The Circular Functions and Their Graphs

Transcription

1 LIALMC_78.QXP // : AM Page 5 The Circular Functions and Their Graphs In August, the planet Mars passed closer to Earth than it had in almost, ears. Like Earth, Mars rotates on its ais and thus has das and nights. The photos here were taken b the Hubble telescope and show two nearl opposite sides of Mars. (Source: In Eercise 9 of Section., we eamine the length of a Martian da. Phenomena such as rotation of a planet on its ais, high and low tides, and changing of the seasons of the ear are modeled b periodic functions. In this chapter, we see how the trigonometric functions of the previous chapter, introduced there in the contet of ratios of the sides of a right triangle, can also be viewed from the perspective of motion around a circle.. Radian Measure. The Unit Circle and Circular Functions. Graphs of the Sine and Cosine Functions. Graphs of the Other Circular Functions Summar Eercises on Graphing Circular Functions.5 Harmonic Motion 5

2 LIALMC_78.QXP // : AM Page 5 5 CHAPTER The Circular Functions and Their Graphs. Radian Measure Radian Measure Converting Between Degrees and Radians Arc Length of a Circle Area of a Sector of a Circle r Figure r = radian TEACHING TIP Students ma resist the idea of using radian measure. Eplain that radians have wide applications in angular motion problems (seen in Section.), as well as engineering and science. Radian Measure In most applications of trigonometr, angles are measured in degrees. In more advanced work in mathematics, radian measure of angles is preferred. Radian measure allows us to treat the trigonometric functions as functions with domains of real numbers, rather than angles. Figure shows an angle in standard position along with a circle of radius r. The verte of is at the center of the circle. Because angle intercepts an arc on the circle equal in length to the radius of the circle, we sa that angle has a measure of radian. Radian An angle with its verte at the center of a circle that intercepts an arc on the circle equal in length to the radius of the circle has a measure of radian. It follows that an angle of measure radians intercepts an arc equal in length to twice the radius of the circle, an angle of measure radian intercepts an arc equal in length to half the radius of the circle, and so on. In general, if is a central angle of a circle of radius r and intercepts an arc of length s, then the s radian measure of is r. Converting Between Degrees and Radians The circumference of a circle the distance around the circle is given b C r, where r is the radius of the circle. The formula C r shows that the radius can be laid off times around a circle. Therefore, an angle of, which corresponds to a complete circle, intercepts an arc equal in length to times the radius of the circle. Thus, an angle of has a measure of radians: An angle of 8 is half the size of an angle of, so an angle of 8 has half the radian measure of an angle of. radians. 8 radians radians Degreeradian relationship TEACHING TIP Emphasize the difference between an angle of a degrees (written a ) and an angle of a radians (written a r, a r, or a). Caution students that although radian measures are often given eactl in terms of, the can also be approimated using decimals. We can use the relationship 8 radians to develop a method for converting between degrees and radians as follows. 8 radians 8 radian Divide b 8. or radian 8 Divide b. Converting Between Degrees and Radians. Multipl a degree measure b radian and simplif to convert to radians Multipl a radian measure b and simplif to convert to degrees.

3 LIALMC_78.QXP // : AM Page 55. Radian Measure 55 EXAMPLE Converting Degrees to Radians Convert each degree measure to radians. (a) 5 (b) 9.8 Some calculators (in radian mode) have the capabilit to convert directl between decimal degrees and radians. This screen shows the conversions for Eample. Note that when eact values involving are required, such as in part (a), calculator approimations are not accept- able. Solution (a) 5 5 Multipl b 8 radian. 8 radian radian (b) Nearest thousandth 8 radian. radians Now tr Eercises and. EXAMPLE Converting Radians to Degrees Convert each radian measure to degrees. (a) 9 (b).5 (Give the answer in decimal degrees.) Solution This screen shows how a calculator converts the radian measures in Eample to degree measures. (a) Multipl b. 9 (b) Use a calculator. Now tr Eercises and. If no unit of angle measure is specified, then radian measure is understood. radians CAUTION Figure shows angles measuring radians and. Be careful not to confuse them. The following table and Figure on the net page give some equivalent angles measured in degrees and radians. Keep in mind that 8 radians. Degrees Radians Degrees Radians Eact Approimate Eact Approimate degrees Figure

4 LIALMC_78.QXP // : AM Page 5 5 CHAPTER The Circular Functions and Their Graphs Looking Ahead to Calculus In calculus, radian measure is much easier to work with than degree measure. If is measured in radians, then the derivative of f sin is f cos. = 5 = 5 = 5 8 = 9 = = 5 = = = However, if is measured in degrees, then the derivative of f sin is f cos. 8 = 7 5 = 5 = 7 = Figure 7 = 5 = = 5 We use radian measure to simplif certain formulas, two of which follow. Each would be more complicated if epressed in degrees. Arc Length of a Circle We use the first formula to find the length of an arc of a circle. This formula is derived from the fact (proven in geometr) that the length of an arc is proportional to the measure of its central angle. In Figure, angle QOP has measure radian and intercepts an arc of length r on the circle. Angle ROT has measure radians and intercepts an arc of length s on the circle. Since the lengths of the arcs are proportional to the measures of their central angles, s r. Multipling both sides b r gives the following result. T Q s r radians radian R r O r P Figure TEACHING TIP To help students appreciate radian measure, show them that in degrees the formula for arc length is s r 8. (See Eercise 5.) r Arc Length The length s of the arc intercepted on a circle of radius r b a central angle of measure radians is given b the product of the radius and the radian measure of the angle, or s r, in radians. CAUTION When appling the formula s r, the value of must be epressed in radians.

5 LIALMC_78.QXP // : AM Page 57. Radian Measure 57 EXAMPLE Finding Arc Length Using s r A circle has radius 8. cm. Find the length of the arc intercepted b a central angle having each of the following measures. (a) (b) 8 radians 8 s r = 8. cm Solution (a) As shown in the figure, r 8. cm and s r s 8. 8 cm s 5. cm. cm 8 Arc length formula Substitute for r and. (b) The formula s r requires that be measured in radians. First, convert to radians b multipling b 8 radian. Convert from degrees to radians. 8 5 radians The length s is given b s r cm Now tr Eercises 9 and 5. Reno s Los Angeles Equator km Figure 5 EXAMPLE Using Latitudes to Find the Distance Between Two Cities Reno, Nevada, is approimatel due north of Los Angeles. The latitude of Reno is N, while that of Los Angeles is N. (The N in N means north of the equator.) The radius of Earth is km. Find the north-south distance between the two cities. Solution Latitude gives the measure of a central angle with verte at Earth s center whose initial side goes through the equator and whose terminal side goes through the given location. As shown in Figure 5, the central angle between Reno and Los Angeles is. The distance between the two cities can be found b the formula s r, after is first converted to radians. 8 radian The distance between the two cities is s r 7 km. Let r and. Now tr Eercise 55.

6 LIALMC_78.QXP // : AM Page CHAPTER The Circular Functions and Their Graphs ft Figure EXAMPLE 5 Finding a Length Using s r A rope is being wound around a drum with radius.875 ft. (See Figure.) How much rope will be wound around the drum if the drum is rotated through an angle of 9.7? Solution The length of rope wound around the drum is the arc length for a circle of radius.875 ft and a central angle of 9.7. Use the formula s r, with the angle converted to radian measure. The length of the rope wound around the drum is approimatel s r ft. 8 Now tr Eercise (a)..5cm Figure 7.8 cm EXAMPLE Finding an Angle Measure Using s r Two gears are adjusted so that the smaller gear drives the larger one, as shown in Figure 7. If the smaller gear rotates through 5, through how man degrees will the larger gear rotate? Solution First find the radian measure of the angle, and then find the arc length on the smaller gear that determines the motion of the larger gear. Since 5 5 radians, for the smaller gear, s r cm. An arc with this length on the larger gear corresponds to an angle measure, in radians, where Substitute for s and.8 for r ; multipl 5 b to solve for. Converting back to degrees shows that the larger gear rotates through Convert 9 to degrees s r Now tr Eercise. r Figure 8 The shaded region is a sector of the circle. Area of a Sector of a Circle A sector of a circle is the portion of the interior of a circle intercepted b a central angle. Think of it as a piece of pie. See Figure 8. A complete circle can be thought of as an angle with measure radians. If a central angle for a sector has measure radians, then the sector makes up the fraction of a complete circle. The area of a complete circle with radius r is A r. Therefore, area of the sector r r, in radians.

7 LIALMC_78.QXP // : AM Page 59. Radian Measure 59 This discussion is summarized as follows. Area of a Sector The area of a sector of a circle of radius r and central angle is given b A r, in radians. CAUTION As in the formula for arc length, the value of must be in radians when using this formula for the area of a sector. m Figure 9 5 EXAMPLE 7 Finding the Area of a Sector-Shaped Field Find the area of the sector-shaped field shown in Figure 9. Solution First, convert 5 to radians radian Now use the formula to find the area of a sector of a circle with radius r. A r,5 m Now tr Eercise 77.. Eercises Convert each degree measure to radians. Leave answers as multiples of. See Eample (a) Convert each degree measure to radians. See Eample (b) Concept Check In Eercises 58, each angle is an integer when measured in radians. Give the radian measure of the angle. 5.. θ θ

8 LIALMC_78.QXP // : AM Page 5 5 CHAPTER The Circular Functions and Their Graphs θ 9. In our own words, eplain how to convert (a) degree measure to radian measure; (b) radian measure to degree measure.. Eplain the difference between degree measure and radian measure. θ Convert each radian measure to degrees. See Eample (a) Convert each radian measure to degrees. Give answers using decimal degrees to the nearest tenth. See Eample (b) Relating Concepts For individual or collaborative investigation (Eercises 5) In anticipation of the material in the net section, we show how to find the trigonometric function values of radian-measured angles. Suppose we want to find sin 5. One wa to do this is to convert radians to 5, and then use the methods of Chapter 5 to evaluate:. (Section 5.) Sine is positive Reference angle in quadrant II. for 5 Use this technique to find each function value. Give eact values. 5. tan. csc 7. cot 8. cos 9. sec. sin 7. cos. tan 9 5 sin 5 sin 5 sin

9 LIALMC_78.QXP // : AM Page 5. Radian Measure cm 5..8 cm m 5. 9 cm 5. The length is r doubled. 5. s km 5. 5 km km km Concept Check angle. Find the eact length of each arc intercepted b the given central.. Concept Check Find the radius of each circle. 5.. Concept Check Find the measure of each central angle (in radians) Unless otherwise directed, give calculator approimations in our answers in the rest of this eercise set. Find the length of each arc intercepted b a central angle Eample. in a circle of radius r. See 9. r. cm, radians 5. r.89 cm, radians 5. r.8 m, 5. r 7.9 cm, 5. Concept Check If the radius of a circle is doubled, how is the length of the arc intercepted b a fied central angle changed? 5. Concept Check Radian measure simplifies man formulas, such as the formula for arc length, s r. Give the corresponding formula when is measured in degrees instead of radians. Distance Between Cities Find the distance in kilometers between each pair of cities, assuming the lie on the same north-south line. See Eample. 55. Panama Cit, Panama, 9 N, and Pittsburgh, Pennslvania, N 5. Farmersville, California, N, and Penticton, British Columbia, 9 N 57. New York Cit, New York, N, and Lima, Peru, S 58. Halifa, Nova Scotia, 5 N, and Buenos Aires, Argentina, S 5

10 LIALMC_78.QXP // : AM Page 5 5 CHAPTER The Circular Functions and Their Graphs 59. N. N. (a). in. (b) cm cm 5. in.. (a) 9, rotations 59. Latitude of Madison Madison, South Dakota, and Dallas, Teas, are km apart and lie on the same north-south line. The latitude of Dallas is N. What is the latitude of Madison?. Latitude of Toronto Charleston, South Carolina, and Toronto, Canada, are km apart and lie on the same north-south line. The latitude of Charleston is N. What is the latitude of Toronto? Work each problem. See Eamples 5 and.. Pulle Raising a Weight (a) How man inches will the weight in the figure rise if the pulle is rotated through an angle of 7 5? (b) Through what angle, to the nearest minute, must the pulle be rotated to raise the weight in.? 9.7 in.. Pulle Raising a Weight Find the radius of the pulle in the figure if a rotation of 5. raises the weight. cm. r. Rotating Wheels The rotation of the smaller wheel in the figure causes the larger wheel to rotate. Through how man degrees will the larger wheel rotate if the smaller one rotates through.? 5. cm 8. cm. Rotating Wheels Find the radius of the larger wheel in the figure if the smaller wheel rotates 8. when the larger wheel rotates cm r 5. Biccle Chain Drive The figure shows the chain drive of a biccle. How far will the biccle move if the pedals are rotated through 8? Assume the radius of the biccle wheel is. in..8 in..7 in.. Pickup Truck Speedometer The speedometer of Terr s small pickup truck is designed to be accurate with tires of radius in. (a) Find the number of rotations of a tire in hr if the truck is driven at 55 mph.

11 LIALMC_78.QXP // : AM Page 5. Radian Measure 5. (b).9 mi; es 7.. km ft m km ft 7.,.9 d 77.. cm m mi 8. 9,85. km m (b) Suppose that oversize tires of radius in. are placed on the truck. If the truck is now driven for hr with the speedometer reading 55 mph, how far has the truck gone? If the speed limit is 55 mph, does Terr deserve a speeding ticket? If a central angle is ver small, there is little difference in length between an arc and the inscribed chord. See the figure. Approimate each of the following lengths b finding the necessar arc length. (Note: When a central angle intercepts an arc, the arc is said to subtend the angle.) Arc length length of inscribed chord Inscribed chord 7. Length of a Train A railroad track in the desert is.5 km awa. A train on the track subtends (horizontall) an angle of. Find the length of the train. 8. Distance to a Boat The mast of Brent Simon s boat is ft high. If it subtends an angle of, how far awa is it? Arc Concept Check Find the area of each sector Concept Check Find the measure (in radians) of each central angle. The number inside the sector is the area sq units 8 sq units Find the area of a sector of a circle having radius r and central angle. See Eample r 9. m, radians 7. r 59.8 km, radians 75. r. ft, radians 7. r 9. d, radians r.7 cm, 78. r 8. m, r. mi, 8. r 9. km, Work each problem. 8. Find the measure (in radians) of a central angle of a sector of area in. in a circle of radius. in. 8. Find the radius of a circle in which a central angle of radian determines a sector of area m.

12 LIALMC_78.QXP // : AM Page 5 5 CHAPTER The Circular Functions and Their Graphs 8. (a) 8. Measures of a Structure The figure shows Medicine Wheel, a Native American ; (b) 8 ft 7 structure in northern Woming. This circular structure is perhaps 5 r old. There are 7 aboriginal spokes in the wheel, all equall spaced. (c) ft (d) approimatel 7 ft in. 85. (a) ft (b) ft (c) ft 8. (a) 55 m (b) 8 m d mi (a) Find the measure of each central angle in degrees and in radians. (b) If the radius of the wheel is 7 ft, find the circumference. (c) Find the length of each arc intercepted b consecutive pairs of spokes. (d) Find the area of each sector formed b consecutive spokes. 8. Area Cleaned b a Windshield Wiper The Ford 95 Model A, built from 98 to 9, had a single windshield wiper on the driver s side. The total arm and blade was in. long and rotated back and forth through an angle of 95. The shaded region in the figure is the portion of the windshield cleaned b the 7-in. wiper blade. What is the area of the region cleaned? 85. Circular Railroad Curves In the United States, circular railroad curves are designated b the degree of curvature, the central angle subtended b a chord of ft. Suppose a portion of track has curvature. (Source: Ha, W., Railroad Engineering, John Wile & Sons, 98.) (a) What is the radius of the curve? (b) What is the length of the arc determined b the -ft chord? (c) What is the area of the portion of the circle bounded b the arc and the -ft chord? 8. Land Required for a Solar-Power Plant A -megawatt solar-power plant requires approimatel 95, m of land area in order to collect the required amount of energ from sunlight. (a) If this land area is circular, what is its radius? (b) If this land area is a 5 sector of a circle, what is its radius? 87. Area of a Lot A frequent problem in surveing cit lots and rural lands adjacent to curves of highwas and railwas is that of finding the area when one or more of the boundar lines is the arc of a circle. Find d the area of the lot shown in the figure. (Source: Anderson, J. and E. Michael, Introduction to Surveing, McGraw-Hill, 985.) d 88. Nautical Miles Nautical miles are used b ships and airplanes. The are different from statute miles, which equal 58 ft. A nautical mile is defined to be the arc length along the equator intercepted b a central angle AOB of min, as illustrated in the figure. If the equatorial radius of Earth is 9 mi, use the arc length formula to approimate the number of statute miles in nautical mile. Round our answer to two decimal places. B Nautical mile in. A O 7in. Not to scale

13 LIALMC_78.QXP // : AM Page 55. The Unit Circle and Circular Functions radius: 97 mi; circumference:,8 mi 9. approimatel 5 mi 9. The area is quadrupled. r 9. A 89. Circumference of Earth The first accurate Sun's ras at noon estimate of the distance around Earth was done b the Greek astronomer Eratosthenes 9 mi 7 ' (795 B.C.), who noted that the noontime Shadow position of the sun at the summer solstice differed b 7 from the cit of Sene to the Aleandria Sene cit of Aleandria. (See the figure.) The distance between these two cities is 9 mi. Use 7 ' the arc length formula to estimate the radius of Earth. Then find the circumference of Earth. (Source: Zeilik, M., Introductor Astronom and Astrophsics, Third Edition, Saunders College Publishers, 99.) 9. Diameter of the Moon The distance to the moon is approimatel 8,9 mi. Use the arc length formula to estimate the diameter d of the moon if angle in the figure is measured to be.57. d Not to scale 9. Concept Check If the radius of a circle is doubled and the central angle of a sector is unchanged, how is the area of the sector changed? 9. Concept Check Give the corresponding formula for the area of a sector when the angle is measured in degrees.. The Unit Circle and Circular Functions Circular Functions Finding Values of Circular Functions Determining a Number with a Given Circular Function Value Angular and Linear Speed = cos s = sin s (, ) Arc of length s (, ) In Section 5., we defined the si trigonometric functions in such a wa that the domain of each function was a set of angles in standard position. These angles can be measured in degrees or in radians. In advanced courses, such as calculus, it is necessar to modif the trigonometric functions so that their domains consist of real numbers rather than angles. We do this b using the relationship between an angle and an arc of length s on a circle. (, ) (, ) (, ) Unit circle + = Figure Circular Functions In Figure, we start at the point, and measure an arc of length s along the circle. If s, then the arc is measured in a counterclockwise direction, and if s, then the direction is clockwise. (If s, then no arc is measured.) Let the endpoint of this arc be at the point,. The circle in Figure is a unit circle it has center at the origin and radius unit (hence the name unit circle). Recall from algebra that the equation of this circle is. (Section.)

14 LIALMC_78.QXP // : AM Page 5 5 CHAPTER The Circular Functions and Their Graphs We saw in the previous section that the radian measure of is related to the arc length s. In fact, for measured in radians, we know that s r. Here, r, so s, which is measured in linear units such as inches or centimeters, is numericall equal to, measured in radians. Thus, the trigonometric functions of angle in radians found b choosing a point, on the unit circle can be rewritten as functions of the arc length s, a real number. When interpreted this wa, the are called circular functions. Looking Ahead to Calculus If ou plan to stud calculus, ou must become ver familiar with radian measure. In calculus, the trigonometric or circular functions are alwas understood to have real number domains. Circular Functions sin s csc s ( ) cos s sec s ( ) tan s cot s ( ) ( ) Since represents the cosine of s and represents the sine of s, and because of the discussion in Section. on converting between degrees and radians, we can summarize a great deal of information in a concise manner, as seen in Figure.* TEACHING TIP Draw a unit circle on the chalkboard and trace a point slowl counterclockwise, starting at,. Students should see that the -value (sine function) starts at, increases to (at radians), decreases to (at radians), decreases again to (at radians), and returns to after one full revolution ( radians). This tpe of analsis will help illustrate the range associated with the sine and cosine functions. (, ) (, (, ) ) 9 ( 5, (, ) ) 5 (, 5 (, ) 5 ) (, ) 8 (, ) 7 (, 5 5 (, ) ) (, (, ) ) 7 (, ) (, ) (, ) Unit circle + = Figure NOTE Since sin s and cos s, we can replace and in the equation and obtain the Pthagorean identit cos s sin s. The ordered pair, represents a point on the unit circle, and therefore and, so cos s and sin s. *The authors thank Professor Marvel Townsend of the Universit of Florida for her suggestion to include this figure.

15 LIALMC_78.QXP // : AM Page 57. The Unit Circle and Circular Functions 57 For an value of s, both sin s and cos s eist, so the domain of these functions is the set of all real numbers. For tan s, defined as, must not equal. The onl wa can equal is when the arc length s is,,,, and so on. To avoid a denominator, the domain of the tangent function must be restricted to those values of s satisfing s n n an integer., The definition of secant also has in the denominator, so the domain of secant is the same as the domain of tangent. Both cotangent and cosecant are defined with a denominator of. To guarantee that, the domain of these functions must be the set of all values of s satisfing In summar, the domains of the circular functions are as follows. s n, n an integer. Domains of the Circular Functions Assume that n is an integer and s is a real number. Sine and Cosine Functions: (, ) Tangent and Secant Functions: Cotangent and Cosecant Functions: s s (n ) {s s n} (, ) (cos s, sin s) = (, ) (, ) (, ) + = Figure s = (, ) Finding Values of Circular Functions The circular functions (functions of real numbers) are closel related to the trigonometric functions of angles measured in radians. To see this, let us assume that angle is in standard position, superimposed on the unit circle, as shown in Figure. Suppose further that is the radian measure of this angle. Using the arc length formula s r with r, we have s. Thus, the length of the intercepted arc is the real number that corresponds to the radian measure of. Using the definitions of the trigonometric functions, we have sin r sin s, and cos r cos s, and so on. As shown here, the trigonometric functions and the circular functions lead to the same function values, provided we think of the angles as being in radian measure. This leads to the following important result concerning evaluation of circular functions. Evaluating a Circular Function Circular function values of real numbers are obtained in the same manner as trigonometric function values of angles measured in radians. This applies both to methods of finding eact values (such as reference angle analsis) and to calculator approimations. Calculators must be in radian mode when finding circular function values.

16 LIALMC_78.QXP // : AM Page CHAPTER The Circular Functions and Their Graphs EXAMPLE Finding Eact Circular Function Values (, ) Find the eact values of and Solution Evaluating a circular function at the real number is equivalent to evaluating it at radians. An angle of radians intersects the unit circle at the point,, as shown in Figure. Since sin, cos, tan. (, ) = (, ) sin s, cos s, and tan s, it follows that (, ) Figure sin, cos, and tan is undefined. Now tr Eercise. TEACHING TIP Point out the importance of being able to sketch the unit circle as in Figure. This will enable students to more easil determine function values of multiples of. EXAMPLE Finding Eact Circular Function Values 7 (a) Use Figure to find the eact values of cos and sin 7. (b) Use Figure to find the eact value of tan 5. (c) Use reference angles and degree/radian conversion to find the eact value of cos Solution. (a) In Figure, we see that the terminal side of circle at,. Thus, radians intersects the unit (b) Angles of radians and radians are coterminal. Their terminal sides intersect the unit circle at so tan,, 5 tan. (c) An angle of radians corresponds to an angle of. In standard position, lies in quadrant II with a reference angle of, so 5 cos 7 and sin 7. Cosine is negative in quadrant II. cos cos cos. 7 Reference angle (Section 5.) Now tr Eercises 7, 7, and. NOTE Eamples and illustrate that there are several methods of finding eact circular function values.

17 LIALMC_78.QXP // : AM Page 59. The Unit Circle and Circular Functions 59 EXAMPLE Approimating Circular Function Values Find a calculator approimation to four decimal places for each circular function value. (a) cos.85 (b) cos.59 (c) cot.9 (d) sec.9 Radian mode This is how a calculator displas the result of Eample (a), correct to four decimal digits. TEACHING TIP Remind students not to use the sin, cos, and tan, kes when evaluating reciprocal functions with a calculator. Solution (a) With a calculator in radian mode, we find cos (b) cos Use a calculator in radian mode. (c) As before, to find cotangent, secant, and cosecant function values, we must use the appropriate reciprocal functions. To find cot.9, first find tan.9 and then find the reciprocal. (d) sec.9 cot.9 tan.9.55 cos.9. Now tr Eercises, 9, and. CAUTION A common error in trigonometr is using calculators in degree mode when radian mode should be used. Remember, if ou are finding a circular function value of a real number, the calculator must be in radian mode. Determining a Number with a Given Circular Function Value Recall from Section 5. how we used a calculator to determine an angle measure, given a trigonometric function value of the angle. The calculator is set to show four decimal digits in the answer. Figure This screen supports the result of Eample (b). The calculator is in radian mode. Figure 5 EXAMPLE Finding a Number Given Its Circular Function Value (a) Approimate the value of s in the interval,, if cos s.985. (b) Find the eact value of s in the interval,, if tan s. Solution (a) Since we are given a cosine value and want to determine the real number in, having this cosine value, we use the inverse cosine function of a calculator. With the calculator in radian mode, we find (Section 5.) See Figure. (Refer to our owner s manual to determine how to evaluate the sin, cos, and tan functions with our calculator.) (b) Recall that tan, and in quadrant III tan s is positive. Therefore, and s 5. cos tan tan 5, Figure 5 supports this result. Now tr Eercises 9 and 55.

18 LIALMC_78.QXP // : AM Page CHAPTER The Circular Functions and Their Graphs CONNECTIONS A convenient wa to see the sine, cosine, and tangent trigonometric ratios geometricall is shown in Figure for in quadrants I and II. The circle shown is the unit circle, which has radius. B remembering this figure and the segments that represent the sine, cosine, and tangent functions, ou can quickl recall properties of the trigonometric functions. Horizontal line segments to the left of the origin and vertical line segments below the -ais represent negative values. Note that the tangent line must be tangent to the circle at,, for an quadrant in which lies. sin cos tan (, ) sin cos (, ) tan in quadrant I in quadrant II Figure PQ sin ; OQ cos ; For Discussion or Writing See Figure 7. Use the definition of the trigonometric functions and similar triangles to show that PQ sin, OQ cos, and AB tan. P(, ) B AB AB AB AO (b similar triangles) tan O Q Figure 7 A(, ) Angular and Linear Speed The human joint that can be fleed the fastest is the wrist, which can rotate through 9, or radians, in.5 sec while holding a tennis racket. Angular speed (omega) measures the speed of rotation and is defined b where is the angle of rotation in radians and t is time. The angular speed of a human wrist swinging a tennis racket is t.5 5 radians per sec. The linear speed v at which the tip of the racket travels as a result of fleing the wrist is given b v r, where r is the radius (distance) from the tip of the racket to the wrist joint. If r ft, then the speed at the tip of the racket is v r 5 7 ft per sec, or about 8 mph. t,

19 LIALMC_78.QXP // : AM Page 55. The Unit Circle and Circular Functions 55 In a tennis serve the arm rotates at the shoulder, so the final speed of the racket is considerabl faster. (Source: Cooper, J. and R. Glassow, Kinesiolog, Second Edition, C.V. Mosb, 98.) EXAMPLE 5 Finding Angular Speed of a Pulle and Linear Speed of a Belt A belt runs a pulle of radius cm at 8 revolutions per min. (a) Find the angular speed of the pulle in radians per second. (b) Find the linear speed of the belt in centimeters per second. TEACHING TIP Mention that, while problems involving linear and angular speed can be solved using the formula for circumference, the formulas presented in this section are much more efficient. Solution (a) In min, the pulle makes 8 revolutions. Each revolution is radians, for a total of 8 radians per min. Since there are sec in min, we find, the angular speed in radians per second, b dividing b. 8 radians per sec (b) The linear speed of the belt will be the same as that of a point on the circumference of the pulle. Thus, v r 8 5. cm per sec. Now tr Eercise 95. Suppose that an object is moving at a constant speed. If it travels a distance s in time t, then its linear speed v is given b v s t. If the object is traveling in a circle, then s r, where r is the radius of the circle and is the angle of rotation. Thus, we can write v r t. The formulas for angular and linear speed are summarized in the table. Angular Speed ( in radians per unit time, in radians) t Linear Speed v s t v r t v r

20 LIALMC_78.QXP // : AM Page CHAPTER The Circular Functions and Their Graphs km km Satellite EXAMPLE Finding Linear Speed and Distance Traveled b a Satellite A satellite traveling in a circular orbit km above the surface of Earth takes hr to make an orbit. The radius of Earth is km. See Figure 8. (a) Find the linear speed of the satellite. (b) Find the distance the satellite travels in.5 hr. Earth Figure 8 Not to scale Solution (a) The distance of the satellite from the center of Earth is (b) For one orbit,, and s r 8 km. (Section.) Since it takes hr to complete an orbit, the linear speed is v s t 8 r 8 km. 8 5, km per hr. s vt 8.5,, km Now tr Eercise 9.. Eercises. (a) (b) (c) undefined. (a) (b) (c). (a) (b) (c). (a) (b) (c) 5. (a) (b) (c). (a) (b) (c) undefined For each value of, find (a) sin, (b) cos, and (c) tan. See Eample Find the eact circular function value for each of the following. See Eample. 7. sin 7 8. cos 5 9. tan. sec. csc.. cos. tan cot cos. sec 7. sin 8. sin sec. csc. tan 5. cos Find a calculator approimation for each circular function value. See Eample.. sin.9. sin.8 5. cos.59. cos tan. 8. tan csc9.9. csc.875. sec.8. sec8.9. cot.. cot.8

21 LIALMC_78.QXP // : AM Page 55. The Unit Circle and Circular Functions negative. negative. negative. positive. positive. negative 5. sin ; cos ; tan ; cot ; sec ; csc. sin 8 ; cos 5 ; 7 7 tan 8 ; cot 5 ; 5 8 sec 7 ; csc sin ; cos 5 ; tan ; cot 5 ; 5 sec ; csc 5 8. sin ; cos ; tan ; cot ; sec ; csc Concept Check The figure displas a unit circle and an angle of radian. The tick marks on the circle are spaced at ever two-tenths radian. Use the figure to estimate each value. 5. cos.8. sin 7. an angle whose cosine is.5 8. an angle whose sine is.95 Concept Check Without using a calculator, decide whether each function value is positive or negative. (Hint: Consider the radian measures of the quadrantal angles.) 9. cos. sin. sin 5. cos. tan.9. tan.9 Concept Check Each figure in Eercises 5 8 shows angle in standard position with its terminal side intersecting the unit circle. Evaluate the si circular function values of. 5.. (, ) 5 8 (, 7 7) radian. radian. radian. radian radian (, ) (, ) Find the value of s in the interval, that makes each statement true. See Eample (a). 9. tan s. 5. cos s sin s cot s sec s.8 5. csc s.9 Find the eact value of s in the given interval that has the given circular function value. Do not use a calculator. See Eample (b). 55. ; sin s 5. ; cos s,,

22 LIALMC_78.QXP // : AM Page CHAPTER The Circular Functions and Their Graphs , , , , I. IV 7. II 8. III 9. radian per sec 7. radian per sec 5 7. min 7. 9 min radian per sec radians radians per sec 5 7. radians per sec radians per sec cm per sec cm 8. d 5 8. sec 8. radian per sec 8. radian per hr 8. radians per min cm per min 7 57., ; tan s 58., ; sin s 59., ;., tan s ; cos s Suppose an arc of length s lies on the unit circle, starting at the point, and terminating at the point,. (See Figure.) Use a calculator to find the approimate coordinates for,. (Hint: cos s and sin s.). s.5. s.. s 7.. s.9 Concept Check For each value of s, use a calculator to find sin s and cos s and then use the results to decide in which quadrant an angle of s radians lies. 5. s 5. s 9 7. s 5 8. s 79 Use the formula to find the value of the missing variable. 9. radians, t 8 sec 7. radians, t sec 5 7. radian, radian per min radians, radian per min 7. radians, t.9 sec 7. radian per min, t.87 min Use the formula v r to find the value of the missing variable. 75. v 9 m per sec, r 5 m 7. v 8 ft per sec, r ft 77. v 7.9 m per sec, r 58.7 m 78. r.95 cm, radian per sec The formula t can be rewritten as. Using t for changes s r to s rt. Use the formula s rt to find the value of the missing variable. 79. r cm, radians per sec, t 9 sec 8. r 9 d, radians per sec, t sec 5 8. s cm, r cm, radian per sec 8. s km, r km, t sec Find for each of the following. 8. the hour hand of a clock 8. a line from the center to the edge of a CD revolving times per min Find v for each of the following. t 5.79 t 85. the tip of the minute hand of a clock, if the hand is 7 cm long

23 LIALMC_78.QXP // : AM Page 555. The Unit Circle and Circular Functions cm per min m per min 88.,88 cm per min 89. sec mph 9.. hr 9. (a) radian 5 (b) radian per hr 8 (c),7 mph 9. (a) radians per da; radian per hr (b) (c),8 km per da or about 5 km per hr (d) about 8, km per da or about km per hr 95. (a). radian per sec (b). cm per sec 9. larger pulle: 5 radians per sec; 8 smaller pulle: 5 8 radians per sec 9 8. a point on the tread of a tire of radius 8 cm, rotating 5 times per min 87. the tip of an airplane propeller m long, rotating 5 times per min (Hint: r.5 m.) 88. a point on the edge of a groscope of radius 8 cm, rotating 8 times per min 89. Concept Check If a point moves around the circumference of the unit circle at the speed of unit per sec, how long will it take for the point to move around the entire circle? 9. What is the difference between linear velocit and angular velocit? Solve each problem. See Eamples 5 and. 9. Speed of a Biccle The tires of a biccle have radius in. and are turning at the rate of revolutions per min. See the figure. How fast is the biccle traveling in miles per hour? (Hint: 58 ft mi.) 9. Hours in a Martian Da Mars rotates on its ais at the rate of about.55 radian per hr. Approimatel how man hours are in a Martian da? (Source: Wright, John W. (General Editor), The Universal Almanac, Andrews and McMeel, 997.) in. 9. Angular and Linear Speeds of Earth Earth travels about the sun in an orbit that is almost circular. Assume that the orbit is a circle with radius 9,, mi. Its angular and linear speeds are used in designing solar-power facilities. (a) Assume that a ear is 5 das, and find the angle formed b Earth s movement in one da. (b) Give the angular speed in radians per hour. (c) Find the linear speed of Earth in miles per hour. 9. Angular and Linear Speeds of Earth Earth revolves on its ais once ever hr. Assuming that Earth s radius is km, find the following. (a) angular speed of Earth in radians per da and radians per hour (b) linear speed at the North Pole or South Pole (c) linear speed at Quito, Ecuador, a cit on the equator (d) linear speed at Salem, Oregon (halfwa from the equator to the North Pole) 95. Speeds of a Pulle and a Belt The pulle shown has a radius of.9 cm. Suppose it takes 8 sec for 5 cm of belt to go around the pulle. (a) Find the angular speed of the pulle in radians per second. (b) Find the linear speed of the belt in centimeters per second. 9. Angular Speed of Pulles The two pulles in the figure have radii of 5 cm and 8 cm, respectivel. The larger pulle rotates 5 times in sec. Find the angular speed of each pulle in radians per second..9 cm 5 cm 8 cm

24 LIALMC_78.QXP // : AM Page CHAPTER The Circular Functions and Their Graphs cm 98. about 9 sec radians per sec. 5 ft per sec 97. Radius of a Spool of Thread A thread is being pulled off a spool at the rate of 59. cm per sec. Find the radius of the spool if it makes 5 revolutions per min. 98. Time to Move Along a Railroad Track A railroad track is laid along the arc of a circle of radius 8 ft. The circular part of the track subtends a central angle of. How long (in seconds) will it take a point on the front of a train traveling mph to go around this portion of the track? 99. Angular Speed of a Motor Propeller A 9-horsepower outboard motor at full throttle will rotate its propeller at 5 revolutions per min. Find the angular speed of the propeller in radians per second.. Linear Speed of a Golf Club The shoulder joint can rotate at about 5 radians per sec. If a golfer s arm is straight and the distance from the shoulder to the club head is 5 ft, estimate the linear speed of the club head from shoulder rotation. (Source: Cooper, J. and R. Glassow, Kinesiolog, Second Edition, C.V. Mosb, 98.). Graphs of the Sine and Cosine Functions Periodic Functions Graph of the Sine Function Graph of the Cosine Function Graphing Techniques, Amplitude, and Period Translations Combinations of Translations Determining a Trigonometric Model Using Curve Fitting Periodic Functions Man things in dail life repeat with a predictable pattern: in warm areas electricit use goes up in summer and down in winter, the price of fresh fruit goes down in summer and up in winter, and attendance at amusement parks increases in spring and declines in autumn. Because the sine and cosine functions repeat their values in a regular pattern, the are periodic functions. Figure 9 shows a sine graph that represents a normal heartbeat. Figure 9 Looking Ahead to Calculus Periodic functions are used throughout calculus, so ou will need to know their characteristics. One use of these functions is to describe the location of a point in the plane using polar coordinates, an alternative to rectangular coordinates. (See Chapter 8). Periodic Function A periodic function is a function f such that f() f( np), for ever real number in the domain of f, ever integer n, and some positive real number p. The smallest possible positive value of p is the period of the function. The circumference of the unit circle is, so the smallest value of p for which the sine and cosine functions repeat is. Therefore, the sine and cosine functions are periodic functions with period.

25 LIALMC_78.QXP // : AM Page 557. Graphs of the Sine and Cosine Functions 557 (, ) (, ) (, ) (, ) Figure (cos s, sin s) s Unit circle + = Graph of the Sine Function In Section. we saw that for a real number s, the point on the unit circle corresponding to s has coordinates (cos s, sin s). See Figure. Trace along the circle to verif the results shown in the table. As s Increases from sin s cos s to Increases from to Decreases from to to Decreases from to Decreases from to to Decreases from to Increases from to to Increases from to Increases from to To avoid confusion when graphing the sine function, we use rather than s; this corresponds to the letters in the -coordinate sstem. Selecting ke values of and finding the corresponding values of sin leads to the table in Figure. To obtain the traditional graph in Figure, we plot the points from the table, use smmetr, and join them with a smooth curve. Since sin is periodic with period and has domain,, the graph continues in the same pattern in both directions. This graph is called a sine wave or sinusoid. SINE FUNCTION f() sin Domain:, Range:, f() = sin, f() = sin Figure The graph is continuous over its entire domain,,. Its -intercepts are of the form n, where n is an integer. Its period is. The graph is smmetric with respect to the origin, so the function is an odd function. For all in the domain, sin sin.

26 LIALMC_78.QXP // : AM Page CHAPTER The Circular Functions and Their Graphs Looking Ahead to Calculus The discussion of the derivative of a function in calculus shows that for the sine function, the slope of the tangent line at an point is given b cos. For eample, look at the graph of sin and notice that a tangent line at 5 will be,,,... horizontal and thus have slope. Now look at the graph of cos and see that for these values, cos. Graph of the Cosine Function We find the graph of cos in much the same wa as the graph of sin. In the table of values shown with Figure for cos, we use the same values for as we did for the graph of sin. Notice that the graph of cos in Figure has the same shape as the graph of sin. It is, in fact, the graph of the sine function shifted, or translated, units to the left. COSINE FUNCTION f() cos Domain:, Range:, f() = cos, f() = cos Figure The graph is continuous over its entire domain,,. Its -intercepts are of the form n, where n is an integer. Its period is. The graph is smmetric with respect to the -ais, so the function is an even function. For all in the domain, cos cos. Notice that the calculator graphs of f sin in Figure and f cos in Figure are graphed in the window, b,, with Xscl and Yscl. This is called the trig viewing window. (Your model ma use a different standard trigonometric viewing window. Consult our owner s manual.) Graphing Techniques, Amplitude, and Period The eamples that follow show graphs that are stretched or compressed either verticall, horizontall, or both when compared with the graphs of sin or cos.

27 LIALMC_78.QXP // : AM Page 559. Graphs of the Sine and Cosine Functions 559 EXAMPLE Graphing a sin Graph sin, and compare to the graph of sin. Solution For a given value of, the value of is twice as large as it would be for sin, as shown in the table of values. The onl change in the graph is the range, which becomes,. See Figure, which includes a graph of sin for comparison. sin sin The thick graph stle represents the function in Eample. Period: = sin = sin Figure The amplitude of a periodic function is half the difference between the maimum and minimum values. Thus, for both the basic sine and cosine functions, the amplitude is Generalizing from Eample gives the following. Amplitude. The graph of a sin or a cos, with a, will have the same shape as the graph of sin or cos, respectivel, ecept with range a, a. The amplitude is a. Now tr Eercise 7. TEACHING TIP Students ma think that the period of sin is. Eplain that a factor of causes values of the argument to increase twice as fast, thereb shortening the length of a period. No matter what the value of the amplitude, the periods of a sin and a cos are still. Consider sin. We can complete a table of values for the interval,. Note that one complete ccle occurs in units, not units. Therefore, the period here is, which equals Now consider sin. Look at the net table.. 8 sin 8 sin

28 LIALMC_78.QXP // : AM Page 5 5 CHAPTER The Circular Functions and Their Graphs These values suggest that a complete ccle is achieved in or units, which is reasonable since In general, the graph of a function of the form sin b or cos b, for b, will have a period different from when b. To see wh this is so, remember that the values of sin b or cos b will take on all possible values as b ranges from to. Therefore, to find the period of either of these functions, we must solve the three-part inequalit sin b b. sin. (Section.7) Divide b the positive number b. Thus, the period is B dividing the interval, b. into four equal parts, we obtain the values for which sin b or cos b is,, or. These values will give minimum points, -intercepts, and maimum points on the graph. Once these points are determined, we can sketch the graph b joining the points with a smooth sinusoidal curve. (If a function has b, then the identities of the net chapter can be used to rewrite the function so that b. ) b NOTE One method to divide an interval into four equal parts is as follows. Step Find the midpoint of the interval b adding the -values of the endpoints and dividing b. Step Find the midpoints of the two intervals found in Step, using the same procedure. EXAMPLE Graphing sin b Graph sin, and compare to the graph of sin. Solution In this function the coefficient of is, so the period is Therefore, the graph will complete one period over the interval,. The endpoints are and, and the three middle points are, which give the following -values., and,,,,,. Left First-quarter Midpoint Third-quarter Right endpoint point point endpoint We plot the points from the table of values given on page 559, and join them with a smooth sinusoidal curve. More of the graph can be sketched b repeating this ccle, as shown in Figure. The amplitude is not changed. The graph of sin is included for comparison.

29 LIALMC_78.QXP // : AM Page 5. Graphs of the Sine and Cosine Functions 5 = sin.5 = sin.5 Figure Now tr Eercise 5. Generalizing from Eample leads to the following result. Period For b, the graph of sin b will resemble that of sin, but with period b. Also, the graph of cos b will resemble that of cos, but with period b. EXAMPLE Graphing cos b Graph cos over one period. Solution The period is. We divide the interval, into four equal parts to get the following -values that ield minimum points, maimum points, and -intercepts. Figure 5 = cos 9,,,, We use these values to obtain a table of ke points for one period. cos 9 9 The amplitude is because the maimum value is, the minimum value is, and half of is. We plot these points and join them with a smooth curve. The graph is shown in Figure 5. Now tr Eercise. This screen shows a graph of the function in Eample. B choosing Xscl =, the -intercepts, maima, and minima coincide with tick marks on the -ais. NOTE Look at the middle row of the table in Eample. The method of dividing the interval, b into four equal parts will alwas give the values,,,, and for this row, resulting in values of,, or for the circular function. These lead to ke points on the graph, which can then be easil sketched.

30 LIALMC_78.QXP // : AM Page 5 5 CHAPTER The Circular Functions and Their Graphs The method used in Eamples is summarized as follows. Guidelines for Sketching Graphs of Sine and Cosine Functions To graph a sin b or a cos b, with b, follow these steps. Step Find the period, b. Start at on the -ais, and la off a distance of b. Step Divide the interval into four equal parts. (See the Note preceding Eample.) Step Evaluate the function for each of the five -values resulting from Step. The points will be maimum points, minimum points, and -intercepts. Step Plot the points found in Step, and join them with a sinusoidal curve having amplitude a. Step 5 Draw the graph over additional periods, to the right and to the left, as needed. The function in Eample has both amplitude and period affected b the values of a and b. EXAMPLE Graphing a sin b Graph sin over one period using the preceding guidelines. = sin Figure Solution Step For this function, b, so the period is The function will be graphed over the interval,.. Step Divide the interval, into four equal parts to get the -values,,,, and. Step Make a table of values determined b the -values from Step. sin sin Step Plot the points,,,,,,,, and,, and join them with a sinusoidal curve with amplitude. See Figure. Step 5 The graph can be etended b repeating the ccle. Notice that when a is negative, the graph of a sin b is the reflection across the -ais of the graph of a sin b. Now tr Eercise 9.