SECTION 6.1 Angles, Arc Length, and Circular Motion 387 4 Convert from Degrees to Radians and from Radians to Degrees Because there are two ways to measure angles, it is important to be able to convert from one measure to the other. To help understand the relationship between the degree measure of an angle and radian measure of an angle, consider Figures 14(a), (b), and (c). In Figure 14(a), a circle of radius r 1 = is drawn where the measure of the central angle is 1 θ = radian.Therefore, the length of the arc subtended on AB is s 1 = unit. In Figure 14(b), a circle of radius r 1 = is drawn where the measure of the central angle is 2 θ = radians, so the length of the arc subtended on AC is s 2 = units. Finally, in Figure 14(c), a circle of radius r 1 = is drawn where the measure of the central angle is 3 θ = radians, so the length of the arc subtended on AD is s 3 = units. Notice that the arc AD almost reaches the negative x-axis.The angle whose initial side is the positive x-axis and whose terminal side is the negative x-axis is 180 . θ = ° This suggests that the measure of an angle, in radians, where the initial side is the positive x-axis and terminal side is the negative x-axis is a little bigger than 3. NOTE Formulas must be consistent with the units used. In formula (4), we write s rθ = To see the units, use equation (3) and write θ θ = = s r s r radians 1 radian length units length units length units length units radians 1 radian The radians divide out, leaving θ ( ) = s r length units length units θ = s r where θ appears to be “dimensionless” but, in fact, is measured in radians. So, in the formula s r ,θ = the dimension for θ is radians, and any convenient unit of length (such as inches or meters) can be used for s and r. j Now Work PROBLEM 71 (b) s = 2 C A u = 2 21 1 1 21 0 (c) s = 3 D A u = 3 21 1 1 21 0 (d) π = 1revolution 2 radians 1 revolution s = 2pr r Figure 14 B s = 1 A u = 1 21 1 1 21 0 (a) Figure 14(d) shows a circle of radius r. A central angle of 1 revolution subtends an arc equal to the circumference of the circle. Because the circumference of a circle of radius r equals r 2 , π we substitute r 2π for s in formula (4) to find that, for an angle θ of 1 revolution, s r r r 2 2 radians θ π θ θ π = = = s r 1revolution; 2 θ π = = Solve for .θ Finding the Length of an Arc of a Circle Find the length of the arc of a circle of radius 2 meters subtended by a central angle of 0.25 radian. Solution EXAMPLE 3 Use formula (4) with r 2 = meters and 0.25. θ = The length s of the arc is s r 2 0.25 0.5 meter θ = = ⋅ =
RkJQdWJsaXNoZXIy NjM5ODQ=