# 4: Radian Measure

- Page ID
- 3222

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

- 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 \(C = 2πr\).*

## Contributors

Michael Corral (Schoolcraft College). The content of this page is distributed under the terms of the GNU Free Documentation License, Version 1.2.