Solution: Solve each equation in x for exact solutions over the interval 30, 2p2 and

CONCEPT PREVIEW Refer to Exercises 1 6 in the previous section, and use those results to solve each equation over the interval 30, 2p2. cos 2x = 1 2

CONCEPT PREVIEW Refer to Exercises 1 6 in the previous section, and use those results to solve each equation over the interval 30, 2p2. 2x = 23 2

CONCEPT PREVIEW Refer to Exercises 1 6 in the previous section, and use those results to solve each equation over the interval 30, 2p2. sin 2x = - 1 2

CONCEPT PREVIEW Refer to Exercises 1 6 in the previous section, and use those results to solve each equation over the interval 30, 2p2. sin 2x = - 23 2

CONCEPT PREVIEW Refer to Exercises 1 6 in the previous section, and use those results to solve each equation over the interval 30, 2p2. cos 2x = -1

CONCEPT PREVIEW Refer to Exercises 1 6 in the previous section, and use those results to solve each equation over the interval 30, 2p2. cos 2x = 0

CONCEPT PREVIEW Refer to Exercises 712 in the previous section, and use those results to solve each equation over the interval 30, 3602. sin u 2 = 0

CONCEPT PREVIEW Refer to Exercises 712 in the previous section, and use those results to solve each equation over the interval 30, 3602. sin u 2 = -1

CONCEPT PREVIEW Refer to Exercises 712 in the previous section, and use those results to solve each equation over the interval 30, 3602. cos u 2 = - 1 2

CONCEPT PREVIEW Refer to Exercises 712 in the previous section, and use those results to solve each equation over the interval 30, 3602. cos u 2 = - 22 2

CONCEPT PREVIEW Refer to Exercises 712 in the previous section, and use those results to solve each equation over the interval 30, 3602. sin u 2 = - 22 2

CONCEPT PREVIEW Refer to Exercises 712 in the previous section, and use those results to solve each equation over the interval 30, 3602. sin u 2 = - 23 2

Suppose solving a trigonometric equation for solutions over the interval 30, 2p2 leads to 2x = 2p 3 , 2p, 8p 3 . What are the corresponding values of x?

Suppose solving a trigonometric equation for solutions over the interval 30, 2p2 leads to 12 x = p 16 , 5p 12 , 5p 8 . What are the corresponding values of x?

Suppose solving a trigonometric equation for solutions over the interval 30, 3602 leads to 3u = 180, 630, 720, 930. What are the corresponding values of u?

Suppose solving a trigonometric equation for solutions over the interval 30, 3602 leads to 13 u = 45, 60, 75, 90. What are the corresponding values of u?

Solve each equation in x for exact solutions over the interval 30, 2p2 and each equation in u for exact solutions over the interval 30, 3602. See Examples 1 4. 2 cos 2x = 23

Solve each equation in x for exact solutions over the interval 30, 2p2 and each equation in u for exact solutions over the interval 30, 3602. See Examples 1 4. 2 cos 2x = -1

Solve each equation in x for exact solutions over the interval 30, 2p2 and each equation in u for exact solutions over the interval 30, 3602. See Examples 1 4. sin 3u = -1

Solve each equation in x for exact solutions over the interval 30, 2p2 and each equation in u for exact solutions over the interval 30, 3602. See Examples 1 4.sin 3u = 0

Solve each equation in x for exact solutions over the interval 30, 2p2 and each equation in u for exact solutions over the interval 30, 3602. See Examples 1 4. 3 tan 3x = 23

Solve each equation in x for exact solutions over the interval 30, 2p2 and each equation in u for exact solutions over the interval 30, 3602. See Examples 1 4. cot 3x = 23

Solve each equation in x for exact solutions over the interval 30, 2p2 and each equation in u for exact solutions over the interval 30, 3602. See Examples 1 4. 22 cos 2u = -1

Solve each equation in x for exact solutions over the interval 30, 2p2 and each equation in u for exact solutions over the interval 30, 3602. See Examples 1 4. 223 sin 2u = 23

Solve each equation in x for exact solutions over the interval 30, 2p2 and each equation in u for exact solutions over the interval 30, 3602. See Examples 1 4. sin x 2 = 22 - sin x 2

Solve each equation in x for exact solutions over the interval 30, 2p2 and each equation in u for exact solutions over the interval 30, 3602. See Examples 1 4. tan 4x = 0

Solve each equation in x for exact solutions over the interval 30, 2p2 and each equation in u for exact solutions over the interval 30, 3602. See Examples 1 4. sin x = sin 2x

Solve each equation in x for exact solutions over the interval 30, 2p2 and each equation in u for exact solutions over the interval 30, 3602. See Examples 1 4. cos 2x - cos x = 0

Solve each equation in x for exact solutions over the interval 30, 2p2 and each equation in u for exact solutions over the interval 30, 3602. See Examples 1 4. 8 sec2 x 2 = 4

Solve each equation in x for exact solutions over the interval 30, 2p2 and each equation in u for exact solutions over the interval 30, 3602. See Examples 1 4. sin2 x 2 - 2 = 0

Solve each equation in x for exact solutions over the interval 30, 2p2 and each equation in u for exact solutions over the interval 30, 3602. See Examples 1 4. sin u 2 = csc u 2

Solve each equation in x for exact solutions over the interval 30, 2p2 and each equation in u for exact solutions over the interval 30, 3602. See Examples 1 4. sec u 2 = cos u 2

Solve each equation in x for exact solutions over the interval 30, 2p2 and each equation in u for exact solutions over the interval 30, 3602. See Examples 1 4. cos 2x + cos x = 0

Solve each equation in x for exact solutions over the interval 30, 2p2 and each equation in u for exact solutions over the interval 30, 3602. See Examples 1 4. sin x cos x = 1 4

Solve each equation (x in radians and u in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. See Examples 1 4. 22 sin 3x - 1 = 0

Solve each equation (x in radians and u in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. See Examples 1 4. -2 cos 2x = 23

Solve each equation (x in radians and u in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. See Examples 1 4. cos u 2 = 1

Solve each equation (x in radians and u in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. See Examples 1 4. sin u 2 = 1

Solve each equation (x in radians and u in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. See Examples 1 4. 223 sin x 2 = 3

Solve each equation (x in radians and u in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. See Examples 1 4. 223 cos x 2 = -3

Solve each equation (x in radians and u in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. See Examples 1 4. 2 sin u = 2 cos 2u

Solve each equation (x in radians and u in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. See Examples 1 4. cos u - 1 = cos 2u

Solve each equation (x in radians and u in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. See Examples 1 4. 1 - sin x = cos 2x

Solve each equation (x in radians and u in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. See Examples 1 4. sin 2x = 2 cos2 x

Solve each equation (x in radians and u in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. See Examples 1 4. 3 csc2 x 2 = 2 sec x

Solve each equation (x in radians and u in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. See Examples 1 4. cos x = sin2 x 2

Solve each equation (x in radians and u in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. See Examples 1 4. 2 - sin 2u = 4 sin 2u

Solve each equation (x in radians and u in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. See Examples 1 4. 4 cos 2u = 8 sin u cos u

Solve each equation (x in radians and u in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. See Examples 1 4. 2 cos2 2u = 1 - cos 2u

Solve each equation (x in radians and u in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. See Examples 1 4. sin u - sin 2u = 0

Solve each equation for solutions over the interval 30, 2p2. Write solutions as exact values or to four decimal places, as appropriate. See Example 4. sin x 2 - cos x 2 = 0

Solve each equation for solutions over the interval 30, 2p2. Write solutions as exact values or to four decimal places, as appropriate. See Example 4. sin x 2 + cos x 2 = 1

Solve each equation for solutions over the interval 30, 2p2. Write solutions as exact values or to four decimal places, as appropriate. See Example 4. tan 2x + sec 2x = 3

Solve each equation for solutions over the interval 30, 2p2. Write solutions as exact values or to four decimal places, as appropriate. See Example 4. tan 2x - sec 2x = 2

The following equations cannot be solved by algebraic methods. Use a graphing calculator to find all solutions over the interval 30, 2p2. Express solutions to four decimal places. 2 sin 2x - x3 + 1 = 0

The following equations cannot be solved by algebraic methods. Use a graphing calculator to find all solutions over the interval 30, 2p2. Express solutions to four decimal places. 3 cos x 2 + 2x - 2 = - 1 2 x + 2

Pressure of a Plucked String If a string with a fundamental frequency of 110 Hz is plucked in the middle, it will vibrate at the odd harmonics of 110, 330, 550, . . . Hz but not at the even harmonics of 220, 440, 660, . . . Hz. The resulting pressure P caused by the string is graphed below and can be modeled by the following equation. P = 0.003 sin 220pt + 0.003 3 sin 660pt + 0.003 5 sin 1100pt + 0.003 7 sin 1540pt (Source: Benade, A., Fundamentals of Musical Acoustics, Dover Publications. Roederer, J., Introduction to the Physics and Psychophysics of Music, Second Edition, Springer-Verlag.) (a) Duplicate the graph shown here. (b) Describe the shape of the sound wave that is produced. (c) At lower frequencies, the inner ear will hear a tone only when the eardrum is moving outward. This occurs when P is negative. Determine the times over the interval 30, 0.034 when this will occur.

Hearing Beats in Music Musicians sometimes tune instruments by playing the same tone on two different instruments and listening for a phenomenon known as beats. Beats occur when two tones vary in frequency by only a few hertz. When the two instruments are in tune, the beats disappear. The ear hears beats because the pressure slowly rises and falls as a result of this slight variation in the frequency. (Source: Pierce, J., The Science of Musical Sound, Scientific American Books.) (a) Consider the two tones with frequencies of 220 Hz and 223 Hz and pressures P1 = 0.005 sin 440pt and P2 = 0.005 sin 446pt, respectively. A graph of the pressure P = P1 + P2 felt by an eardrum over the 1-sec interval 30.15, 1.154 is shown here. How many beats are there in 1 sec? (b) Repeat part (a) with frequencies of 220 and 216 Hz. (c) Determine a simple way to find the number of beats per second if the frequency of each tone is given

Hearing Difference Tones When a musical instrument creates a tone of 110 Hz, it also creates tones at 220, 330, 440, 550, 660, . . . Hz. A small speaker cannot reproduce the 110-Hz vibration but it can reproduce the higher frequencies, which are the upper harmonics. The low tones can still be heard because the speaker produces difference tones of the upper harmonics. The difference between consecutive frequencies is 110 Hz, and this difference tone will be heard by a listener. (Source: Benade, A., Fundamentals of Musical Acoustics, Dover Publications.) (a) In the window 30, 0.034 by 3-1, 14, graph the upper harmonics represented by the pressure P = 1 2 sin32p12202t4 + 1 3 sin32p13302t4 + 1 4 sin32p14402t4. (b) Estimate all t-coordinates where P is maximum. (c) What does a person hear in addition to the frequencies of 220, 330, and 440 Hz? (d) Graph the pressure produced by a speaker that can vibrate at 110 Hz and above.

Daylight Hours in New Orleans The seasonal variation in length of daylight can be modeled by a sine function. For example, the daily number of hours of daylight in New Orleans is given by h = 35 3 + 7 3 sin 2px 365 , where x is the number of days after March 21 (disregarding leap year). (Source: Bushaw, D., et al., A Sourcebook of Applications of School Mathematics, Mathematical Association of America.) (a) On what date will there be about 14 hr of daylight? (b) What date has the least number of hours of daylight? (c) When will there be about 10 hr of daylight?

Average Monthly Temperature in Vancouver The following function approximates average monthly temperature y (in F) in Vancouver, Canada. Here x represents the month, where x = 1 corresponds to January, x = 2 corresponds to February, and so on. (Source: www.weather.com) 1x2 = 14 sin c p 6 1x - 42 d + 50 When is the average monthly temperature (a) 64F (b) 39F?

Average Monthly Temperature in Phoenix The following function approximates average monthly temperature y (in F) in Phoenix, Arizona. Here x represents the month, where x = 1 corresponds to January, x = 2 corresponds to February, and so on. (Source: www.weather.com) 1x2 = 19.5 cos c p 6 1x - 72 d + 70.5 When is the average monthly temperature (a) 70.5F (b) 55F?

(Modeling) Alternating Electric Current The study of alternating electric current requires solving equations of the form i = Imax sin 2pt, for time t in seconds, where i is instantaneous current in amperes, Imax is maximum current in amperes, and f is the number of cycles per second. (Source: Hannon, R. H., Basic Technical Mathematics with Calculus, W. B. Saunders Company.) Find the least positive value of t, given the following data. i = 40, Imax = 100, = 60

(Modeling) Alternating Electric Current The study of alternating electric current requires solving equations of the form i = Imax sin 2pt, for time t in seconds, where i is instantaneous current in amperes, Imax is maximum current in amperes, and f is the number of cycles per second. (Source: Hannon, R. H., Basic Technical Mathematics with Calculus, W. B. Saunders Company.) Find the least positive value of t, given the following data. i = 50, Imax = 100, = 120

(Modeling) Alternating Electric Current The study of alternating electric current requires solving equations of the form i = Imax sin 2pt, for time t in seconds, where i is instantaneous current in amperes, Imax is maximum current in amperes, and f is the number of cycles per second. (Source: Hannon, R. H., Basic Technical Mathematics with Calculus, W. B. Saunders Company.) Find the least positive value of t, given the following data. i = Imax , = 60

(Modeling) Alternating Electric Current The study of alternating electric current requires solving equations of the form i = Imax sin 2pt, for time t in seconds, where i is instantaneous current in amperes, Imax is maximum current in amperes, and f is the number of cycles per second. (Source: Hannon, R. H., Basic Technical Mathematics with Calculus, W. B. Saunders Company.) Find the least positive value of t, given the following data. i = 1 2 Imax , = 60