598 CHAPTER 6 The Circular Functions and Their Graphs EXAMPLE 3 Finding Arc Length Using s =r U A circle has radius 18.20 cm. Find the length of the arc intercepted by a central angle having each measure. (a) 3p 8 radians (b) 144° SOLUTION (a) As shown in Figure 6, r = 18.20 cm and u = 3p 8 . s = r u Arc length formula s = 18.20 a 3p 8 b Let r = 18.20 and u = 3p 8 . s ≈21.44 cm Use a calculator. (b) The formula s = r u requires that u be measured in radians. First, convert u to radians by multiplying 144° by p 180 radian. 144° = 144 a p 180b = 4p 5 radians Convert from degrees to radians. The length s is found using s = r u. s = r u = 18.20 a 4p 5 b ≈45.74 cm Let r = 18.20 and u = 4p 5 . S Now Try Exercises 67 and 71. r = 18.20 cm s 3P 8 Figure 6 Be sure to use radians for u in s = r u. EXAMPLE 4 Finding the Distance betweenTwo Cities Reno, Nevada, is approximately due north of Los Angeles. The latitude of Reno is 40° N, and that of Los Angeles is 34° N. (The N in 34° N means north of the equator.) The radius of Earth is 6400 km. Find the north-south distance between the two cities. SOLUTION As in shown Figure 7, the central angle between Reno and Los Angeles is 40° - 34° = 6°. The distance between the two cities can be found using the formula s = r u, after 6° is converted to radians. 6° = 6 a p 180b = p 30 radian The distance between the two cities is given by s. s = r u = 6400 a p 30b ≈670 km Let r = 6400 and u = p 30 . S Now Try Exercise 75. Equator 348 408 68 Los Angeles Reno s 6400 km Figure 7 Latitude gives the measure of a central angle with vertex at Earth’s center whose initial side goes through the equator and whose terminal side goes through the given location. As an example, see Figure 7.
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