Pre-Calc Exam Notes 90

Pre-Calc Exam Notes 90 - 1, as in ±igure 4.2.1(a)....

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90 Chapter 4 Radian Measure §4.2 4.2 Arc Length In Section 4.1 we saw that one revolution has a radian measure of 2 π rad. Note that 2 π is the ratio of the circumference (i.e. total arc length) C of a circle to its radius r : Radian measure of 1 revolution = 2 π = 2 π r r = C r = total arc length radius Clearly, that ratio is independent of r . In general, the radian measure of an angle is the ratio of the arc length cut off by the corresponding central angle in a circle to the radius of the circle, independent of the radius. To see this, recall our formal deFnition of a radian: the central angle in a circle of radius r which intercepts an arc of length r . So suppose that we have a circle of radius r and we place a central angle with radian measure 1 on top of another central angle with radian measure
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Unformatted text preview: 1, as in ±igure 4.2.1(a). Clearly, the combined central angle of the two angles has radian measure 1 + 1 = 2, and the combined arc length is r + r = 2 r . r r 1 1 2 r (a) 2 radians r /2 r /2 1/2 1 r (b) 1 2 radian Figure 4.2.1 Radian measure and arc length Now suppose that we cut the angle with radian measure 1 in half, as in ±igure 4.2.1(b). Clearly, this cuts the arc length r in half as well. Thus, we see that Angle = 1 radian ⇒ arc length = r , Angle = 2 radians ⇒ arc length = 2 r , Angle = 1 2 radian ⇒ arc length = 1 2 r , and in general, for any θ ≥ 0, Angle = θ radians ⇒ arc length = θ r , so that θ = arc length radius ....
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This note was uploaded on 01/21/2012 for the course MAC 1130 taught by Professor Dr.cheun during the Fall '11 term at FSU.

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