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All Textbook Solutions for Precalculus: Mathematics for Calculus - 6th Edition
64E65E66E67E68EInterference Two identical tuning forks are struck, one a fraction of a second after the other. The sounds produced are modeled by f1(t) = C sin t and f2(t) = C sin(t + ). The two sound waves interfere to produce a single sound modeled by the sum of these functions f(t)=Csint+Csin(t+) (a) Use the Addition Formula for Sine to show that f can be written in the form f(t) = A sin t + B cos t, where A and B are constants that depend on . (b) Suppose that C = 10 and = /3. Find constants k and so that f(t) = k sin(t + ).70E71EIf we know the values of sin x and cos x, we can find the value of sin 2x by using the _____ Formula for Sine. State the formula: sin 2x = __________.If we know the value of cos x and the quadrant in which x/2 lies, we can find the value of sin(x/2) by using the _____ Formula for Sine. State the formula: sin(x/2) = __________.3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46E47E48E49E50E51E52E53E54E55E56E57E58E59E60E61E62E63E64E65E66E67E68E69E70E71E72E73E74E75E76E77E78E79E80E81E82E83E84E85E86E87E88E89E90E91E92E93E94E95E96E97E98E99E100E101E102E103E104ESawing a Wooden Beam A rectangular beam is to be cut from a cylindrical log of diameter 20 in. (a) Show that the cross-sectional area of the beam is modeled by the function A()=200sin2where is as shown in the figure. (b) Show that the maximum cross-sectional area of such a beam is 200 in2. [Hint: Use the fact that sin u achieves its maximum value at u = /2.]106E107ETouch-Tone Telephones When a key is pressed on a touchtone telephone, the keypad generates two pure tones, which combine to produce a sound that uniquely identifies the key. The figure shows the low frequency f1 and the high frequency f2 associated with each key. Pressing a key produces the sound wave y = sin(2f1t) + sin(2f2t). (a) Find the function that models the sound produced when the 4 key is pressed. (b) Use a Sum-to-Product Formula to express the sound generated by the 4 key as a product of a sine and a cosine function. (c) Graph the sound wave generated by the 4 key from t = 0 to t = 0.006 s.109EBecause the trigonometric functions are periodic, if a basic trigonometric equation has one solution, it has _____ (several/infinitely many) solutions.The basic equation sin x = 2 has _____ (no/one/infinitely many) solutions, whereas the basic equation sin x = 0.3 has _____ (no/one/infinitely many) solutions.We can find some of the solutions of sin x = 0.3 graphically by graphing y = sin x and y = _____. Use the graph below to estimate some of the solutions.4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46E47E48E49E50E51E52E53E54E55E56ERefraction of Light It has been observed since ancient times that light refracts, or bends, as it travels from one medium to another (from air to water, for example). If v1, is the speed of light in one medium and v2 is its speed in another medium, then according to Snells Law, sin1sin2=v1v2 where 1 is the angle of incidence and 2 is the angle of refraction (see the figure). The number v1/v2 is called the index of refraction. The index of refraction for several substances is given in the table. If a ray of light passes through the surface of a lake at an angle of incidence of 70, what is the angle of refraction? Substance Refraction from air to substance Water 1.33 Alcohol 1.36 Glass 1.52 Diamond 2.41Total Internal Reflection When light passes from a more-dense to a less-dense mediumfrom glass to air, for examplethe angle of refraction predicted by Snells Law (see Exercise 57) can be 90 or larger. In this case the light beam is actually reflected back into the denser medium. This phenomenon, called total internal reflection, is the principle behind fiber optics. Set 2 = 90 in Snells Law, and solve for 1 to determine the critical angle of incidence at which total internal reflection begins to occur when light passes from glass to air. (Note that the index of refraction from glass to air is the reciprocal of the index from air to glass.)Phases of the Moon As the moon revolves around the earth, the side that faces the earth is usually just partially illuminated by the sun. The phases of the moon describe how much of the surface appears to be in sunlight. An astronomical measure of phase is given by the fraction F of the lunar disc that is lit. When the angle between the sun, earth, and moon is (0 360), then F=12(1cos) Determine the angles that correspond to the following phases: (a) F = 0 (new moon) (b) F = 0.25 (a crescent moon) (c) F = 0.5 (first or last quarter) (d) F = 1 (full moon)60EWe can use identities to help us solve trigonometric equations. 1. Using a Pythagorean identity we see that the equation sin x + sin2x + cos2x = 1 is equivalent to the basic equation __________ whose solutions are x = _____.We can use identities to help us solve trigonometric equations. 2. Using a Double-Angle Formula we see that the equation sin x + sin 2x = 0 is equivalent to the equation _____. Factoring, we see that solving this equation is equivalent to solving the two basic equations _____ and _____.3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46E47E48E49E50E51E52E53E54E55E56E57E58E59E60E61E62E63EDamped Vibrations The displacement of a spring vibrating in damped harmonic motion is given by y=4e3tsin2t Find the times when the spring is at its equilibrium position (y = 0).Hours of Daylight In Philadelphia the number of hours of daylight on day t (where t is the number of days after January 1) is modeled by the function L(t)=12+2.83sin(2365(t80)) (a) Which days of the year have about 10 h of daylight? (b) How many days of the year have more than 10 h of daylight?Belts and Pulleys A thin belt of length L surrounds two pulleys of radii R and r, as shown in the figure to the right. (a) Show that the angle (in rad) where the belt crosses itself satisfies the equation +2cot2=LR+r [Hint: Express L in terms of R, r, and by adding up the lengths of the curved and straight parts of the belt.] (b) Suppose that R = 2.42 ft, r = 1.21 ft, and L = 27.78 ft. Find by solving the equation in part (a) graphically. Express your answer both in radians and in degrees.67E1RCC2RCC3RCC4RCC5RCC6RCC7RCC8RCC1RE2RE3RE4RE5RE6RE7RE8RE9RE10RE11RE12RE13RE14RE15RE16RE17RE18RE19RE20RE21RE22RE23RE24RE25RE26RE27RE28RE29RE30RE31RE32RE33RE34RE35RE36RE37RE38RE39RE40RE41RE42RE43RE44RE45RE46RE47RE48RE