Pre-Calc Exam Notes 92

Pre-Calc Exam Notes 92 - 92 Chapter 4 Radian Measure 4.2...

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92 Chapter 4 Radian Measure §4.2 Example 4.5 A central angle in a circle of radius 5 m cuts off an arc of length 2 m. What is the measure of the angle in radians? What is the measure in degrees? Solution: Letting r = 5 and s = 2 in formula (4.4), we get: θ = s r = 2 5 = 0.4 rad In degrees, the angle is: θ = 0.4 rad = 180 π · 0.4 = 22.92 For central angles θ > 2 π rad, i.e. θ > 360 , it may not be clear what is meant by the inter- cepted arc, since the angle is larger than one revolution and hence “wraps around” the circle more than once. We will take the approach that such an arc consists of the full circumference plus any additional arc length determined by the angle. In other words, formula (4.4) is still valid for angles θ > 2 π rad. What about negative angles? In this case using s = r θ would mean that the arc length is negative, which violates the usual concept of length. So we will adopt the convention of only using nonnegative central angles when discussing arc length.
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