Trigonometric Functions 2.1 Angles and Their Measure

Transcription

1 Ch. Trigonometric Functions.1 Angles and Their Measure 1 Convert between Decimals and Degrees, Minutes, Seconds Measures for Angles MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Draw the angle. 1) 0 ) 15 Page 1

2 ) π ) - π Page

3 5) -150 ) 0 Page

4 7) - 7π ) 5π Page

5 9) ) 7π Convert the angle to a decimal in degrees. Round the answer to two decimal places. 11) 91 '1'' Page 5

6 1) 11 0'1'' ) 55 ''' ) 7'7'' ) 1 17''' Convert the angle to D M' S'' form. Round the answer to the nearest second. 1) ''' 11 0'0'' 11 0'1'' 11 0''' 17) ''' ''' 179 5'9'' 179 5'9'' 1) ''' 79 5'9'' 79 '9'' 79 57''' Find the Length of an Arc of a Circle MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. If s denotes the length of the arc of a circle of radius r subtended b a central angle θ, find the missing quantit. 1) r = 1.07 centimeters, θ =.7 radians, s =? 59.5 cm 0.5 cm 5.5 cm 1.5 cm ) r =.7 inches, θ = 5, s =?. in.. in.. in..5 in. ) r = 5 feet, s = 1 feet, θ =? 0 radians 5 radians 0 5 ) s =. meters, θ = radians, r =?. m 0.5 m 1. m 1.1 m Find the length s. Round the answer to three decimal places. 5) s π 9 d 9.5 d 1.07 d 1.5 d.59 d Page

7 ) π 5 s 10 cm. cm cm 1.5 cm cm 7) s 5 11 m 1.79 m 1.77 m 11.1 m 9.9 m ) s 5 ft. ft.7 ft.199 ft 1.95 ft Solve the problem. 9) For a circle of radius feet, find the arc length s subtended b a central angle of 0. Round to the nearest hundredth ft. ft.09 ft.19 ft 10) For a circle of radius feet, find the arc length s subtended b a central angle of 0. Round to the nearest hundredth..19 ft.5 ft.5 ft.0 ft 11) A ship in the Atlantic Ocean measures its position to be 1 0' north latitude. Another ship is reported to be due north of the first ship at 1' north latitude. Approimatel how far apart are the two ships? Round to the nearest mile. Assume that the radius of the Earth is 90 miles. 15 mi 7,1 mi 15 mi 7,00 mi SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 1) Salt Lake Cit, Utah, is due north of Flagstaff, Arizona. Find the distance between Salt Lake Cit (0 5' north latitude) and Flagstaff (5 1' north latitude). Assume that the radius of the Earth is 90 miles. Round to nearest whole mile. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) The minute hand of a clock is 5 inches long. How far does the tip of the minute hand move in 0 minutes? If necessar, round the answer to two decimal places in. 1. in. 1.9 in in. Page 7

8 1) A pendulum swings though an angle of 0 each second. If the pendulum is 5 inches long, how far does its tip move each second? If necessar, round the answer to two decimal places. 1. in..5 in..71 in in. Convert from Degrees to Radians and from Radians to Degrees MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Convert the angle in degrees to radians. Epress the answer as multiple of π. 1) 90 π π π π ) π - π - π - π ) 1 π 5 5π π 5π ) -1-9π 10-10π 11 - π 9-10π 9 5) 7 9π 0 9π 90 9π 0 9π 10 ) π 0 π 0 π 15 π 1 Convert the angle in radians to degrees. 7) 10π ) - 5π ) π 0 1 0π 10) - π π Page

9 11) 11π π 1) π π ) π ) 11π Convert the angle in degrees to radians. Epress the answer in decimal form, rounded to two decimal places. 15) ) Convert the angle in radians to degrees. Epress the answer in decimal form, rounded to two decimal places. 17) ) ) Find the Area of a Sector of a Circle MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. If A denotes the area of the sector of a circle of radius r formed b the central angle θ, find the missing quantit. If necessar, round the answer to two decimal places. 1) r = 1 inches, θ = π radians, A =? 19.5 in 9.1 in 9. in 1. in ) r = 1 feet, A = 90 square feet, θ =? 0.9 radians 0. radians 0 radians 17,0 radians ) θ = 5 radians, A = square meters, r =? 5. m 0 m 10 m 1.9 m ) r = 19 inches, θ = 90, A =?.9 in 5.77 in 1.9 in 9. in 5) r = feet, A = 51 square feet, θ =? , ,10. Page 9

10 ) θ = 0, A = square meters, r =? 9. m 9.9 m.07 m.91 m 7) r =. centimeters, θ = π 10 radians, A =? 11. cm. cm 5.9 cm 5. cm ) r =. feet, θ =.7, A =? ft 5.79 ft 50.5 ft 50.5 ft Find the area A. Round the answer to three decimal places. 9) 10) π cm.09 cm 1.07 cm.19 cm 1. cm π 11) d.5 d 0.75 d.71 d 1.5 d 70 cm 5.9 cm 1. cm cm 1.75 cm Page 10

11 1) 5 cm cm 1. cm cm.5 cm Solve the problem. 1) A circle has a radius of 11 centimeters. Find the area of the sector of the circle formed b an angle of 0. If necessar, round the answer to two decimal places.. cm 5.7 cm 1.71 cm 0.17 cm 1) An irrigation sprinkler in a field of lettuce spras water over a distance of 5 feet as it rotates through an angle of 10. What area of the field receives water? If necessar, round the answer to two decimal places ft.7 ft 0.5 ft 5. ft 15) As part of an eperiment to test different liquid fertilizers, a sprinkler has to be set to cover an area of 10 square ards in the shape of a sector of a circle of radius 0 ards. Through what angle should the sprinkler be set to rotate? If necessar, round the answer to two decimal places ) The blade of a windshield wiper sweeps out an angle of 15 in one ccle. The base of the blade is 1 inches from the pivot point and the tip is inches from the pivot point. What area does the wiper cover in one ccle? (Round to the nearest 0.1 square inch.) 10.7 in in 101. in 9. in 5 Find the Linear Speed of an Object Traveling in Circular Motion MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) An object is traveling around a circle with a radius of 10 centimeters. If in 0 seconds a central angle of 1 radian is swept out, what is the linear speed of the object? 1 cm/sec cm/sec 1 radians/sec radians/sec ) An object is traveling around a circle with a radius of 0 meters. If in 10 seconds a central angle of 1 5 radian is swept out, what is the linear speed of the object? 5 m/sec 1 5 m/sec 1 m/sec 1 m/sec ) An object is traveling around a circle with a radius of 10 meters. If in 15 seconds a central angle of radians is swept out, what is the linear speed of the object? m/sec m/sec m/sec 1 m/sec ) A weight hangs from a rope 0 feet long. It swings through an angle of 7 each second. How far does the weight travel each second? Round to the nearest 0.1 foot. 9. feet.7 feet 9.0 feet.1 feet Page 11

12 5) A gear with a radius of centimeters is turning at π radians/sec. What is the linear speed at a point on the outer 9 edge of the gear? π 9π cm/sec 9 cm/sec π cm/sec π cm/sec ) A wheel of radius 5. feet is moving forward at 11 feet per second. How fast is the wheel rotating?.1 radians/sec 0. radians/sec.5 radians/sec 0.59 radians/sec 7) A car is traveling at mph. If its tires have a diameter of inches, how fast are the car's tires turning? Epress the answer in revolutions per minute. If necessar, round to two decimal places. 50. rpm 1 rpm. rpm 100. rpm ) A pick-up truck is fitted with new tires which have a diameter of inches. How fast will the pick-up truck be moving when the wheels are rotating at 5 revolutions per minute? Epress the answer in miles per hour rounded to the nearest whole number. mph 7 mph 7 mph mph 9) The Earth rotates about its pole once ever hours. The distance from the pole to a location on Earth north latitude is about 9. miles. Therefore, a location on Earth at north latitude is spinning on a circle of radius 9. miles. Compute the linear speed on the surface of the Earth at north latitude. 75 mph 11 mph 1,197 mph 75 mph 10) To approimate the speed of a river, a circular paddle wheel with radius 0. feet is lowered into the water. If the current causes the wheel to rotate at a speed of 1 revolutions per minute, what is the speed of the current? If necessar, round to two decimal places. 0. mph 0.07 mph 7. mph 0.1 mph SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 11) The four Galilean moons of Jupiter have orbital periods and mean distances from Jupiter given b the following table. Distance (km) Period (Earth hours) Io Europa Ganmeade Callisto Find the linear speed of each moon. Which is the fastest (in terms of linear speed)? MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) In a computer simulation, a satellite orbits around Earth at a distance from the Earth's surface of. 10 miles. The orbit is circular, and one revolution around Earth takes 10.7 das. Assuming the radius of the Earth is 90 miles, find the linear speed of the satellite. Epress the answer in miles per hour to the nearest whole mile. 7 mph 1,1 mph 1 mph 5 mph 1) A carousel has a radius of 1 feet and takes 5 seconds to make one complete revolution. What is the linear speed of the carousel at its outside edge? If necessar, round the answer to two decimal places.. ft/sec 0.51 ft/sec 1. ft/sec 11.1 ft/sec Page 1

13 . Trigonometric Functions: Unit Circle Approach 1 Find the Eact Values of the Trigonometric Functions Using a Point on the Unit Circle MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. In the problem, t is a real number and P = (, ) is the point on the unit circle that corresponds to t. Find the eact value of the indicated trigonometric function of t. 1) ( 9, 77 ) Find sin t ) (, 7 ) Find tan t ) ( 55, ) Find sec t ) (- 11, 5 ) Find cos t ) (- 11, 5 ) Find cot t ) (- 1 5, - ) Find sin t ) (- 7, - ) Find cot t ) ( 7, - ) Find csc t Page 1

14 9) ( 7, - ) Find cos t ) ( 7, - 10 ) Find csc t Find the Eact Values of the Trigonometric Functions of Quadrantal Angles MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the eact value. Do not use a calculator. 1) sin π 0 1 undefined ) cos undefined ) tan undefined ) cot undefined 5) cot π undefined ) tan π undefined 7) cos π undefined ) cot π undefined 9) tan (1π) undefined 10) cot (- π ) undefined Page 1

15 11) tan (-π) undefined Find the Eact Values of the Trigonometric Functions of π/ = 5 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the eact value. Do not use a calculator. 1) sec π - ) csc 5 Find the eact value of the epression if θ = 5. Do not use a calculator. ) f(θ) = tan θ Find f(θ) ) g(θ) = sin θ Find [g(θ)]. 1-5) f(θ) = sin θ Find 5f(θ) ) g(θ) = sin θ Find 10g(θ) Solve the problem. 7) If friction is ignored, the time t (in seconds) required for a block to slide down an inclined plane is given b the formula a t = g sinθ cosθ where a is the length (in feet) of the base and g feet per second per second is the acceleration of gravit. How long does it take a block to slide down an inclined plane with base a = 1 when θ = 5? If necessar, round the answer to the nearest tenth of a second. 1. sec 1. sec 0. sec 1.5 sec ) The force acting on a pendulum to bring it to its perpendicular resting point is called the restoring force. The restoring force F, in Newtons, acting on a string pendulum is given b the formula F = mg sinθ where m is the mass in kilograms of the pendulum's bob, g 9. meters per second per second is the acceleration due to gravit, and θ is angle at which the pendulum is displaced from the perpendicular. What is the value of the restoring force when m = 0. kilogram and θ = 5? If necessar, round the answer to the nearest tenth of a Newton.. N 5 N N.9 N Page 15

16 Find the Eact Values of the Trigonometric Functions of π/ = 0 and π/ = 0 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the eact value. Do not use a calculator. 1) tan 0 1 ) sin 0 1 ) csc π 1 ) csc π Find the eact value of the epression. Do not use a calculator. 5) cot 5 - cos ) cot 0 - cos ) cos 0 + tan ) sin π - cos π ) tan π - cos π Find the eact value of the epression if θ = 0. Do not use a calculator. 10) f(θ) = cos θ Find f(θ). Page 1

17 11) g(θ) = cos θ Find g(θ) ) f(θ) = sin θ Find [f(θ)] ) g(θ) = sin θ Find 1g(θ) ) f(θ) = cos θ Find 5f(θ) Find the eact value of the epression if θ = 0. Do not use a calculator. 15) f(θ) = sin θ Find f(θ). 1 1) g(θ) = cos θ Find [g(θ)]. 1 17) f(θ) = sin θ Find f(θ) ) g(θ) = cos θ Find 11g(θ) Solve the problem. 19) If friction is ignored, the time t (in seconds) required for a block to slide down an inclined plane is given b the formula a t = g sinθ cosθ where a is the length (in feet) of the base and g feet per second per second is the acceleration of gravit. How long does it take a block to slide down an inclined plane with base a = when θ = 0? If necessar, round the answer to the nearest tenth of a second. 1.1 sec 1. sec 0. sec 1. sec 0) The force acting on a pendulum to bring it to its perpendicular resting point is called the restoring force. The restoring force F, in Newtons, acting on a string pendulum is given b the formula F = mg sinθ where m is the mass in kilograms of the pendulum's bob, g 9. meters per second per second is the acceleration due to gravit, and θ is angle at which the pendulum is displaced from the perpendicular. What is the value of the restoring force when m = 0. kilogram and θ = 0? If necessar, round the answer to the nearest tenth of a Newton.. N. N.9 N.5 N Page 17

18 5 Find the Eact Values for Integer Multiples of π/ = 0, π/ = 5, and π/ = 0 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the eact value. Do not use a calculator. 1) cos 1π ) sec 19π ) sin ) cot Find the eact value of the epression. Do not use a calculator. 5) tan 7π 5π + tan ) sin 15 - sin ) cos π + tan 5π ) cos 10 tan ) tan 150 cos ) sin 0 sin Page 1

19 Use a Calculator to Approimate the Value of a Trigonometric Function MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use a calculator to find the approimate value of the epression rounded to two decimal places. 1) sin ) cos ) tan ) cos π ) sec π ) csc ) cot π ) cot ) cos ) cos ) tan Solve the problem. 1) If friction is ignored, the time t (in seconds) required for a block to slide down an inclined plane is given b the formula a t = g sinθ cosθ where a is the length (in feet) of the base and g feet per second per second is the acceleration of gravit. How long does it take a block to slide down an inclined plane with base a = 11 when θ = 57? If necessar, round the answer to the nearest tenth of a second. 1. sec 1. sec 0. sec 1.5 sec Page 19

20 1) The force acting on a pendulum to bring it to its perpendicular resting point is called the restoring force. The restoring force F, in Newtons, acting on a string pendulum is given b the formula F = mg sinθ where m is the mass in kilograms of the pendulum's bob, g 9. meters per second per second is the acceleration due to gravit, and θ is angle at which the pendulum is displaced from the perpendicular. What is the value of the restoring force when m = 0.7 kilogram and θ = 7? If necessar, round the answer to the nearest tenth of a Newton..5 N 1.7 N.1 N.7 N SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 1) The strength S of a wooden beam with rectangular cross section is given b the formula S = kd sin θ cos θ where d is the diagonal length, θ the angle illustrated, and k is a constant that varies with the tpe of wood used. Let d = 1 and epress the strength S in terms of the constant k for θ = 5, 50, 55, 0, and 5. Does the strength alwas increase as θ gets larger? 7 Use a Circle of Radius r to Evaluate the Trigonometric Functions MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. A point on the terminal side of an angle θ is given. Find the eact value of the indicated trigonometric function of θ. 1) (-, ) Find sin θ ) (-, ) Find cos θ ) (- 1 5, 1 ) Find cos θ ) (-5, -) Find tan θ ) (, -5) Find cot θ Page 0

21 ) (-, -1) Find csc θ ) (-, -1) Find sec θ Solve the problem. ) If sin θ = 0., find sin (θ + π) ) If sin θ = 1, find csc θ undefined 10) A racetrack curve is banked so that the outside of the curve is slightl elevated or inclined above the inside of the curve. This inclination is called the elevation of the track. The maimum speed on the track in miles per hour is given b r( tan θ) where r is the radius of the track in miles and θ is the elevation in degrees. Find the maimum speed for a racetrack with an elevation of 5 and a radius of 0.5 miles. Round to the nearest mile per hour. 155 mph,05 mph 1,5 mph 1 mph 11) The path of a projectile fired at an inclination θ to the horizontal with an initial speed v o is a parabola. The range R of the projectile, the horizontal distance that the projectile travels, is found b the formula R = v o sin θ where g g =. feet per second per second or g = 9. meters per second per second. Find the range of a projectile fired with an initial velocit of 17 feet per second at an angle of to the horizontal. Round our answer to two decimal places. 91. ft 550. ft ft 91. ft. Properties of the Trigonometric Functions 1 Determine the Domain and the Range of the Trigonometric Functions MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) What is the domain of the cosine function? all real numbers all real numbers from -1 to 1, inclusive all real numbers, ecept odd multiples of π (90 ) all real numbers, ecept integral multiples of π (10 ) ) For what numbers θ is f(θ) = sec θ not defined? odd multiples of π (90 ) integral multiples of π (10 ) odd multiples of π (10 ) all real numbers Page 1

22 ) For what numbers θ is f(θ) = csc θ not defined? integral multiples of π (10 ) odd multiples of π (90 ) odd multiples of π (10 ) all real numbers ) What is the range of the cosine function? all real numbers from -1 to 1, inclusive all real numbers all real numbers greater than or equal to 1 or less than or equal to -1 all real numbers greater than or equal to 0 5) What is the range of the cotangent function? all real numbers all real numbers from -1 to 1, inclusive all real numbers greater than or equal to 1 or less than or equal to -1 all real numbers, ecept integral multiples of π(10) ) What is the range of the cosecant function? all real numbers greater than or equal to 1 or less than or equal to -1 all real numbers from -1 to 1, inclusive all real numbers all real numbers, ecept integral multiples of π(10) Determine the Period of the Trigonometric Functions MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the fact that the trigonometric functions are periodic to find the eact value of the epression. Do not use a calculator. 1) sin ) tan 90 - ) csc ) cot ) cot undefined ) tan undefined Page

23 7) cos π ) sin 17π ) tan 1π ) sec 11π Solve the problem. 11) If sin θ = 0., find the value of sin θ + sin (θ + π) + sin (θ + π)... + π. 0. 1) If tan θ =., find the value of tan θ + tan (θ + π) + tan (θ + π) π 1. undefined 1) If f(θ) = cos θ and f(a) = 1, find the eact value of f(a) + f(a + π) + f(a + π) π 5 1 1) If f(θ) = tan θ and f(a) =, find the eact value of f(a) + f(a + π) + f(a + π) π undefined 15) If f(θ) = cos θ and f(a) = - 1, find the eact value of f(a) + f(a - π) + f(a + π) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 1) If f(θ) = sin θ and f(a) = - 1, find the eact value of f(a) + f(a - π) + f(a - π). 9 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 17) If sin θ = 0.5, find the value of sin θ + sin (θ + π) + sin (θ + π) π Page

24 1) If tan θ = 1.0, find the value of tan θ + tan (θ + π) + tan (θ + π). + π 5 undefined Determine the Signs of the Trigonometric Functions in a Given Quadrant MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Name the quadrant in which the angle θ lies. 1) tan θ > 0, sin θ < 0 I II III IV ) cos θ < 0, csc θ < 0 I II III IV ) sin θ > 0, cos θ < 0 I II III IV ) cot θ < 0, cos θ > 0 I II III IV 5) csc θ > 0, sec θ > 0 I II III IV ) sec θ < 0, tan θ < 0 I II III IV 7) tan θ < 0, sin θ < 0 I II III IV ) cos θ > 0, csc θ < 0 I II III IV 9) cot θ > 0, sin θ < 0 I II III IV 10) sin θ > 0, cos θ > 0 I II III IV Solve the problem. 11) Which of the following trigonometric values are negative? I. sin(-9 ) II. tan(-19 ) III. cos(-07 ) IV. cot II and III III onl II, III, and IV I and III SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 1) Determine the sign of the trigonometric values listed below. (i) sin 50 (ii) tan 0 (iii) cos(-0 ) Page

25 Find the Values of the Trigonometric Functions Using Fundamental Identities MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. In the problem, sin θ and cos θ are given. Find the eact value of the indicated trigonometric function. 5 1) sin θ =, cos θ = Find tan θ ) sin θ = 7, cos θ = Find cot θ ) sin θ = 5, cos θ = Find sec θ ) sin θ = 5, cos θ = Find csc θ Use the properties of the trigonometric functions to find the eact value of the epression. Do not use a calculator. 5) sin 0 + cos ) sec 0 - tan ) cos 5 sec sin 0 ) tan 0 - cos undefined 5 Find Eact Values of the Trig Functions of an Angle Given One of the Functions and the Quadrant of the Angle MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the eact value of the indicated trigonometric function of θ. 1) tan θ = - 10, θ in quadrant II Find cos θ Page 5

26 ) csc θ = - 9, θ in quadrant III Find cot θ ) sec θ =, θ in quadrant IV Find tan θ ) tan θ = 15, 10 < θ < 70 Find cos θ ) cos θ = 7 5, π - 7 < θ < π Find cot θ ) cos θ =, tan θ < 0 Find sin θ ) sin θ = -, tan θ > 0 Find sec θ ) cot θ = - 7, cos θ < 0 Find csc θ ) sin θ = 1, sec θ < 0 Find cos θ and tan θ. cos θ = -, tan θ = - 10 cos θ = -, tan θ = - cos θ = cos θ = -, tan θ =, tan θ = SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 10) sin θ = 1, sec θ < 0 Find cos θ and tan θ. Page

27 Use Even-Odd Properties to Find the Eact Values of the Trigonometric Functions MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the even-odd properties to find the eact value of the epression. Do not use a calculator. 1) sin (-0 ) ) sin (-0 ) ) sec (-0 ) - - ) cot (-0 ) - - 5) cos (-150 ) ) cos - π - - 7) sec - π - - ) cot - π - - 9) csc - π undefined 10) sec (-π) undefined Page 7

28 11) cot - π Solve the problem. 1) If f(θ) = cos θ and f(a) = 1, find the eact value of f(-a) ) If f(θ) = sec θ and f(a) =, find the eact value of f(-a) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 1) Is the function f(θ) = sin θ + cos θ even, odd, or neither? 15) Is the function f(θ) = sin θ + tan θ even, odd, or neither? Page

29 . Graphs of the Sine and Cosine Functions 1 Graph Functions of the Form = A sin(ω) Using Transformations MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use transformations to graph the function. 1) = sin Page 9

30 ) = sin ( + π) Page 0

31 ) = sin Page 1

32 ) = - sin Page

33 5) = sin (π) Page

34 ) = sin Page

35 7) = -5 sin ( + π ) Page 5

36 ) = sin (π - ) Solve the problem. 9) For what numbers, 0 π, does sin = 0? 0, π, π π, π 10) For what numbers, 0 π, does sin = 1? π π, π 0, 1 0, 1, 0, π none Page

37 11) For what numbers, 0 π, does sin = -1? π π, π π none Graph Functions of the Form = A cos(ω) Using Transformations MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use transformations to graph the function. 1) = cos Page 7

38 ) = cos ( - π ) Page

39 ) = cos Page 9

40 ) = - cos Page 0

41 5) = cos ( π ) Page 1

42 ) = cos Page

43 7) = - cos ( - π ) Page

44 ) = cos (π - ) Solve the problem. 9) What is the -intercept of = cos? 1 0 π π 10) For what numbers, 0 π, does cos = 0? π, π 0, π, π 0, 1 0, 1, Page

45 11) For what numbers, 0 π, does cos = 1? 0, π π, π π none 1) For what numbers, 0 π, does cos = -1? π π, π π none Determine the Amplitude and Period of Sinusoidal Functions MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Without graphing the function, determine its amplitude or period as requested. 1) = - sin Find the amplitude. -π π π ) = sin 1 Find the amplitude. π π π ) = - sin Find the amplitude. π π ) = sin Find the period. π π 1 5) = 5 cos 1 Find the amplitude. 5 5π π 5 π ) = cos Find the period. π π 1 7) = - cos 1 Find the period. π - π π ) = 5 cos Find the period. π 5 π 5 π Page 5

46 9) = 5 7 π sin (- ) Find the period. 7 1π 7 5π 5 10) = 9 9 π cos (- ) Find the amplitude. π 9 π SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 11) Wildlife management personnel use predator-pre equations to model the populations of certain predators and their pre in the wild. Suppose the population M of a predator after t months is given b M = sin π t while the population N of its primar pre is given b N = 1, cos π t Find the period for each of these functions. 1) The average dail temperature T of a cit in the United States is approimated b T = 55 - cos π (t -0) 5 where t is in das, 1 t 5, and t = 1 corresponds to Januar 1. Find the period of T. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) The current I, in amperes, flowing through a particular ac (alternating current) circuit at time t seconds is I = 110 sin (0πt) What is the period and amplitude of the current? period = 1 0 second, amplitude = 110 period = 0π seconds, amplitude = 1 0 period = 1 10 second, amplitude = 10 period = π second, amplitude = Page

47 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 1) The current I, in amperes, flowing through an ac (alternating current) circuit at time t, in seconds, is I = 0 sin(50πt) What is the amplitude? What is the period? Graph this function over two periods beginning at t = 0. I t 15) A mass hangs from a spring which oscillates up and down. The position P of the mass at time t is given b P = cos(t) What is the amplitude? What is the period? Graph this function over two periods beginning at t = 0. P t - Page 7

48 1) Before eercising, an athlete measures her air flow and obtains a = 0.5 sin π 5 t where a is measured in liters per second and t is the time in seconds. If a > 0, the athlete is inhaling; if a < 0, the athlete is ehaling. The time to complete one complete inhalation/ehalation sequence is a respirator ccle. What is the amplitude? What is the period? What is the respirator ccle? Graph a over two periods beginning at t = 0. 1 a 5 10 t -1 17) A bo is fling a model airplane while standing on a straight line. The plane, at the end of a twent-five foot wire, flies in circles around the bo. The directed distance of the plane from the straight line is found to be d = 5 cos π t where d is measured in feet and t is the time in seconds. If d > 0, the plane is in front of the bo; if d < 0, the plane is behind him. What is the amplitude? What is the period? Graph d over two periods beginning at t = 0. d 5 1 t -5 Page

49 Graph Sinusoidal Functions Using Ke Points MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Match the given function to its graph. 1) 1) = sin ) = cos ) = -sin ) = -cos A B C D C, A, B, D 1A, D, C, B 1B, D, C, A 1A, B, C, D Page 9

50 ) 1) = sin ) = cos ) = sin ) = cos A B C D B, D, C, A 1A, D, C, B 1A, C, D, B 1A, B, C, D Page 50

51 ) 1) = sin ( - π ) ) = cos ( + π ) ) = sin ( + π ) ) = cos ( - π ) A B C D C, A, B, D 1A, D, C, B 1B, D, C, A 1A, B, C, D Page 51

52 ) 1) = 1 + sin ) = 1 + cos ) = -1 + sin ) = -1 + cos A B C D B, D, C, A 1A, D, C, B 1A, C, D, B 1A, B, C, D Page 5

53 5) 1) = sin ( 1 ) ) = 1 cos ) = 1 sin ) = cos ( 1 ) A B C D B, D, C, A 1A, D, C, B 1A, C, D, B 1A, B, C, D Page 5

54 ) 1) = sin () ) = sin ( 1 ) ) = cos () ) = cos ( 1 ) A B C D A, C, D, B 1C, A, B, D 1C, A, D, B 1D, B, A, C Page 5

55 7) 1) = - sin ( π ) ) = - sin (1 ) ) = - cos ( π ) ) = - cos ( 1 ) B, D, A, C 1A, C, B, D 1C, A, D, B 1A, C, D, B Graph the sinusoidal function. ) = - sin (π) Page 55

56 ) = - cos (π) Page 5

57 ) = 5 sin () Page 57

58 ) = cos (π) Page 5

59 ) = - sin ( 1 ) Page 59

60 ) = 7 cos (- 1 ) Page 0

61 Answer the question. 1) Which one of the equations below matches the graph? = cos 1 = cos = sin 1 = - sin Page 1

62 15) Which one of the equations below matches the graph? = cos = sin 1 = cos 1 = cos 1 1) Which one of the equations below matches the graph? = sin 1 = - sin 1 = cos 1 = cos 17) Which one of the equations below matches the graph? = - sin = - sin 1 = sin 1 = - cos 5 Find an Equation for a Sinusoidal Graph MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Write the equation of a sine function that has the given characteristics. 1) Amplitude: Period: π = sin = sin () = sin () + = sin 1 Page

63 ) Amplitude: Period: = sin 1 π = sin () = sin (π) + = sin π Find an equation for the graph. ) = sin 1 = sin () = sin 1 = sin () ) = 5 cos 1 = 5 cos () = cos 1 = cos (5) 5 5) = -5 sin 1 = -5 sin () = -5 cos 1 = -5 cos () Page

64 ) = sin () = sin 1 = sin 1 = sin () 7) = cos () = cos 1 = cos 1 = cos () ) = - cos () = - cos 1 = - sin 1 = - sin () Page

65 9) = 5 sin (π) = 5 sin π = sin (5π) = sin π 5 10) = cos π = cos (π) = cos (π) = cos π 11) = cos () = cos 1 = sin () = - cos () Page 5

66 1) = -5 sin 1 = 5 cos 1 = -5 sin () = -5 sin 1) = - cos 1 = cos 1 = - sin () = - cos () 1) = 1 cos () = 1 cos 1 = 1 cos 1 = cos () Page

67 .5 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions 1 Graph Functions of the Form = A tan(ω) + B and = A cot(ω) + B MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Match the function to its graph. 1) = tan Page 7

68 ) = tan + π Page

69 ) = tan ( + π) Page 9

70 ) = tan - π Graph the function. 5) = -cot Page 70

71 ) = -tan ( + π) Page 71

72 ) = tan () Page 7

73 ) = - tan Page 7

74 ) = -cot (π) Page 7

75 ) = cot() Page 75

76 ) = - cot () Page 7

77 ) = - tan + π Page 77

78 ) = 1 cot + π Page 7

79 ) = -tan - π Page 79

80 ) = tan - π Page 0

81 ) = -cot - π Page 1

82 Solve the problem. 17) What is the -intercept of = sec? 1 0 π none 1) What is the -intercept of = cot? π 0 1 none 19) For what numbers, -π π, does the graph of = tan have vertical asmptotes.? - π, - π, π, π -π, -π, 0, π, π -, -1, 0, 1, none 0) For what numbers, -π π, does the graph of = csc have vertical asmptotes? -π, -π, 0, π, π - π, - π, π, π -, -1, 0, 1, none Page

83 Graph Functions of the Form = A csc(ω) + B and = A sec(ω) + B MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Graph the function. 1) = csc + π Page

84 ) = sec + π Page

85 ) = -sec Page 5

86 ) = csc () Page

87 5) = sec Page 7

88 ) = csc Page

89 7) = sec () Page 9

90 ) = - csc + π Page 90

91 9) = sec - π Page 91

92 10) = csc π 5 + π Solve the problem. 11) A rotating beacon is located 1 ft from a wall. If the distance from the beacon to the point on the wall where the beacon is aimed is given b a = 1 sec πt, where t is in seconds, find a when t = 0.1 seconds. Round our answer to the nearest hundredth..0 ft.91 ft 1.7 ft -.0 ft Page 9

93 . Phase Shift; Sinusoidal Curve Fitting 1 Graph Sinusoidal Functions of the Form = A sin (ω - φ) + B MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the phase shift of the function. 1) = sin - π π units to the right π units to the left units up units down ) = cos + π π units to the left π units to the right units up units down ) = sin - π π units to the right π units to the left π units up π units down ) = 5 cos ( + π) π units to the left π 5 units to the left 5π units to the right π units to the right 5) = 5 sin 1 - π π units to the right π units to the right π 1 units to the left π 5 units to the left ) = cos 1 + π π units to the left π units to the left π units to the right π units to the right 7) = sin (π - ) π units to the right units to the left units to the right units to the left ) = cos - - π π units to the left π units to the right π units to the left π units to the right Page 9

94 Graph the function. Show at least one period. 9) = sin (π - ) Page 9

95 10) = sin( - π) Page 95

96 11) = cos 5 + π Page 9

97 1) = - sin + π Page 97

98 1) = sin(π + ) Page 9

99 1) = cos - + π Page 99

100 15) = sin(- - π) Page 100

101 1) = - cos( - π) Solve the problem. 17) For the equation = - 1 sin( + π), identif (i) the amplitude, (ii) the phase shift, and (iii) the period. (i) (ii) π (iii) π (i) - 1 (ii) - π (i) 1 (iii) (i) 1 (ii) - π (ii) - π (iii) π (iii) Page 101

102 1) For the equation = - 1 cos( - π), identif (i) the amplitude, (ii) the phase shift, and (iii) the period. (i) 1 (ii) π (iii) π (i) 1 (ii) π (iii) π (i) (ii) π (iii) π (i) (ii) π (iii) π Write the equation of a sine function that has the given characteristics. 19) Amplitude: Period: π Phase Shift: π = sin 1-1 π = sin + 1 π = sin π = sin + π 0) Amplitude: Period: 5π Phase Shift: - π 5 = sin π = sin 5-5 π = sin 5-5 π = sin 5 - π 5 1) Amplitude: Period: π Phase Shift: - = sin ( + 1) = sin 1-1 = sin ( - ) = sin ( + ) ) Amplitude: Period: π Phase Shift: = sin ( - ) = sin 1 - = sin + = sin ( + ) Page 10

103 Build Sinusoidal Models from Data MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) An eperiment in a wind tunnel generates cclic waves. The following data is collected for 0 seconds: Time (in seconds) Wind speed (in feet per second) Let V represent the wind speed (velocit) in feet per second and let t represent the time in seconds. Write a sine equation that describes the wave. π V = 1 sin 0 t - π + 7 V = 5 sin(0t - 0) + 1 V = 5 sin π 0 t - π + 1 V = sin (0t - 0) + 1 ) A town's average monthl temperature data is represented in the table below: Month, Januar, 1 Februar, March, April, Ma, 5 June, Jul, 7 August, September, 9 October, 10 November, 11 December, 1 Average Monthl Temperature, F Find a sinusoidal function of the form = A sin (ω - φ) + B that fits the data. = 7.5 sin π - π =. sin π - π + 9. = 5.95 sin π - π = 9. sin π - π +. Page 10

104 ) The number of hours of sunlight in a da can be modeled b a sinusoidal function. In the northern hemisphere, the longest da of the ear occurs at the summer solstice and the shortest da occurs at the winter solstice. In 000, these dates were June (the 17nd da of the ear) and December 1 (the 5th da of the ear), respectivel. A town eperiences 11.1 hours of sunlight at the summer solstice and.1 hours of sunlight at the winter solstice. Find a sinusoidal function = A sin (ω - φ) + B that fits the data, where is the da of the ear. (Note: There are das in the ear 000.) = 1.55 sin = 1.55 sin π 1-11π π 1 - π = 11.1 sin π - π = 11.1 sin 17π 5 - π ) The data below represent the average monthl cost of natural gas in an Oregon home. Month Aug Sep Oct Nov Dec Jan Cost Month Feb Mar Apr Ma Jun Jul Cost Above is the graph of 5.05 sin superimposed over a scatter diagram of the data. Find the sinusoidal function of the form = A sin (ω - φ) + B which best fits the data. = 5.05 sin π - π +.5 = 5.05 sin π t = 5.05 sin π - π = 5.05 sin π - π Page 10

105 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 5) The data below represent the average monthl cost of natural gas in an Oregon home. Month Aug Sep Oct Nov Dec Jan Cost Month Feb Mar Apr Ma Jun Jul Cost Above is the graph of 7.5 sin. Make a scatter diagram of the data. Find the sinusoidal function of the form = A sin (ω - φ) + B which fits the data. Page 105

106 ) The following data represents the normal monthl precipitation for a certain cit in California. Month, Januar, 1 Februar, March, April, Ma, 5 June, Jul, 7 August, September, 9 October, 10 November, 11 December, 1 Normal Monthl Precipitation, inches Draw a scatter diagram of the data for one period. Find a sinusoidal function of the form = A sin (ω - φ) + B that fits the data. Draw the sinusoidal function on the scatter diagram. Use a graphing utilit to find the sinusoidal function of best fit. Draw the sinusoidal function of best fit on the scatter diagram. Page 10

107 7) The following data represents the normal monthl precipitation for a certain cit in Arkansas. Month, Januar, 1 Februar, March, April, Ma, 5 June, Jul, 7 August, September, 9 October, 10 November, 11 December, 1 Normal Monthl Precipitation, inches Draw a scatter diagram of the data for one period. Find the sinusoidal function of the form = A sin (ω - φ) + B that fits the data. Draw the sinusoidal function on the scatter diagram. Use a graphing utilit to find the sinusoidal function of best fit. Draw the sinusoidal function of best fit on the scatter diagram. Page 107

108 ) The following data represents the average monthl minimum temperature for a certain cit in California. Month, Januar, 1 Februar, March, April, Ma, 5 June, Jul, 7 August, September, 9 October, 10 November, 11 December, 1 Average Monthl Minimum Temperature, F Draw a scatter diagram of the data for one period. Find a sinusoidal function of the form = A sin (ω - φ) + B that fits the data. Draw the sinusoidal function on the scatter diagram. Use a graphing utilit to find the sinusoidal function of best fit. Draw the sinusoidal function of best fit on the scatter diagram. Page 10

109 9) The following data represents the average percent of possible sunshine for a certain cit in Indiana. Month, Januar, 1 Februar, March, April, Ma, 5 June, Jul, 7 August, September, 9 October, 10 November, 11 December, 1 Average Percent of Possible Sunshine Draw a scatter diagram of the data for one period. Find the sinusoidal function of the form = A sin (ω - φ) + B that fits the data. Draw the sinusoidal function on the scatter diagram. Use a graphing utilit to find the sinusoidal function of best fit. Draw the sinusoidal function of best fit on the scatter diagram. Page 109

110 Ch. Trigonometric Functions Answer Ke.1 Angles and Their Measure 1 Convert between Decimals and Degrees, Minutes, Seconds Measures for Angles 1) A ) A ) A ) A 5) A ) A 7) A ) A 9) A 10) A 11) A 1) A 1) A 1) A 15) A 1) A 17) A 1) A Find the Length of an Arc of a Circle 1) A ) A ) A ) A 5) A ) A 7) A ) A 9) C 10) A 11) A 1) 79 mi 1) A 1) A Convert from Degrees to Radians and from Radians to Degrees 1) A ) A ) A ) A 5) A ) A 7) A ) A 9) A 10) A 11) A 1) A 1) A 1) A 15) A 1) A 17) A 1) A Page 110

111 19) A Find the Area of a Sector of a Circle 1) A ) A ) A ) A 5) A ) A 7) A ) A 9) A 10) A 11) A 1) A 1) A 1) A 15) A 1) A 5 Find the Linear Speed of an Object Traveling in Circular Motion 1) A ) A ) A ) A 5) A ) A 7) A ) A 9) A 10) A 11). 10 kmp; kmp;.9 10 kmp; kmp; Io 1) A 1) A. Trigonometric Functions: Unit Circle Approach 1 Find the Eact Values of the Trigonometric Functions Using a Point on the Unit Circle 1) A ) A ) A ) A 5) A ) A 7) A ) A 9) A 10) A Find the Eact Values of the Trigonometric Functions of Quadrantal Angles 1) A ) A ) A ) D 5) A ) A 7) A ) A 9) A 10) A 11) A Page 111

112 Find the Eact Values of the Trigonometric Functions of π/ = 5 1) A ) A ) A ) A 5) A ) A 7) A ) A Find the Eact Values of the Trigonometric Functions of π/ = 0 and π/ = 0 1) A ) A ) A ) A 5) A ) A 7) A ) A 9) A 10) A 11) A 1) A 1) A 1) A 15) A 1) A 17) A 1) A 19) A 0) A 5 Find the Eact Values for Integer Multiples of π/ = 0, π/ = 5, and π/ = 0 1) A ) A ) A ) A 5) A ) A 7) A ) A 9) A 10) A Use a Calculator to Approimate the Value of a Trigonometric Function 1) A ) A ) A ) A 5) A ) A 7) A ) A 9) A 10) A 11) A 1) A 1) A 1) 0.5k; 0.77k; 0.5k; 0.75k and 0.7k; No, it reaches a maimum near 55. Page 11

113 7 Use a Circle of Radius r to Evaluate the Trigonometric Functions 1) A ) A ) A ) A 5) A ) A 7) A ) A 9) A 10) A 11) A. Properties of the Trigonometric Functions 1 Determine the Domain and the Range of the Trigonometric Functions 1) A ) A ) A ) A 5) A ) A Determine the Period of the Trigonometric Functions 1) A ) A ) A ) A 5) D ) A 7) A ) A 9) A 10) A 11) A 1) A 1) A 1) A 15) A 1) ) A 1) A Determine the Signs of the Trigonometric Functions in a Given Quadrant 1) C ) C ) B ) D 5) A ) B 7) D ) D 9) C 10) A 11) A 1) (i) negative (ii) negative (iii) positive Find the Values of the Trigonometric Functions Using Fundamental Identities 1) A Page 11

114 ) A ) A ) A 5) A ) A 7) A ) A 5 Find Eact Values of the Trig Functions of an Angle Given One of the Functions and the Quadrant of the Angle 1) A ) A ) A ) A 5) A ) A 7) A ) A 9) C 5 10) cos θ = -, tan θ = Use Even-Odd Properties to Find the Eact Values of the Trigonometric Functions 1) A ) A ) A ) A 5) A ) A 7) A ) A 9) A 10) A 11) A 1) A 1) A 1) neither 15) odd. Graphs of the Sine and Cosine Functions 1 Graph Functions of the Form = A sin(ω) Using Transformations 1) A ) A ) A ) A 5) A ) A 7) A ) A 9) A 10) A 11) A Graph Functions of the Form = A cos(ω) Using Transformations 1) A ) A ) A ) A 5) A ) A 7) A ) A Page 11

115 9) A 10) A 11) A 1) A Determine the Amplitude and Period of Sinusoidal Functions 1) A ) A ) A ) A 5) A ) A 7) A ) A 9) A 10) A 11) 1, 1 1) 5 das 1) A 1) amplitude = 0, period = 1 5 I I = 0sin(50πt) t -0 15) amplitude =, period = π P P = cos(t) t - Page 115

116 1) amplitude = 0.5, period = 5, respirator ccle = 5 seconds a a = 0.5sin π 5 t t ) amplitude = 5, period = / 5 d d = 5 cos π t 1 t -5 Graph Sinusoidal Functions Using Ke Points 1) A ) A ) A ) A 5) A ) A 7) A ) A 9) A 10) A 11) A 1) A 1) A 1) A 15) A 1) A 17) A Page 11

117 5 Find an Equation for a Sinusoidal Graph 1) A ) A ) A ) A 5) A ) A 7) A ) A 9) A 10) A 11) A 1) A 1) A 1) A.5 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions 1 Graph Functions of the Form = A tan(ω) + B and = A cot(ω) + B 1) A ) A ) A ) A 5) A ) A 7) A ) A 9) A 10) A 11) A 1) A 1) A 1) A 15) A 1) A 17) A 1) D 19) A 0) A Graph Functions of the Form = A csc(ω) + B and = A sec(ω) + B 1) A ) A ) A ) A 5) A ) A 7) A ) A 9) A 10) A 11) A. Phase Shift; Sinusoidal Curve Fitting 1 Graph Sinusoidal Functions of the Form = A sin (ω - φ) + B 1) A ) A ) A ) A 5) A ) A Page 117

118 7) A ) A 9) A 10) A 11) A 1) A 1) A 1) A 15) A 1) A 17) B 1) A 19) A 0) A 1) A ) A Build Sinusoidal Models from Data 1) A ) A ) A ) A 5) = 7.5 sin ( π - π ) +.5 ) =.1 sin ( ) +.1 7) =.17 sin (0.9-1.) +.0 Page 11

119 ) =. sin ( ) ) = sin ( ) + 0. Page 119