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All Textbook Solutions for Precalculus: Mathematics for Calculus (Standalone Book)
Terminal Points Find the terminal point P(x, y) on the unit circle determined by the given value of t. 32. t=53Terminal Points Find the terminal point P(x, y) on the unit circle determined by the given value of t. 33. t=74Terminal Points Find the terminal point P(x, y) on the unit circle determined by the given value of t. 34. t=43Terminal Points Find the terminal point P(x, y) on the unit circle determined by the given value of t. 35. t=34Terminal Points Find the terminal point P(x, y) on the unit circle determined by the given value of t. 36. t=116Reference Numbers Find the reference number for each value of t. 37. (a) t=43 (b) t=53 (c) t=76 (d) t = 3.5Reference Numbers Find the reference number for each value of t. 38. (a) t = 9 (b) t=54 (c) t=256 (d) t = 4Reference Numbers Find the reference number for each value of t. 39. (a) t=57 (b) t=79 (c) t = 3 (d) t = 5Reference Numbers Find the reference number for each value of t. 40. (a) t=115 (b) t=97 (c) t = 6 (d) t = 7Terminal Points and Reference Numbers Find (a) the reference number for each value of t and (b) the terminal point determined by t. 41. t=11642E43E44ETerminal Points and Reference Numbers Find (a) the reference number for each value of t and (b) the terminal point determined by t. 45. t=2346ETerminal Points and Reference Numbers Find (a) the reference number for each value of t and (b) the terminal point determined by t. 47. t=13448E49E50E51E52E53E54E55E56E57E58E59E60EDISCOVER PROVE: Finding the Terminal Point for /6 Suppose the terminal point determined by t = /6 is P(x, y) and the points Q and R are as shown in the figure. Why are the distances PQ and PR the same? Use this fact, together with the Distance Formula, to show that the coordinates of P satisfy the equation 2y=x2+(y1)2. Simplify this equation using the fact that x2 + y2 = 1. Solve the simplified equation to find P(x, y).DISCOVER PROVE: Finding the Terminal Point for /3 Now that you know the terminal point determined by t = /6, use symmetry to find the terminal point determined by t = /3 (see the figure). Explain your reasoning.Let P(x, y) be the terminal point on the unit circle determined by t. Then sin t = __________, cos t = __________, and tan t = __________.If P(x, y) is on the unit circle, then x2 + y2 = __________. So for all t we have sin2 t + cos2 t = __________.Evaluating Trigonometric Functions Find sin t and cos t for the values of t whose terminal points are shown on the unit circle in the figure. In Exercise 3, t increases in increments of /4; in Exercise 4, t increases in increments of /6. (See Exercises 21 and 22 in Section 5.1.) 3.Evaluating Trigonometric Functions Find sin t and cos t for the values of t whose terminal points are shown on the unit circle in the figure. In Exercise 3, t increases in increments of /4: in Exercise 4, t increases in increments of /6. (See Exercises 21 and 22 in Section 5.1.) 4.5EEvaluating Trigonometric Functions Find the exact value of the trigonometric function at the given real number. 6. (a) sin53 (b) cos113 (c) tan53Evaluating Trigonometric Functions Find the exact value of the trigonometric function at the given real number. 7. (a) sin114 (b) sin(4) (c) sin548E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31EEvaluating Trigonometric Functions The terminal point P(x, y) determined by a real number t is given. Find sin t, cos t, and tan t. 32. (4041,941)33E34EEvaluating Trigonometric Functions The terminal point P(x, y) determined by a real number t is given. Find sin t, cos t, and tan t. 35. (2029,2129)36EValues of Trigonometric Functions Find an approximate value of the given trigonometric function by using (a) the figure and (b) a calculator. Compare the two values. 37. sin 138EValues of Trigonometric Functions Find an approximate value of the given trigonometric function by using (a) the figure and (b) a calculator. Compare the two values. 39. sin 1.2Values of Trigonometric Functions Find an approximate value of the given trigonometric function by using (a) the figure and (b) a calculator. Compare the two values. 40. cos 541E42E43EValues of Trigonometric Functions Find an approximate value of the given trigonometric function by using (a) the figure and (b) a calculator. Compare the two values. 44. sin(5.2)45E46E47E48E49E50E51E52E53E54E55E56E57E58E59E60E61EWriting One Trigonometric Expression in Terms of Another Write the first expression in terms of the second if the terminal point determined by t is in the given quadrant. 62. sec2 t, sin2 t, cos t; any quadrant63E64EUsing the Pythagorean Identities Find the values of the trigonometric functions of t from the given information. 65. sec t = 3, terminal point of t is in Quadrant IV66E67E68E69E70E71E72E73EEven and Odd Functions Determine whether the function is even, odd, or neither. (See page 204 for the definitions of even and odd functions.) 74. f(x) = sin x + cos x75E76E77E78EHarmonic Motion The displacement from equilibrium of an oscillating mass attached to a spring is given by y(t) = 4 cos 3t where y is measured in inches and t in seconds. Find the displacement at the limes indicated in the table. t y(t) 0 0.25 0.50 0.75 1.00 1.25Circadian Rhythms Everybodys blood pressure varies over the course of the day. In a certain individual the resting diastolic blood pressure at lime t is given by B(t) = 80 + 7 sin(t/12), where t is measured in hours since midnight and B(t) in mmHg (millimeters of mercury). Find this persons resting diastolic blood pressure at (a) 6:00 a.m. (b) 10:30 a.m. (c) Noon (d) 8:00 p.m.Electric Circuit After the switch is closed in the circuit shown, the current t seconds later is I(t) = 0.8e3t sin 10t. Find the current at the times (a) t = 0.1 s and (b) t = 0.5 s.Bungee Jumping A bungee jumper plummets from a high bridge to the river below and then bounces back over and over again. At time t seconds after her jump, her height H (in meters) above the river is given by H(t)=100+75et/20cos(4t). Find her height at the times indicated in the table. t H(t) 0 1 2 4 6 8 12DISCOVER PROVE: Reduction Formulas A reduction formula is one that can be used to reduce the number of terms in the input for a trigonometric function. Explain how the figure shows that the following reduction formulas are valid: sin(t+)=sintcos(t+)=costtan(t+)=tantDISCOVER PROVE: More Reduction Formulas By the Angle-Side-Angle Theorem from elementary geometry, triangles CDO and AOB in the figure to the right arc congruent. Explain how this proves that if B has coordinates (x, y), then D has coordinates ( y, x). Then explain how the figure shows that the following reduction formulas are valid: sin(t+2)=costcos(t+2)=sinttan(t+2)=cottIf a function f is periodic with period p, then f(t + p) = __________ for every t. The trigonometric functions y = sin x and y = cos x are periodic, with period __________ and amplitude __________. Sketch a graph of each function on the interval [0, 2].To obtain the graph of y = 5 + sin x, we start with the graph of y = sin x, then shift it 5 units __________ (upward/downward). To obtain the graph of y = cos x, we start with the graph of y = cos x, then reflect it in the __________-axis.The sine and cosine curves y = a sin kx and y = a cos kx, k 0, have amplitude __________ and period __________. The sine curve y = 3 sin 2x has amplitude __________ and period __________.The sine curve y = a sin k(x b) has amplitude __________, period __________, and horizontal shift __________. The sine curve y=4sin3(x6) has amplitude __________, period __________, and horizontal shift __________.Graphing Sine and Cosine Functions Graph the function. 5. f(x) = 2 + sin x6E7E8E9E10E11E12E13E14E15E16E17E18EAmplitude and Period Find the amplitude and period of the function, and sketch its graph. 19. y = cos 2x20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38EHorizontal Shifts Find the amplitude, period, and horizontal shift of the function, and graph one complete period. 39.y=5cos(3x4)40E41E42E43E44E45EHorizontal Shifts Find the amplitude, period, and horizontal shift of the function, and graph one complete period. 46. y=cos(2x)47EEquations from a Graph The graph of one complete period of a sine or cosine curve is given. (a) Find the amplitude, period, and horizontal shift. (b) Write an equation that represents the curve in the form y=asink(xb)ory=acosk(xb) 48.Equations from a Graph The graph of one complete period of a sine or cosine curve is given. (a) Find the amplitude, period, and horizontal shift. (b) Write an equation that represents the curve in the form y=asink(xb)ory=acosk(xb) 49.Equations from a Graph The graph of one complete period of a sine or cosine curve is given. (a) Find the amplitude, period, and horizontal shift. (b) Write an equation that represents the curve in the form y=asink(xb)ory=acosk(xb) 50.51E52E53E54E55E56EGraphing Trigonometric Functions Determine an appropriate viewing rectangle for each function, and use it to draw the graph. 57. f(x) = sin(x/40)58E59E60E61E62E63E64E65E66E67E68E69E70E71E72E73E74EMaxima and Minima Find the maximum and minimum values of the function. 75. y = 2 sin x + sin2x76E77E78E79E80E81E82EHeight of a Wave As a wave passes by an offshore piling, the height of the water is modeled by the function h(t)=3cos(10t) where h(t) is the height in feet above mean sea level at time t seconds. (a) Find the period of the wave. (b) Find the wave height, that is, the vertical distance between the trough and the crest of the wave.Sound Vibrations A tuning fork is struck, producing a pure tone as its tines vibrate. The vibrations are modeled by the function v(t)=0.7sin(880t) where v(t) is the displacement of the tines in millimeters at time t seconds. (a) Find the period of the vibration. (b) Find the frequency of the vibration, that is, the number of times the fork vibrates per second. (c) Graph the function v.Blood Pressure Each time your heart beats, your blood pressure first increases and then decreases as the heart rests between beats. The maximum and minimum blood pressures are called the systolic and diastolic pressures, respectively. Your blood pressure reading is written as systolic/diastolic. A reading of 120/80 is considered normal. A certain persons blood pressure is modeled by the function p(t)=115+25sin(160t) where p(t) is the pressure in mmHg (millimeters of mercury), at time t measured in minutes. (a) Find the period of p. (b) Find the number of heartbeats per minute. (c) Graph the function p. (d) Find the blood pressure reading. How does this compare to normal blood pressure?Variable Stars Variable stars are ones whose brightness varies periodically. One of the most visible is R Leonis; its brightness is modeled by the function b(t)=7.92.1cos(156t) where t is measured in days. (a) Find the period of R Leonis. (b) Find the maximum and minimum brightness. (c) Graph the function b.87EDISCUSS: Periodic Functions I Recall that a function f is periodic if there is a positive number p such that f(t + p) = f(t) for every t, and the least such p (if it exists) is the period of f. The graph of a function of period p looks the same on each interval of length p, so we can easily determine the period from the graph. Determine whether the function whose graph is shown is periodic: if it is periodic, find the period. (a) (b) (c) (d)89EDISCUSS: Sinusoidal Curves The graph of y = sin x is the same as the graph of y = cos x shifted to the right /2 units. So the sine curve y = sin x is also at the same time a cosine curve: y=cos(x2). In fact, any sine curve is also a cosine curve with a different horizontal shift, and any cosine curve is also a sine curve. Sine and cosine curves are collectively referred to as sinusoidal. For the curve whose graph is shown, find all possible ways of expressing it as a sine curve y = a sin(x b) or as a cosine curve y = a cos(x b). Explain why you think you have found all possible choices for a and b in each case.The trigonometric function y = tan x has period __________ and asymptotes x = __________. Sketch a graph of this function on the interval (/2, /2).The trigonometric function y = csc x has period __________ and asymptotes x = __________. Sketch a graph of this function on the interval (, ).3EGraphs of Trigonometric Functions Match the trigonometric function with one of the graphs IVI. 4. f(x) = sec 2xGraphs of Trigonometric Functions Match the trigonometric function with one of the graphs IVI. 5. f(x) = cot 4xGraphs of Trigonometric Functions Match the trigonometric function with one of the graphs IVI. 6. f(x) = tan x7E8E9E10E11E12E13E14E15E16E17E18E19E20E21EGraphs of Trigonometric Functions with Different Periods Find the period, and graph the function. 22. y = 3 tan 4x23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46EGraphs of Trigonometric Functions with Horizontal Shifts Find the period, and graph the function. 47. y=cot(2x2)48E49EGraphs of Trigonometric Functions with Horizontal Shifts Find the period, and graph the function. 50. y=3sec(14x6)51E52E53E54E55E56E57E58E59E60ELighthouse The beam from a lighthouse completes one rotation every 2 min. At time t, the distance d shown in the figure below is d(t)=3tant where t is measured in minutes and d in miles. (a) Find d(0.15), d(0.25), and d(0.45). (b) Sketch a graph of the function d for 0t12. (c) What happens to the distance d as t approaches 12?Length of a Shadow On a day when the sun passes directly overhead at noon, a 6-ft-tall man casts a shadow of length S(t)=6|cot12t| where S is measured in feet and t is the number of hours since 6 a.m. (a) Find the length of the shadow al 8:00 a.m., noon, 2:00 p.m., and 5:45 p.m. (b) Sketch a graph of the function S for 0 t 12. (c) From the graph, determine the values of t at which the length of the shadow equals the mans height. To what lime of day does each of these values correspond? (d) Explain what happens to the shadow as the time approaches 6 p.m. (that is, as t 12).PROVE: Periodic Functions (a) Prove that if f is periodic with period p, then 1/f is also periodic with period p. (b) Prove that cosecant and secant both have period 2.64EPROVE: Reduction Formulas Use the graphs in Figure 5 to explain why the following formulas are true. tan(x2)=cotxsec(x2)=cscx(a) To define the inverse sine function, we restrict the domain of sine to the interval __________. On this interval the sine function is one-to-one, and its inverse function sin1 is defined by sin1 x = y sin _____ = _____. For example, sin112= because sin _____ = _____. (b) To define the inverse cosine function, we restrict the domain of cosine to the interval __________. On this interval the cosine function is one-to-one and its inverse function cos1 is defined by cos1 x = y cos _____ = _____. For example, cos112= because cos _____ = _____.The cancellation property sin1(sin x) = x is valid for x in the interval __________. Which of the following is not true? (i) sin1(sin3)=3 (ii) sin1(sin103)=103 (iii) sin1(sin(4))=43E4E5E6EEvaluating Inverse Trigonometric Functions Find the exact value of each expression, if it is defined. 7. (a) tan1(1) (b) tan13 (c) tan1338E9E10E11E12E13E14E15EInverse Trigonometric Functions with a Calculator Use a calculator to find an approximate value of each expression correct to five decimal places, if it is defined. 16. sin1(0.13844)17E18E19E20E21EInverse Trigonometric Functions with a Calculator Use a calculator to find an approximate value of each expression correct to five decimal places, if it is defined. 22. tan1(0.25713)23E24E25ESimplifying Expressions Involving Trigonometric Functions Find the exact value of the expression, if it is defined. 26. sin(sin1 5)27E28E29E30E