
# 4: Radian Measure


• 4.1: Radians and Degrees
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: $$ω = \frac{θ}{ t}$$.
• 4.E: Radian Measure (Exercises)
These are homework exercises to accompany Corral's "Elementary Trigonometry" Textmap. This is a text on elementary trigonometry, designed for students who have completed courses in high-school algebra and geometry. Though designed for college students, it could also be used in high schools. The traditional topics are covered, but a more geometrical approach is taken than usual. Also, some numerical methods (e.g. the secant method for solving trigonometric equations) are discussed.

Thumbnail: Angle $$θ$$ and intercepted arc $$\overparen{AB}$$ on circle of circumference $$C = 2πr$$.