5TRIGONOMETRIC AND CIRCULAR FUNCTIONS 5.1 Angles and Units of Measure 5.2 Trigonometric Functions and the Unit Circle 5.3 Evaluation of Trigonometric Functions 5.4 Properties and Graphs 5.5 Inverse Trigonometric Functions N CHAPTER 2 WE INTRODUCED the concept of functions. Chapters 3 and 4 explored important special functions, namely polynomial, exponential, and logarithmic functions. Equally important in both applications and theoretical mathematics are the trigonometric functions we introduce in this chapter. Trigonometry (meaning “triangle measurement”) has a long and remarkable history. Some of its roots and applications go back to antiquity, but it continues to find new applications through the space age and beyond. Trigonometry has provided tools for surveying and navigation for thousands of years. Today it is built into sophisticated devices that, for example, help satellites navigate among the planets or determine how fast the spreading ocean floor is pushing continents apart. Partly because it has served so many different uses, trigonometry may appear somewhat schizophrenic in its presentation. Triangle and circle measurement commonly use degree measure, while all modern applications of trigonometry that describe periodic phenomena—from tides to orbiting satellites to the wave nature of quantum physics— require functions of real numbers, not degrees. In Section 5.1 we introduce both modes of angle measure because it is important to become familiar with both. In Section 5.2 the trigonometric functions are also defined in both modes. I 253 Degree measure. For brevity. The measure of an angle is described by the amount of rotation. The modes are related by their measures of one complete revolution. we say the angle is positive if its measure is positive. Figure 1 illustrates the labeling of angles and rotation. An angle is the union of two rays together with a rotation. Figure 2 illustrates several angles and their degree measures. More critically. . and seconds. with numeration based on multiples and fractions of 60. For most purposes. or 2 radians. The initial position is called the initial side and the final position is called the terminal side of the angle. An angle. An angle has positive measure if the rotation is counterclockwise. just as all other scientific branches.254 Chapter 5 Trigonometric and Circular Functions 5. minutes. Yi Lin I intended to take either physics or mathematics . The measure of one revolution is 360 degrees. from an initial position to a final position. A ϭ 90Њ to denote “the measure of angle A is 90Њ. The point about which the ray rotates is called the vertex of the angle. . for instance. In geometry. seconds) have the same historical basis. Frederick Mosteller The study of plane geometry considers all geometric figures as sets of points in a plane. Degree measure is part of our legacy from Babylonian mathematics. I found myself very excited by a course called Physical Measurements. The rotation in angle B is greater than one revolution. mathematics. the measure of an angle involves the notion of rotation. Terminal side A Initial side B FIGURE 1 C Units of Angular Measure Your calculator operates both in degree mode and in radian mode. we consider an angle as being generated by rotating a ray in the plane about its endpoint. however. For brevity. angles are measured most often in degrees. is the union of two rays with a common endpoint. . Units of time (hours. verifying. and negative measure if the rotation is clockwise. and intended to become a high school teacher. for example.” . is developed in the process of examining.1 ANGLES AND UNITS OF MEASURE . In trigonometry we talk about angles of a triangle as the union of two line segments that have a common endpoint. Angles A and B are positive while angle is negative. reflecting two different ways of measuring angles. and modifying itself. The curved arrow indicates the direction and amount of rotation. or decimal fractions of degrees. We kept measuring things to more and more decimal places by more and more ingenious methods. we write. minutes. . written 1Ј.24Њ is 64Њ14Ј24 Љ .24 ͒͑60Ј ͒ ϭ 14.24Њ. the central angle is 1 radian. is 3600 60 1 degree. Next convert the 0. with the initial side meeting the circle at A and the terminal side at B. ᭣ r r 2 radians 1 radian r FIGURE 3 An arc length of one radius measures 1 radian Radian measure. ͪ ᭤EXAMPLE 2 Decimal form to DMS and seconds. s is negative. ᭤EXAMPLE 1 DMS to decimal form Express 36Њ16Ј23 Љ in decimal form and round the result to three decimal places. so s can be any real number. Note that this definition is independent of any particular circle.4Ј ϭ ͑0. if we have a segment of length equal to the radius and lay it out along the circle. take a circle of a radius r with center at the vertex. minutes.1 Angles and Units of Measure 255 Degrees. 1 minute. the radian measure of one revolution is 2 radians.4Ј into seconds. 0. of a minute. or 1 second. The radian measure of is defined as the ratio of s to r . An arc length of two radii (or a diameter of the circle) measures a central angle of 2 radians.4 ͒͑60 Љ ͒ ϭ 24 Љ . D = – 180° Solution First convert the decimal part. written 1Њ. Greek letters such as (theta) and (phi) are often used to refer to angles.24Њ in degrees.24Њ ϭ ͑0. Think of a point P moving around the circle from A to B. Solution A = 90° B = 45° 16Ј is C = 450° 16 60 of a degree. and seconds are related by the following: 1 of a complete rotation. 64.) For counterclockwise rotation. For an arbitrary angle . That is. for clockwise rotation. s is positive. Since 1Њ is 60Ј. . but you can also use the relations above to change between DMS and decimal forms. and 23 Љ is 36Њ16Ј23 Љ ϭ 36 ϩ ͩ 23 3600 of a degree. The directed distance s that P travels is the directed arc length associated with (see Figure 4. The radian measure of an angle is determined as a ratio of arc length to radius.5. 360 1 of a degree. so we have 23 Њ 16 ϩ Ϸ 36. minutes. written 1 Љ . 0. ᭣ 60 3600 Express 64. is 60 1 1 of a degree. If the rotation is greater than one revolution.273Њ. See Figure 3. FIGURE 2 0.4Ј. Since the total arc length of a circle (its circumference) is 2 r . Therefore. into minutes. the arc length is greater than 2 r (positive or negative). as in the next example. is Your calculator may allow you to enter angles in degree–minute–second (DMS) form (see your instruction manual).  = 6. ᭣ Degree–Radian Relationships s = 13. ϭ s 2.3. This requires a technique for conversion. Determine the measures of ␣ and  in radians.6 In many cases we may have the measure of an angle in degrees when we need the radian measure. ϭ 2 cm r s = 2. Since one complete rotation is measured by either 360Њ or 2 radians. Degree–radian conversions 180Њ ϭ radians. Solution c r = 2 cm. as the arrow in the figure indicates.256 B P Chapter 5 Trigonometric and Circular Functions s Definition: radian measure of an angle Terminal side O Initial side A Suppose is any angle and C is a circle of radius r with its center at the vertex of . ϭ ␣ = 1.3 radians. s ϭ . When the measure of an angle is given as a real number. then the radian measure of is s r .8 radians. r In taking the ratio of two lengths. is an angle such that in a circle of radius 2 centimeters. The basic relationship 360Њ ϭ 2 radians connects degree and radian measures.6 cm. ͩ ͪ FIGURE 6 See Figure 7 for equivalent measures of selected angles. we have the necessary equivalence. If s is the directed arc length associated with . We could emphasize that the radian measure of is 1.3.8 2 (1) 1Њ ϭ radians or 1Њ Ϸ 0. for example.6 for ␣ and s ϭ 13. ␣ and  are central angles of a circle of radius 2.8 ϭ 2 r The measure of ␣ is 1. Then the measure of in radians is given by FIGURE 4 Directed arc length s is the distance P travels along the circle from A to B.6 ϭ 6.017453 radians. The units cancel. but our normal convention is that radians need not be written. If. 180 180 Њ 1 radian ϭ or 1 radian Ϸ 57.6 centimeters (see Figure 5). thus radian measure is simply a real number (no units). the associated arc length is 2.6 s␣ 3.8 radians. . it is understood that the measure is radians. The measure of  is greater than one revolution (2 Ϸ 6.28).296Њ. and we write simply ϭ 1. units cancel. The lengths of the subtended arcs are s␣ ϭ 3.8 2 The measure of  is 6.3 by writing ϭ 1.8 ϭ 2 r s 13.6 ϭ 1. ᭤EXAMPLE 3 Radian measure of central angles In Figure 6. that is. or vice versa.6 cm ϭ 1.6 for  . FIGURE 5 ␣ϭ s␣ = 3. measure. To convert from degree measure to radian . ϭ 60Њ. (a) ␣ ϭ 210Њ (b)  ϭ 585Њ (c) ␥ ϭ Ϫ150Њ Solution Diagrams in Figure 10 show the angles. as in Figure 8. you need to develop a feeling for radian measure expressed simply as numbers.3Њ. a visual reminder of these relations is helpful. almost 60Њ. an angle of 3 radians is very nearly a straight angle (remember that 3. Similarly. In addition to thinking in terms of fractions of radians. we need to recognize them immediately in both degree and radian measure: ϭ 180Њ. 5 radians 6 radians 1 radian 2 radians 3 radians 4 radians FIGURE 9 Integer multiples of 1 radian.57 will become very familiar. 2 3 4 6 For most of us. For example. See Figure 9.14 Ϸ ϭ 180Њ). multiply by 180 . we need to learn some convenient equivalences. and because radian measure is so important in calculus. ᭤EXAMPLE 4 Degree–radian conversion Draw a diagram that shows the angle and then find the corresponding radian measure. ϭ 45Њ. ϭ 30Њ. ϭ 90Њ. Right angles are so common that the decimal approximation of ͞2 as 1.5. Most of us are familiar with degree measure. 6 = 30° 4 = 45° 3 = 60° 2 = 90° = 180° FIGURE 8 Degree and radian measure for some familiar angles. Give the result both in exact form and as a decimal approximation rounded off to two decimal places. 1 radian is about 57.1 Angles and Units of Measure 257 2 = 360° = 180° (a) (b) 2 = 90° 3 = 60° (c) (d) FIGURE 7 Degree–radian measure for some familiar angles. There are a few angles that are used so often. 47 radians ϭ 2. suppose the length of the subtended arc is s and the area of the sector is A. For the sector shown in the figure with central angle . Let s denote the length of the subtended arc and let A denote the area of the sector. 2.62 ᭣ ϭϪ (c) ␥ ϭ Ϫ150Њ ϭ Ϫ150 180 6 ᭤EXAMPLE 5 Radian–degree conversion If the radian measure of is 2. 2 r .47 radians Ϸ 141.47 radians Ϸ 141Њ31Ј.47. Rounded off to one decimal place. If is measured in radians. ϭ 2 r 2 Solving for s and A .258 Chapter 5 Trigonometric and Circular Functions  = 585° ␣ = 210° ␥ = – 150° (a) (b) FIGURE 10 (c) (a) ␣ ϭ 210Њ ϭ 210 7 Ϸ 3. 2 FIGURE 11 s ϭ r . that is. 2. 2 r 2 1 A ϭ r 2 . Arc length and the area of a circular sector Suppose is a central angle of a circle of radius r . Hence.21 180 4 5 Ϸ Ϫ2.67 ϭ 180 6 13 (b)  ϭ 585Њ ϭ 585 ϭ Ϸ 10. s . then s and A are given by s ϭ r 1 A ϭ r 2 2 (2) 1 A ϭ rs 2 (3) . find its degree measure rounded off to one decimal place and then to the nearest minute.5205754Њ.5Њ. is approximately 141Њ31Ј. Convert the decimal part of a degree to minutes by multiplying by 60. s. and r 2 for the entire circle are equal. Follow the strategy. and A to the respective measures 2 . 2 r r 2 1 A ϭ rs. ᭣ Applications of Radian Measure: Arc Length and Area Sector s A r In a circle a given central angle between 0 and 2 determines a portion of the circle called a sector. to the nearest minute. 2 A s ϭ . ͩ ͪ ͩ ͪ ͩ ͪ ͩ ͪ Strategy: To get the degree Solution measure. as indicated by the shaded region shown in Figure 11. multiply by 180 . 2. A ϭ . The ratios of .47 ͩ ͪ 180 Њ Ϸ 141. we get an area of approximately 42. Either way. Draw a graph of A͑r͒ in the window ͓0.4 Ϸ 16. ᭣ ͩ ͪ ͩ ͪ FIGURE 12 ᭤EXAMPLE 7 r 30° Given a circular sector with central angle of 30Њ. 20] by [0.6 (one decimal place). (c) Find r when A ϭ 25. First convert 150Њ to radians and then use Equations ͑2 ͒ and ͑3 ͒. 20 ͔ ϫ ͓0. A ϭ ͑3. ϭ 150Њ ϭ 150 ͩ ͪ 5 . 50] (b) x 2 A= 12 FIGURE 13 Assume that the moon travels around the earth in a circular path of radius 239. ᭣ ᭤EXAMPLE 8 Circular motion [0.7)2͞12 . then r ϭ ͙͑12 ͒͑25. (a) Find a formula for the distance D͑ x͒ (in thousands of miles) that the moon travels in x days.6 Round off to two significant digits to get an arc length of 9. 6 1 5 ϭ 5. (b) Evaluate A when r ϭ 12. (a) Give a formula for the area A of the sector as a function of the radius r . ϭ 180 6 5 for in Equations ͑2 ͒ and ͑3 ͒ 6 Substitute 3. and the area (see Figure 13a) is given by A͑r͒ ϭ 2 r 12 (a) Graphing Y ϭ X2͞12 in the specified window gives us a calculator graph as shown in Figure 13b.5.6 ͒2 2 6 s ϭ 3.6 ͒͞ Ϸ 9.6. clearly part of a parabola.96. We can zoom in for more accuracy.9. How far does the moon travel (b) in 10 days? in 21 days and 6 hours? (c) How many days does it take for the moon to travel a million miles? A billion miles? .7. Draw a diagram to show the sector and find the arc length and the area of the sector.6 for r and = 150° r = 3.1 Angles and Units of Measure 259 ᭤EXAMPLE 6 Arc length/area of circular sector The radius of a circle is 3. Draw a calculator graph.2. 50 ͔. (c) If r 2͞12 ϭ 25.6 centimeters and the central angle of a circular sector is 150Њ. Solution The sector is the shaded region in Figure 12. Strategy: Equations ͑2 ͒ and ͑3 ͒ require to be in radians. Following the strategy.4 cm and an area of 17 cm2.42.6 5 ϭ 3 Ϸ 9. (b) Tracing on the calculator graph does not allow us to read with sufficient accuracy the value of A when r ϭ 12.000 miles and that it makes one complete revolution every 28 days. or we can return to the home screen and evaluate (12.7 (one decimal place). we use 30Њ ϭ ͞6. Area as a function Solution (a) Since the formula for area ͑Equation ͑3 ͒͒ requires radian measure for the central angle. 1 of a complete Alexandria to be 50 Travelers reported that at revolution. ϭ summer solstice (the longest day of 50 C the year). A represents Alexandria and real genius Eratosthenes lay in his analysis of the B.50 million).000 miles. As his name suggests.000 miles.25 ͒ Ϸ 1. 6 hours. . leading to the equation. the circumference of the earth to be sun’s rays were coming directly toward the center C ϭ 50 · 5000 ϭ 250.000 miles. that at Syene on that date. Syene (the modern city Aswan). problem and his recognition that geometric figures the arc length s. in 21 days. Eratosthenes the distance from Alexandria to was of Greek descent. the sun By the best estimates of the day. but Eratosthenes’ estimate would same day. every 28 days. where the radius is 239 thousand miles. 14 A calculator graph in ͓0. D͑21. dramatic applications of s ϭ trigonometry was made by C One revolution To Eratosthenes a little before 200 B. the stadium. of the sun’s rays at Alexandria at noon on the varied in size.140.260 Chapter 5 Trigonometric and Circular Functions HISTORICAL NOTE MEASUREMENT OF THE CIRCUMFERENCE OF THE EARTH One of the earliest and most Symbolically.C. but the In the diagram. of the Earth.000 miles ( Ϸ 1. Eratosthenes realized that he could use compare to something near 24. Several geometric relationships to find the circumference compensating errors probably contributed to the of the Earth.000 stadia. the 2 1 5000 sun cast no shadow at noon on the . a distance his life in Egypt during the reign of s estimated from travel by camel Ptolemy II and later became the A caravans by surveyors trained to B head of the greatest scientific count paces of constant length. 1600͔ is shown in Figure 14. (b) Either from the graph or from the formula. One day’s distance is 2 r͞28. Six hours is a quarter of a day. 1600] 239 x D(x) = 14 FIGURE 14 239 x. Syene. from which he calculated the circumthen. Solution (a) The distance traveled in one revolution is 2 r . truly remarkable accuracy of his figure. but he spent Syene was 5000 stadia. Eratosthenes reasoned. By measuring the angle difficult because the measuring unit. Alexandria was supposed to be Comparison with modern measurements is directly north of Syene. to the entire circumference C can tell us something about the nature of the must equal the ratio of angle to the entire world that we can learn in no other way. library in the ancient world at Eratosthenes measured angle at Alexandria. central angle (a complete revolution). 30] by [0. or 478. 30 ͔ ϫ ͓0. from which the number of thousands of miles is given by D͑ x͒ ϭ [0. D͑10 ͒ Ϸ 536. The ratio of the distance from A to B. FIGURE 15 Solution (a) For the top of the cup.000. less than three weeks. You can make a conical cup by sliding OA to any other radius OB. so we want to solve for x when D͑ x͒ ϭ 1000: x ϭ 14 · 1000͑͞239 ͒ Ϸ 18. so r ͑ x͒ ϭ C͑ x͒͞2 ϭ 4 ͑2 Ϫ x͒ 2 ϭ ͑2 Ϫ x͒. Assuming 365 days in a year. so r 2 ϩ h 2 ϭ 42. (a) Using the circumference of the cone given by C͑ x͒ ϭ 4 ͑2 Ϫ x͒. ᭣ ᭤EXAMPLE 9 Making a paper cup Draw a circle of radius 4 on a piece of paper.6 days. as in the diagram. h 2 ϭ 16 Ϫ 4 4 2 ͑4 2 Ϫ 4 x ϩ x 2͒. What is the maximum volume? Amount of overlap is 4x B x 0 4 A Solid arc AB. (c) Find the approximate value of x giving the maximum volume.1 Angles and Units of Measure 261 (c) A million miles is a thousand thousand ͑10 6 ϭ 10 310 3͒. 4(2– x) r h 4 A. 2 Looking at a cross-section of the cup. r and h are legs of a right triangle with hypotenuse 4 (the radius of the original circle). becomes rim of the cone.5. Cut from a point A on the circle to the center O. See Figure 15. of length 4(2 – x). it will take more than 51 years for the moon to travel a billion miles in circling the earth.000͑͞239 ͒ Ϸ 18. x ϭ 14. 2 ͑2 Ϫ x͒ ϭ 16 Ϫ 2 . C ϭ 2 r . find a formula for the radius r ͑ x͒ at the top of the cone and the height h͑ x͒. effectively cutting out a sector with central angle x ϭ Є AOB. When D͑ x͒ ϭ 10 6. B coincide to make cone. (b) Express the volume V ͑ x͒ as a function of x and draw a calculator graph.600 days. A billion miles is 10 9͞10 3 ϭ 10 6 thousands. If the central angle AOB is . or about 66Њ. One measure of speed. Tracing and zooming to find the high point. 2 (1.1 ͒͑80 ͒ ϭ 88 . A volume of 25.1 feet.4 inches. then the linear and angular speeds are vϭ FIGURE 17 s and ϭ . The linear speed of a point on the ft Ϸ 276 ft/min. t t . so we have V ͑ x͒ ϭ Thus. why do you suppose paper cups are not made to hold the maximum volume? ᭣ Linear and Angular Speed There are two kinds of speeds associated with rotational motion.8) 2 ͙4 x Ϫ x 2. so that its radius r is 13. and P moves from A to B in time t . angular speed. or V ϭ 1 3 r h.15 radians. 30 ͔ gives the graph shown in Figure 16. Since that is the shape with the maximum volume and the least waste (overlap). In 1 minute a point on the circumference turns through an angle equal to 80 radians. 3 2 4 2 2 ͑r ͑ x͒͒2h͑ x͒ ϭ ͙4 x Ϫ x 2. For a particle moving in a circular path at a uniform rate. Suppose a bicycle wheel is rotating at a constant rate of 40 revolutions per minute (40 rpm).8 cubic inches is just over 14 ounces.262 Chapter 5 Trigonometric and Circular Functions or h2 ϭ Thus. Suppose that such a particle P moves from point A to point B along a circle of radius r . 2 ͑2 Ϫ x͒ 3 3 [0. the angular speed can also be expressed as 40͑2 ͒ or 80 radians per minute. the point travels 88 feet in 1 minute.8 cubic inches when x is about 1. frequently denoted by the Greek letter . gives the rate of rotation.15. (b) The volume of a cone is a third of the volume of the cylinder with the same 2 base. We suggest that you make a cone with x Ϸ 66Њ and observe that your cone is not standard. the linear and angular speeds are obviously related. we find that the maximum volume is about 25. circumference is 88 min A O Relationship between linear and angular speeds. consider an example. 25. 10] by [0. V ͑ x͒ ϭ 8 ͑2 Ϫ x͒2͙4 x Ϫ x 2.2 inches or 1. s B Hence. and the distance s traveled by point on the circumference in 1 minute is s ϭ r ϭ ͑1. Since one revolution is equivalent to 2 radians. Suppose the diameter of the bicycle wheel is 26. as indicated in Figure 17. 30] FIGURE 16 (c) Graphing V in ͓0. the distance (arc length) from A to B is s. or 4800 radians per hour. 10 ͔ ϫ ͓0. The bicycle wheel’s angular speed is 40 rpm by one measure. h͑ x͒ ϭ 4 ͑4 x Ϫ x 2͒ . To introduce the basic ideas. 0 Ͻ x Ͻ 2 . 9 ft/sec. in a direction tangent to the circular path of motion. vϭ Velocity (v) s r ϭ ϭr ϭ r t t t ͩͪ Linear and angular speed relationship For a particle moving in a circular path of radius r at a uniform rate. The wheel of a grindstone with (a) Find the angular speed in radians per second. Strategy: First convert r and to units that are consistent with feet per second. or ϭ 30 30 ϭ 2 ϭ 45 rad/sec. meaning that it has both a magnitude and a direction (see Section 7. (c) For a certain job. Velocity is a vector quantity. it is desirable to have the linear speed of the grinding edge of the wheel at 30 ft/sec. the linear speed v and angular speed are related by the equation v ϭ r. r ϭ 8 in ϫ and ͩ ͪ ͩ 2 1 ft ϭ ft 3 12 in ϭ 300 rev 2 rad 1 min ϫ ϫ ϭ 10 rad/sec min rev 60 sec rad sec ͪ ͩ ͪ . the linear speed of a point on the grinding edge of the wheel is 20.5). (b) Find the linear speed of a point on the circumference of the wheel in feet per second. v ϭ r ϭ ͩͪ 20 2 Ϸ 20. O (4) FIGURE 18 Linear speed and angular speed are sometimes called linear velocity and angular velocity. (c) For v to be 30 ft sec . (a) The angular speed is 10 (b) Using Equation ͑4 ͒.1 Angles and Units of Measure 263 Since s ϭ r (where is measured in radians). In uniform circular motion. ͑10 ͒ ϭ 3 3 Hence. r 3 . as indicated by the arrow in Figure 18. but we reserve the term velocity for directed speeds. ᭤EXAMPLE 10 Linear and angular speed a radius of 8 inches is rotating at 300 rpm. What change in angular speed (in revolutions per minute) is required? Solution Given that r is 8 inches and is 300 rpm. specify r ϭ 30.pg [ ] g 5. follow the strategy and express r and in units of feet and seconds. the magnitude of the linear velocity vector is the linear speed defined above.9. results given as decimal approximations should be rounded off to the number of significant digits consistent with the given data. Give reasons. (a) Ͻ Ͻ Ͻ Ͻ Ϫ 4 5 (c) 0. (a) 23Њ38Ј (b) 143Њ16Ј23 Љ (c) Ϫ95Њ31Ј 8. convert radians per second. almost half again as fast as the present angular speed. 2. Include a curved arrow and also illustrate the range of position for the terminal side with dashed rays.264 Chapter 5 Trigonometric and Circular Functions To express in revolutions per minute. if s ϭ 48 and r ϭ 24. 5. 1. if possible. (a) A ϭ 3 4 5 (c) C ϭ Ϫ2.5 5 3 (b) Ϫ 6. If the central angle of a circle measures 1 radian. 10.7 Ͻ Ͻ 2. if s ϭ 12 and ϭ 60Њ. In Example 10. . Exercises 1–4 Draw a diagram to show the angle. for instance. then rϭ . In a circular sector. (a) A ϭ 540Њ (b) B ϭ Ϫ135Њ (c) C ϭ 67Њ30Ј 7 2 (b) B ϭ Ϫ (c) C ϭ 1. (a) A ϭ (b) B ϭ Ϫ3 3 Exercises 5–6 Sketch an angle that satisfies the inequality.16 radians.79 Ͻ Ͻ 1. Assume that the planets travel in circular orbits about the sun. 7. we found in part (b) that when is 300 rpm. Include a curved arrow to indicate the amount and direction of rotation from the initial side to the terminal side. the speed is near 20 ft/sec. An angle of 22 7 radians is equal to an angle of 180Њ. The sector of a circle with a central angle of 1 radian 2 2 and radius r cm has an area of 1 2 r cm . the wheel speed must increase to about 430 rpm. An angular speed of 5 rpm is equal to Develop Mastery rad min . in part (c) an angular speed of 30 ft/sec should correspond to about 3 2 of 300 rpm. An angle of 180Њ is greater than an angle of 3. giving an estimate of about 450 rpm. 1. then the length of the arc subtended by is equal to the length of the radius. 5 is equal to an angle of degrees. ᭣ Does this answer make sense? Always ask yourself if a solution is reasonable and make an independent check or a simple estimate. 9. (a) A ϭ 240Њ (b) B ϭ 720Њ (c) C ϭ Ϫ210Њ 2. Exercises 6–10 Fill in the blank so that the resulting statement is true. The result of 430 rpm in part (c) is entirely reasonable. In a circular sector. (a) 60Њ (b) 330Њ (c) 22Њ30Ј (d) 105Њ 10. (a) 57Њ34Ј (b) 241Њ15Ј51 Љ (c) Ϫ73Њ43Ј Exercises 9–10 Find the radian measure of the angle and give the result in both exact form (involving the number ) and decimal form rounded off to two decimal places.05 Exercises 7–8 DMS to Decimal Express the angle as a decimal number of degrees rounded off to three decimal places.8 3. (b) Ϫ Ͻ Ͻ Ϫ 5. If the planet Venus takes 225 days for one revolution about the sun and the Earth takes 365 days. then the angular speed of Venus is less than the angular speed of the Earth. then ϭ radians. Therefore. ϭ 45 rad 1350 rev 1 rev 60 sec ϫ Ϸ 430 rpm ϫ ϭ sec 2 rad min min ͩ ͪ ͩ ͪ For a linear speed of 30 ft/sec. 9. 6. 4. 3. EXERCISES 5. (a) 90Њ (b) 450Њ (c) 67Њ30Ј (d) Ϫ165Њ Unless otherwise specified.36 4. 8. (a) Ͻ Ͻ 2 2 (c) 1.1 Check Your Understanding Exercises 1–5 True or False. An angle of 4 7. An angle of 210Њ is equal to an angle of radians. r ϭ 164. (a) Through what angle (in radians) will the radial line from the sun to the Earth sweep in 73 days? (b) How far does the Earth travel in its orbit about the sun in 73 days? . The radius of a circular sector is 12. ϭ 23. 17. Find the central angle (a) in radians and (b) in degrees. ϭ 30Њ 3 24. each of which is then formed into a cone. A ϭ . ϭ 25. ␣ ϭ 7 113 21 16. ϭ 96. C ϭ 84Њ37Ј 5 2 20. A circular piece of tin of radius 12 inches is cut into three equal sectors. (a) What is the height of each cone? (b) What is the volume of each cone? Exercises 13–16 Ordering Angles Order angles ␣ . ␣ ϭ 120Њ36Ј.48Њ.  ϭ 154. What is the measure in degrees of the smaller angle between the hour and minute hands of a clock (a) At 2:30? (b) At 2:45? 32.M.” look it up in the dictionary. B ϭ 73Њ 18.Bϭ 19. ␥ ϭ 10 Exercises 17–20 Triangle Angles Two of the three angles A.705 22 355 .5 centimeters and its area is 182 square centimeters. 28. (a) How far does the tip of the hand travel in 15 minutes? (b) How far does the tip of the hand travel between 8:00 A.3 8 27. (a) 4 5 12 11 (b) 12 (b) (c) 4 (c) Ϫ5 (d) 3. 31.824 14. 40. C ϭ 4 12 3 15 Exercises 21–26 Arc Length.1. Draw a diagram that shows the sector and determine (a) the arc length s and (b) the area A for the sector.2. ␣ ϭ 154Њ35Ј. rounded off. ␥ ϭ 2. ϭ 4. and 4:15 P. Assuming that the Earth is a sphere of radius 3960 miles.3Њ 21. B. Assume that the Earth travels a circular orbit of radius 93 million miles about the sun and that it takes 365 days to complete an orbit. ␣ Ͻ ␥ Ͻ  ).5Њ and area 6. r ϭ 24.  ϭ 47.1 Angles and Units of Measure 265 Exercises 11–12 Radians to Degrees Express the angle in decimal degree form. The minute hand of a clock is 6 inches long.M. a nautical mile is equal to how many ordinary miles (5280 ft)? 37. Area The radius r and the central angle of a circular sector are given. ␥ ϭ 0. Through what angle does a spoke of the wheel rotate when the bicycle moves forward 24 feet? Give your result in radians to two significant digits. and ␥ from smallest to largest (as. Nautical Mile A nautical mile is the length of an arc of a great circle subtended on the surface of the Earth by an angle of one minute (1Ј) at the center of the Earth.4 30. r ϭ 16. (a) How fast is it moving in miles per hour? (b) How far does it travel in 4 hours in nautical miles? In ordinary miles? (c) Through what angle does a line from the center of the Earth to the ship revolve in 4 hours? (Hint: If you are not familiar with the word “knots. Find the volume of the cone. (a) What is the linear speed (in inches per hour) of the tip of the minute hand in Exercise 33? (b) What is the linear speed of a point 1 inch from the tip of the minute hand? 35. What is the angular speed in radians per minute of (a) the hour hand of a clock? (b) the minute hand? 36.6 (d) 5.32Њ. r ϭ 32. if necessary. 13. Speed in Knots A ship is traveling along a great circle route at a speed of 20 knots. of the same day? 34.5.␥ ϭ 15. (a) 2 3 7 12. ␣ ϭ 47Њ24Ј.ϭ . B ϭ 37Њ41Ј.  ϭ 120. A ϭ 58Њ. A circular sector with central angle 90Њ is cut out of a circular piece of tin of radius 15 inches.80 square feet? 29. Repeat Exercise 38 reducing the central angle of the sector cut out of the piece of tin to 60Њ. r ϭ 36. The remaining piece is formed into a cone (see Example 9).) 38. Find the third angle. for example. A ϭ . The diameter of a bicycle wheel is 26 inches.53Њ. ϭ 256Њ 5 7 26. 22. and C of a triangle are given. r ϭ 47. At what times to the nearest tenth of a minute between 1:00 and 2:00 is the smaller angle between the hour and minute hands 15Њ? 33. to one decimal place. Remember that the sum of the three angles of any triangle is equal to 180Њ (or ). 39. What is the radius of a circular sector with central angle 37.  . 11. The diameter of a bicycle wheel is 26 inches. When the bicycle moves at a speed of 30 mph. It makes one complete revolution every 150 minutes. (b) circular sector BCD. In the diagram C is the center and AD is a diameter of the circle with radius 24 cm. 45. Solve for x . we can show that the x -value in Example 9 giving a maximum volume is a root of the equation 3 x 2 Ϫ 12 x ϩ 4 2 ϭ 0. (a) What is the blade’s angular speed in revolutions per minute? (b) What is the linear speed (in kilometers per hour) of the tip of the blade? 52. (a) What is the average angular speed of the rope in revolutions per minute? (b) Assuming that the rope forms an arc so that its midpoint travels in a circular path of radius 3.299 turns in 5 hours and 33 minutes. The blade of a rotary lawnmower is 34 cm long and rotates at 31 rad/sec. Does this agree with the value of x found in Example 9? 43.5 feet. what is the speed of the current in miles per hour? 51.266 Chapter 5 Trigonometric and Circular Functions 41. are connected by a belt (see 50. (a) What is its angular speed in revolutions per hour and in radians per hour? (b) What is its linear speed? Assume that the radius of the Earth is 3960 miles. The face of a windmill is 4.0 meters in diameter and a wind is causing it to rotate at 30 rev/min. a circular paddle wheel with a radius of 3 feet is lowered into the water just far enough to cause it to rotate. Looking Ahead to Calculus Using calculus. What is the linear speed of the tip of one of the blades (in meters per minute)? 46. determine the angular speed of the wheel in revolutions per minute. 47. Find in exact form the area of (a) ᭝ ABC. 44. Find the linear speed (in miles per hour) of the Earth in its orbit about the sun. Є BCD measures 60Њ. In Example 9 find the value of x (2 decimal places) for which V ϭ 20. If the wheel rotates at a speed of 12 rev/min. one with a radius of 4 inches and the other with a radius of 12 inches.7 cubic inches. 8 rpm Belt 12 in. A record was set in rope turning with 49. Find the linear speed (in miles per hour) of the moon in its orbit about the Earth. what is the average linear speed (in feet per minute) of the midpoint of the rope? (c) How far (in miles) did the midpoint travel during the record-setting turning session? . (b) What is the linear speed of a point on the belt? 4 in. (a) determine the angular speed of the larger pulley (in revolutions per minute). To measure the approximate speed of the current of a river. A satellite travels in a circular orbit 140 miles above the surface of the Earth. B the diagram). Assume that the moon follows a circular orbit about the Earth with a radius of 239.3 days. Two pulleys. 42. 48. 60Њ C 24 A D 49. If the smaller pulley is being driven by a motor at 8 rev/min. Assume that the Earth travels about the sun in a circular orbit with a radius of 93 million miles and that one revolution takes 365 days.000 miles and that one revolution takes 27. (c) the shaded region.