597 6.1 Radian Measure y x 908 = P 2 608 = P 3 458 = P 4 308 = P 6 08 = 0 3308 = 11P 6 3158 = 7P 4 3008 = 5P 3 2708 = 3P 2 2408 = 4P 3 2258 = 5P 4 2108 = 7P 6 1808 = P 1508 = 5P 6 1358 = 3P 4 1208 = 2P 3 Figure 4 LOOKING AHEAD TO CALCULUS In calculus, radian measure is much easier to work with than degree measure. If x is measured in radians, then the derivative of f 1x2 = sin x is f ′1x2 = cos x. However, if x is measured in degrees, then the derivative of f 1x2 = sin x is f ′1x2 = p 180 cos x. Learn the equivalences in Figure 4. They appear often in trigonometry. y r r O P U radians x r s 1 radian T Q R Figure 5 Arc Length on a Circle The formula for finding the length of an arc of a circle follows directly from the definition of an angle u in radians, where u = s r . In Figure 5, we see that angle QOP has measure 1 radian and intercepts an arc of length r on the circle. We also see that angle ROT has measure u radians and intercepts an arc of length s on the circle. From plane geometry, we know that the lengths of the arcs are proportional to the measures of their central angles. s r = u 1 Set up a proportion. Multiplying each side by r gives s = r u. Solve for s. Arc Length The length s of the arc intercepted on a circle of radius r by a central angle of measure u radians is given by the product of the radius and the radian measure of the angle. s =r U, where U is in radians CAUTION When the formula s =r U is applied, the value of U MUST be expressed in radians, not degrees.
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