find the period. c. Write the domain in interval notation. d. Write the range in interval notation.Given gx=sinx and kx=6x , find gkx and gkx a. Find the amplitude. b. Find the period. c. Write the domain in interval notation. d. Write the range in interval notation.81PEWrite a function of the form y=AcosBxC+D that has period 4 , amplitude 2 , phase shift 3 , and vertical shift 7 .Write a function of the form y=AcosBxC+D that has period 16 , phase shift 4 , and range 3y7 .Write a function of the form y=AsinBxC+D that has period 8 , phase shift 2 , and range 14y6 .85PEFor Exercises 85-86 a. Write an equation of the form y=AcosBxC+D with A0 to model the graph. b. Write an equation of the form y=AsinBxC+D with A0 to model the graph.For Exercises 87-90, explain how to graph the given function by performing transformations on the "parent" graphs y=sinx and y=cosx . a. y=sin2x b. y=2sinxFor Exercises 87-90, explain how to graph the given function by performing transformations on the "parent" graphs y=sinx and y=cosx . a. y=13cosx b. y=cos13xFor Exercises 87-90, explain how to graph the given function by performing transformations on the "parent" graphs y=sinx and y=cosx . a. y=sinx+2 b. y=sinx+2For Exercises 87-90, explain how to graph the given function by performing transformations on the "parent" graphs y=sinx and y=cosx . a. y=cosx4 b. y=cosx491PEIs fx=cosx one-to-one? Explain why or why not.93PEExplain why a function that is increasing on its entire domain cannot be periodic.For Exercises 95-96, find the average rate of change on the given interval. Give the exact value and an approximation to 4 decimal places. Verify that your results are reasonable by comparing the results to the slopes of the lines given in the graph. fx=sinx a. 0,6 b. 6,3 c. 3,2For Exercises 95-96, find the average rate of change on the given interval. Give the exact value and an approximation to 4 decimal places. Verify that your results are reasonable by comparing the results to the slopes of the lines given in the graph. fx=cosx a. 0,6 b. 6,3 c. 3,2For exercises 97-100, use your knowledge of the graphs of the sine function and linear functions to determine the number of solutions to the equation. sinx=x2For exercises 97-100, use your knowledge of the graphs of the sine function and linear functions to determine the number of solutions to the equation. cosx=x99PEFor exercises 97-100, use your knowledge of the graphs of the sine function and linear functions to determine the number of solutions to the equation. 2sin2x=2For Exercises 101-102, graph the piecewise-defined function. gx=sinxfor0xsinxforx2For Exercises 101-102, graph the piecewise-defined function. fx=cosxfor0x4sinxfor4x2Functions a and m approximate the duration of daylight, respectively, for Albany, New York, and Miami, Florida, for a given year for day t . The value t=1 represents January 1 , t=2 represents February 1 , and so on. at=12+3.1sin2365t80mt=12+1.6sin2365t80 a. Graph the two functions with a graphing utility and comment on the difference between the two graphs. b. Both functions have a constant term of 12. What does this represent graphically and in the context of this problem? c. What do the factors 3.1 and 1.6 represent in the two functions? d. What is the period of each function? e. What does the horizontal shift of 80 units represent in the context of this problem. f. Use the Intersect feature to approximate the points of intersection. g. Interpret the meaning of the points of intersection.For Exercises 104-105, we demonstrate that trigonometric functions can be approximated by polynomial functions over a given interval in the domain. Graph functions f,g,h , and k on the viewing window 4x4,4y4 . Then use a Table feature on a graphing utility to evaluate each function for the given values of x . How do functions g,h , and k compare to function f for x values farther from 0 ?For Exercises 104-105, we demonstrate that trigonometric functions can be approximated by polynomial functions over a given interval in the domain. Graph functions f,g,h , and k on the viewing window 4x4,4y4 . Then use a Table feature on a graphing utility to evaluate each function for the given values of x . How do functions g,h , and k compare to function f for x values farther from 0 ?For Exercises 106-107, use a graph to solve the equation on the given interval. cos2x3=0.5 on 0, Viewing window; 0,,3 by 1,1,12For Exercises 106-107, use a graph to solve the equation on the given interval. sin2x+4=1 on 0,2 Viewing window; 0,2,8 by 2,2,1For Exercises 108-109, use a graph to solve the equation on the given interval. Round the answer to 2 decimal places sinx4=ex on , Viewing window; ,,2 by 2,2,1109PEGraph the functions on the window provided. y=2y=2y=2sinx a. Viewing window: 2,2,2 by 3,3,1 b. y=0.5xy=0.5xy=0.5xsinx Viewing window: 5,5,by8,8,1 c.y=cosxy=cosxy=cosxsin12x Viewing window: 0,2.5,0.25 by 1,1,0.5 d. Explain the relationship among the three functions in parts (a), (b), and (c).Graph y=2sec4x .Graph y=cscx+2 .Graph y=4tan2x .4SPAt values of x for which sinx=0 , the graph of y=cscx will have a . This occurs for x= for all integers n .2PEThe relative maxima on the graph of y=sinx correspond to the on the graph of y=cscx .4PEIf a function is an odd function, then each point x,y in Quadrant I will have a corresponding point , in Quadrant .The range of y=tanx and y=cotx is .The graphs of both y=tanx and y=cotx are symmetric with respect to the .For the functions y=AtanBxC and y=AcotBxC with B0 , the vertical scaling factor is , the period is , and the phase shift is .Sketch the graph of y=cscx from memory. Use the graph of y=sinx for reference.Sketch the graph of y=secx from memory. Use the graph of y=cosx for reference.For Exercises 11-16, identify the statements among a-h that follow directly from the given condition about x . a. cscx is undefined. b. secx is undefined. c. The graph of y=secx has a relative maximum at x . d. The graph of y=cscx has a relative minimum at x . e. The graph of y=secx has a vertical asymptote. f. The graph of y=cscx has a vertical asymptote. g. The graph of y=cscx has a relative maximum at x . h. The graph of y=secx has a relative minimum at x . sinx=0For Exercises 11-16, identify the statements among a-h that follow directly from the given condition about x . a. cscx is undefined. b. secx is undefined. c. The graph of y=secx has a relative maximum at x . d. The graph of y=cscx has a relative minimum at x . e. The graph of y=secx has a vertical asymptote. f. The graph of y=cscx has a vertical asymptote. g. The graph of y=cscx has a relative maximum at x . h. The graph of y=secx has a relative minimum at x . cosx=0For Exercises 11-16, identify the statements among a-h that follow directly from the given condition about x . a. cscx is undefined. b. secx is undefined. c. The graph of y=secx has a relative maximum at x . d. The graph of y=cscx has a relative minimum at x . e. The graph of y=secx has a vertical asymptote. f. The graph of y=cscx has a vertical asymptote. g. The graph of y=cscx has a relative maximum at x . h. The graph of y=secx has a relative minimum at x . The graph of y=cosx has a relative maximum at x .14PEFor Exercises 11-16, identify the statements among a-h that follow directly from the given condition about x . a. cscx is undefined. b. secx is undefined. c. The graph of y=secx has a relative maximum at x . d. The graph of y=cscx has a relative minimum at x . e. The graph of y=secx has a vertical asymptote. f. The graph of y=cscx has a vertical asymptote. g. The graph of y=cscx has a relative maximum at x . h. The graph of y=secx has a relative minimum at x . The graph of y=cosx has a relative maximum at x .For Exercises 11-16, identify the statements among a-h that follow directly from the given condition about x . a. cscx is undefined. b. secx is undefined. c. The graph of y=secx has a relative maximum at x . d. The graph of y=cscx has a relative minimum at x . e. The graph of y=secx has a vertical asymptote. f. The graph of y=cscx has a vertical asymptote. g. The graph of y=cscx has a relative maximum at x . h. The graph of y=secx has a relative minimum at x . The graph of y=sinx has a relative maximum at x .For Exercises 17-32, graph one period of the function. (See Examples 1-2) y=2cscxFor Exercises 17-32, graph one period of the function. (See Examples 1-2) y=14secx19PEFor Exercises 17-32, graph one period of the function. (See Examples 1-2) y=13cscxFor Exercises 17-32, graph one period of the function. (See Examples 1-2) y=3cscx3For Exercises 17-32, graph one period of the function. (See Examples 1-2) y=4secx223PEFor Exercises 17-32, graph one period of the function. (See Examples 1-2) y=csc3xFor Exercises 17-32, graph one period of the function. (See Examples 1-2) y=cscx4For Exercises 17-32, graph one period of the function. (See Examples 1-2) y=secx+3For Exercises 17-32, graph one period of the function. (See Examples 1-2) y=2sec2x+For Exercises 17-32, graph one period of the function. (See Examples 1-2) y=csc3x+2For Exercises 17-32, graph one period of the function. (See Examples 1-2) y=2csc2x+4+130PE31PEFor Exercises 17-32, graph one period of the function. (See Examples 1-2) y=cscx++4For Exercises 33-34, write the range of the function in interval notation. a. y=4csc2x+7 b. y=2csc3x10 .34PE35PEWrite a function of the form y=cscBxC for the given graph.A plane flying at an altitude of 5mi travels on a path directly over a radar tower. a. Express the distance d (in miles) between the plane and the tower as a function of the angle in standard position from the tower to the plane. b. If d=5 , what is the measure of the angle and where is the plane located relative to the tower? c. Can the value of be ? Explain your answer in terms of the function d .The distance dx (in feet) between an observer 30 ft from a straight highway and a police car traveling down the highway is given bydx=30secx , where x is the angle (in degrees) between the observer and the police car. a. Use a calculator to evaluate dx for the given values of x . Round to the nearest foot. b. Try experimenting with values of x closer to 90 . What happens as x90 ?a. Graph y=tanx on the interval , b. How many periods of the tangent function are shown on the interval , ?a. Graph y=cotx on the interval , . b. How many periods of the cotangent function are shown on the interval , ?For Exercises 41-42, graph one complete period of the function. Identify the x- intercept and evaluate the function for x values midway between the x- intercept and the asymptotes. (See Example 3) a. y=12tanx b. y=3tanxFor Exercises 41-42, graph one complete period of the function. Identify the x- intercept and evaluate the function for x values midway between the x- intercept and the asymptotes. (See Example 3) a. y=2cotx b. y=14cotxFor Exercises 43-58, graph the function. (See Example 3-4) y=tan2xFor Exercises 43-58, graph the function. (See Example 3-4) y=cot3xFor Exercises 43-58, graph the function. (See Example 3-4) y=cot2xFor Exercises 43-58, graph the function. (See Example 3-4) y=tanx47PEFor Exercises 43-58, graph the function. (See Example 3-4) y=cot13xFor Exercises 43-58, graph the function. (See Example 3-4) y=4cot2x50PE51PEFor Exercises 43-58, graph the function. (See Example 3-4) y=5cot4x