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So far we have been using degrees as our unit of measurement for angles. However, there is another way of measuring angles that is often more convenient. The idea is simple: associate a central angle of a circle with the arc that it intercepts.
• 4.2: Arc Length
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 1, as in Figure 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 = 2r.
• 4.3: Area of a Sector
In geometry you learned that the area of a circle of radius $$r$$ is $$πr^ 2$$ . We will now learn how to find the area of a sector of a circle. A sector is the region bounded by a central angle and its intercepted arc, such as the shaded region in Figure 4.3.1.
• 4.4: Circular Motion- Linear and Angular Speed
So suppose that an object moves along a circle of radius r, traveling a distance s over a period of time t, as in Figure 4.4.1. Then it makes sense to define the (average) linear speed ν of the object as: $$v=\frac{s}{t}$$. Let θ be the angle swept out by the object in that period of time. Then we define the (average) angular speed ω of the object as: $$ω = \frac{θ}{ t}$$.
Thumbnail: Angle $$θ$$ and intercepted arc $$\overparen{AB}$$ on circle of circumference $$C = 2πr$$.