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satellites satellites at at DRTE. DRTE. Dr.Dr. Franklin Franklin moved moved on on to to thethe Department Department of of Communications Communications in 1969 in 1969 and and became became the project the project manager manager for the for Hermes the Hermes communications communications satellite satellite in the in early the early 1970s. 1970s. He remained involved in developing space communications technology, both at home and internationally, and in the mid-1980s helped draft a submission for the establishment of the Canadian Space Agency. Following a stint in the late 1980s as a visiting professor of electrical engineering in New Zealand, Dr. Franklin spent two years as chief scientist at Spar Aerospace, the company that designed and built the Canadarm. He was appointed to the Order of Canada in 1990. 16–1 c, Ra diu s, r Ar al ntr Ce le, g an S Radian Measure A central angle is one whose vertex is at the centre of the circle (Fig. 16–1). An arc is a portion of the circle. In Chapter 6 we introduced the radian: If an arc is laid off along a circle with a length equal to the radius of the circle, the central angle subtended by this arc is defined as one radian, as shown in Fig. 16–2. Angle Conversion An arc having a length equal to the radius of the circle subtends a central angle of one radian; an arc having a length of twice the radius subtends a central angle of two radians; and so on. Thus an arc with a length of 2 times the radius (the entire circumference) subtends a central angle of 2 radians. Therefore 2 radians is equal to 1 revolution, or 360. This gives us the following conversions for angular measure: FIGURE 16–1 Angle Conversions 1 rev 360 2 rad 117 We convert angular units in the same way that we converted other units in Chapter 1. One radian ◆◆◆ Example 1: Convert 47.6 to radians and revolutions. Solution: By Eq. 117, 2 rad 47.6 p q 0.831 rad 360 r r Note that 2 and 360 are exact numbers, so we keep the same number of digits in our answer as in the given angle. Now converting to revolutions, we obtain r 1 rev 47.6 p q 0.132 rev 360 FIGURE 16–2 ◆◆◆ Example 2: Convert 1.8473 rad to degrees and revolutions. Solution: By Eq. 117, If we divide 360 by 2 we get 1 rad ⬇ 57.3 Some prefer to use Eq. 117 in this approximate form instead of the one given. 426 ◆◆◆ 360 1.8473 rad p q 105.84 2 rad and 1 rev 1.8473 rad p q 0.294 01 rev 2 rad ◆◆◆ Section 16–1 ◆◆◆ ◆ 427 Radian Measure Example 3: Convert 1.837 520 rad to degrees, minutes, and seconds. Solution: We first convert to decimal degrees. 360 1.837 520 rad p q 105.2821 2 rad Converting the decimal part (0.2821) to minutes, we obtain 60 0.2821 p q 16.93 1 Converting the decimal part of 16.93 to seconds yields 60 0.93 p q 56 1 So 1.837 520 rad 1051656. ◆◆◆ Radian Measure in Terms of Not only can we express radians in decimal form, but it is also very common to express radian measure in terms of . We know that 180 equals radians, so 90 rad 2 45° rad 4 15° rad 12 and so on. Thus to convert an angle from degrees to radians in terms of , multiply the angle by ( rad/180), and reduce to lowest terms. ◆◆◆ Example 4: Express 135 in radian measure in terms of . Solution: 135 3 rad 135 p q rad rad 180 4 180 ◆◆◆ Students sometimes confuse the decimal value of with the degree equivalent of radians. What is the value of ? 180 or 3.1416 . . .? Remember that the approximate decimal value of is always Common Error ⬵ 3.1416 but that radians converted to degrees equals radians 180 Note that we write “radians” or “rad” after when referring to the angle, but not when referring to the decimal value. To convert an angle from radians to degrees, multiply the angle by (180/ rad). Cancel the in numerator and denominator, and reduce. ◆◆◆ Example 5: Convert 7/9 rad to degrees. Solution: 7 9 p 180 q rad 7(180) rad deg 140 9 ◆◆◆ 428 Chapter 16 ◆ Radian Measure, Arc Length, and Circular Motion Trigonometric Functions of Angles in Radians We use a calculator to find the trigonometric functions of angles in radians just as we did for angles in degrees. However, we first switch the calculator into radian mode. Consult your calculator manual for how to switch to this mode. ◆◆◆ Example 6: Use your calculator to verify the following to four decimal places: (a) sin 2.83 rad 0.3066 (b) cos 1.52 rad 0.0508 (c) tan 0.463 rad 0.4992 ◆◆◆ To find the cotangent, secant, or cosecant, we use the reciprocal relations, just as when working in degrees. ◆◆◆ Example 7: Find sec 0.733 rad to three decimal places. Solution: We put the calculator into radian mode. The reciprocal of the secant is the cosine, so we take the cosine of 0.733 rad and get cos 0.733 rad 0.7432 Then taking the reciprocal of 0.7432 gives 1 1 sec 0.733 1.346 cos 0.733 0.7432 ◆◆◆ ◆◆◆ Example 8: Use your calculator to verify the following to four decimal places: (a) csc 1.33 rad 1.0297 (b) cot 1.22 rad 0.3659 (c) sec 0.726 rad 1.3372 ◆◆◆ The Inverse Trigonometric Functions The inverse trigonometric functions are found the same way as when working in degrees. Just be sure that your calculator is in radian mode. As with the trigonometric functions, some calculators require that you press the function key before entering the number, and some require that you press it after entering the number. ◆◆◆ Example 9: Use your calculator to verify the following, in radians, to four decimal places: (a) arcsin 0.2373 0.2396 rad (b) cos1 0.5152 1.0296 rad (c) arctan 3.246 1.2720 rad ◆◆◆ To find the arccot, arcsec, and arccsc, we first take the reciprocal of the given function and then find the inverse function, as shown in the following example. ◆◆◆ Remember that the inverse trigonometric function can be written in two different ways. Thus the inverse sine can be written arcsin or 1 sin Example 10: If cot1 2.745, find in radians to four decimal places. Solution: If the cotangent of is 2.745, then cot 2.745 so 1 tan Also recall that there are infinitely many angles that have a particular value of a trigonometric function. Of these,we are finding just the smallest positive angle. 2.745 Taking reciprocals of both sides gives 1 tan 0.3643 2.745 Section 16–1 ◆ 429 Radian Measure So tan1 0.3643 0.3494 ◆◆◆ To find the trigonometric function of an angle in radians expressed in terms of using the calculator, it is necessary first to convert the angle to decimal form. ◆◆◆ Example 11: Find cos(5/12) to four significant digits. Solution: Converting 5/12 to decimal form, we have 5()/12 1.3090 With our calculator in radian mode, we then take the cosine. cos 1.3090 rad 0.2588 ◆◆◆ ◆◆◆ Example 12: Evaluate to four significant digits: 5 cos(2/5) 4 sin2(3/7) Solution: From the calculator, 2 cos 0.309 02 and 5 3 sin 0.974 93 7 so 3 2 5 cos 4 sin2 5(0.309 02) 4(0.974 93)2 7 5.347 5 ◆◆◆ Areas of Sectors and Segments Sectors and segments of a circle (Fig. 16–3) were defined in Sec. 6–4. Now we will compute their areas. The area of a circle of radius r is given by r2, so the area of a semicircle, of course, is r2/2; the area of a quarter circle is r2/4; and so on. The sector area is the same fractional part of the whole area as the central angle is of the whole. Thus if the central angle is 1/4 revolution, the sector area is also 1/4 of the total circle area. If the central angle (in radians) is /2 revolution, the sector area is also /2 the total circle area. So area of sector r2 p q 2 Area of a Sector ◆◆◆ r2 Area 2 where is in radians h r Sector FIGURE 16–3 Segment and sector of a circle. 116 Example 13: Find the area of a sector having a radius of 8.25 m and a central angle of 46.8. Solution: We first convert the central angle to radians. 46.8( rad/180) 0.8168 rad Then, by Eq. 116, (8.25)2 (0.8168) area 27.8 m2 2 Segment ◆◆◆ 430 Chapter 16 ◆ Radian Measure, Arc Length, and Circular Motion The area of a segment of a circle (Fig. 16–3) is rh Area r2 arccos (r h) 兹2rh h2 r Area of a Segment rh where arccos is in radians r An alternate equation for calculating the area of a segment is: r2( sin ) Area 2 Area of a Segment (alternate equation) [ ] (r h) where 2 arccos (in radians) r ◆◆◆ Example 14: Compute the area of a segment having a height of 10.0 cm in a circle of radius 25.0 cm. Solution: Substituting into the given equation gives 25.0 10.0 Area (25.0)2 arccos (25.0 10.0) 兹 2(25.0)(10.0) (10.0)2 25.0 625 arccos 0.600 15.0 兹500 _ 100 625(0.9273) 15.0(20.0) 280 cm2 The same solution can be found using the alternative equation for the area of a segment: First, find 2 arccos [(25.0 10.0)/25.0] 106.3 106.3 (/180) rad 1.855 rad Then, substituting into the equation r2( sin )/2, we obtain: 2 2 Area 25.0 _ (1.855 sin 106.5)/2 279.7 cm 280 cm2 Exercise 1 ◆ Radian Measure Convert to radians. 1. 47.8 4. 0.370 rev 2. 18.7 5. 1.55 rev 3. 3515 6. 1.27 rev 8. 2.30 rad 11. 1.12 rad 9. 3.12 rad 12. 0.766 rad 14. 4.275 rad 17. 1.14 rad 15. 0.372 rad 18. 0.116 rad Convert to revolutions. 7. 1.75 rad 10. 0.0633 rad Convert to degrees (decimal). 13. 2.83 rad 16. 0.236 rad ◆◆◆ Section 16–1 ◆ 431 Radian Measure Convert each angle given in degrees to radian measure in terms of . 19. 60 20. 130 21. 66 22. 240 23. 126 24. 105 25. 78 26. 305 27. 400 28. 150 29. 81 30. 189 Convert each angle given in radian measure to degrees. 2 31. 32. 8 3 3 34. 35. 5 9 5 7 38. 37. 8 9 6 40. 41. 7 12 9 33. 11 4 36. 5 2 39. 15 8 42. 9 Calculator Use a calculator to evaluate to four significant digits. 44. tan 0.442 43. sin 3 3 2 46. tan p q 47. cos 3 5 4 49. sec 0.355 50. csc 3 9 6 p 52. tan 53. cos q 11 5 55. cos 1.832 56. cot 2.846 58. csc 0.8163 59. arcsin 0.7263 62. arccot 1.546 61. cos1 0.2320 1 65. arctan 3.7253 64. csc 2.6263 2 68. 7 tan2 67. sin cos 9 6 6 70. sin 71. sin tan 6 8 8 6 45. cos 1.063 7 48. sin p q 8 8 51. cot 9 54. sin 1.075 57. 60. 63. 66. sin 0.6254 arccos 0.6243 sin1 0.2649 arcsec 2.8463 3 69. cos2 4 72. 3 sin cos2 9 9 73. 74. 75. 76. 77. Find the area of a sector having a radius of 5.92 cm and a central angle of 62.5. Find the area of a sector having a radius of 3.15 m and a central angle of 28.3. Find the area of a segment of height 12.4 cm in a circle of radius 38.4 cm. Find the area of a segment of height 55.4 cm in a circle of radius 122.6 cm. A weight bouncing on the end of a spring moves with simple harmonic motion according to the equation y 4 cos 25t, where y is in inches. Find the displacement y when t 2.00 s. (In this equation, the angle 25t must be in radians.) 78. The angle D (measured at the earth’s centre) between two points on the earth’s surface is found by cos D sin L1 sin L2 cos L1 cos L2 cos(M1 M2) where L1 and M1 are the latitude and longitude, respectively, of one point, and L2 and M2 are the latitude and longitude of the second point. Find the angle between Yarmouth, NS (latitude 44 N, longitude 66 W), and Whitehorse, YT (61 N, 135 W), by substituting into this equation. 432 Chapter 16 ◆ Radian Measure, Arc Length, and Circular Motion 79. A grinding machine for granite uses a grinding disk that has four abrasive pads with the dimensions shown in Fig. 16–4. Find the area of each pad. 80. A partial pulley is in the form of a sector with a cylindrical hub, as shown in Fig. 16–5. Using the given dimensions, find the volume of the pulley. Disk, 62.8 cm radius Hole, 8.50 cm radius 5.270 in. radius Hole dia. 0.500 in. 67.5˚ 55.8˚ Abrasive pad FIGURE 16–4 16–2 Hub dia. 1.250 in. 0.250 in. FIGURE 16–5 Arc Length We have seen that arc length, radius, and central angle are related to each other. In Fig. 16–6, if the arc length s is equal to the radius r, we have equal to 1 radian. For other lengths s, the angle is equal to the ratio of s to r. r s Central Angle s r where is in radians 115 The central angle in a circle (in radians) is equal to the ratio of the intercepted arc and the radius of the circle. FIGURE 16–6 Relationship between arc length, radius, and central angle. When you are dividing s by r to obtain , s and r must have the same units. The units cancel, leaving as a dimensionless ratio of two lengths. Thus the radian is not a unit of measure like the degree or inch, although we usually carry it along as if it were. We can use Eq. 115 to find any of the quantities , r, or s when the other two are known. ◆◆◆ Example 15: Find the angle that would intercept an arc of 27.0 m in a circle of radius 21.0 m. Solution: From Eq. 115, 27.0 m s 1.29 rad r 21.0 m ◆◆◆ ◆◆◆ Example 16: Find the arc length intercepted by a central angle of 62.5 in a 10.4-cm-radius circle. Solution: Converting the angle to radians, we get rad 62.5 p q 1.09 rad 180 By Eq. 115, s r 10.4 cm(1.09) 11.3 cm ◆◆◆