SECTION 6.1 Angles, Arc Length, and Circular Motion 389 Converting from Radians to Degrees Convert each angle in radians to degrees. (a) π 6 radian (b) π3 2 radians (c) π − 7 4 radians (d) π7 3 radians (e) 3 radians Solution (a) π π π π = ⋅ = ⋅ = ° 6 radian 6 1 radian 6 180 degrees 30 (b) π π π = ⋅ = ° 3 2 radians 3 2 180 degrees 270 (c) π π π − = − ⋅ = − ° 7 4 radians 7 4 180 degrees 315 (d) π π π = ⋅ = ° 7 3 radians 7 3 180 degrees 420 (e) π = ⋅ ≈ ° 3 radians 3 180 degrees 171.89 Now Work PROBLEM 35 Table 1 lists the degree and radian measures of some common angles.You should learn to feel equally comfortable using either measure. EXAMPLE 5 Degrees 0° 30° 45° 60° 90° 120° 135° 150° 180° Radians 0 6 π 4 π 3 π 2 π 2 3 π 3 4 π 5 6 π π Degrees 210° 225° 240° 270° 300° 315° 330° 360° Radians 7 6 π 5 4 π 4 3 π 3 2 π 5 3 π 7 4 π 11 6 π 2π Table 1 Field Width of a DSLR Camera Lens For small angles, the length of the arc subtended by a central angle is approximately equal to the length of the chord that is subtended. Use this fact to approximate the field width (the width of scenery the lens can image) of a 400mm camera lens at a distance of 750 feet if the viewing angle of the lens is 6 12 . ° ′ Solution See Figure 15.The measure of the central angle is 6 12 , ° ′ but remember that the angle in formula (4) must be in radians. So begin by converting the angle to radians. 6 12 6.2 6.2 180 radian 0.108 radian θ π = ° ′ = ° = ⋅ ≈ ↑ 12 0.2 ′ = ° Use 0.108 θ = radian and r 750 = feet in formula (4). The field width for the lens at a distance of 750 feet from the camera is approximately s r 750 0.108 81feet θ = = ⋅ = EXAMPLE 6 Figure 15 Camera viewing angle Viewing angle Field width Distance to object NOTE If the measure of an angle is given as 5, it is understood to mean 5 radians; if the measure of an angle is given as 5 ,° it means 5 degrees. j When an angle is measured in degrees, the degree symbol is always shown. However, when an angle is measured in radians, we usually omit the word radians. So if the measure of an angle is given as 6 , π it is understood to mean 6 π radian. Now Work PROBLEM 107
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