Pre-Calc Exam Notes 91

Pre-Calc Exam Notes 91 - In a circle of radius r = 10 ft...

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Arc Length Section 4.2 91 Intuitively, it is obvious that shrinking or magnifying a circle preserves the measure of a central angle even as the radius changes. The above discussion says more, namely that the ratio of the length s of an intercepted arc to the radius r is preserved, precisely because that ratio is the measure of the central angle in radians (see Figure 4.2.2). O r s = r θ θ (a) Angle θ , radius r O r s = r θ θ (b) Angle θ , radius r Figure 4.2.2 Circles with the same central angle, different radii We thus get a simple formula for the length of an arc: In a circle of radius r , let s be the length of an arc intercepted by a central angle with radian measure θ 0. Then the arc length s is: s = r θ (4.4) Example4.3 In a circle of radius r = 2 cm, what is the length s of the arc intercepted by a central angle of measure θ = 1 . 2 rad ? Solution: Using formula (4.4), we get: s = r θ = (2)(1 . 2) = 2 . 4 cm Example4.4
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Unformatted text preview: In a circle of radius r = 10 ft, what is the length s of the arc intercepted by a central angle of measure θ = 41 ◦ ? Solution: Using formula (4.4) blindly with θ = 41 ◦ , we would get s = r θ = (10)(41) = 410 ft. But this impossible, since a circle of radius 10 ft has a circumference of only 2 π (10) ≈ 62.83 ft! Our error was in using the angle θ measured in degrees , not radians . So ±rst convert θ = 41 ◦ to radians, then use s = r θ : θ = 41 ◦ = π 180 · 41 = 0.716 rad ⇒ s = r θ = (10)(0.716) = 7.16 ft Note that since the arc length s and radius r are usually given in the same units, radian measure is really unitless, since you can think of the units canceling in the ratio s r , which is just θ . This is another reason why radians are so widely used....
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