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Chapter 1: Problem 3

How is the radian measure of an angle determined?

Short Answer

Expert verified
Answer: The radian measure of a 60-degree angle is pi/3 radians.

Step by step solution

01

Definition of Radians

A radian is a unit of measurement that represents the angle created when the length of the arc formed by the angle is equal to the length of the radius of the circle from which it is being measured. In other words, if we put the angle vertex at the center of the circle and measure the angle in such a way that the arc length is equal to the radius length, then that angle will be equal to one radian.
02

Radian Measure Formula

To determine the radian measure of an angle, we can use the following formula: \[ \text{Radian Measure} = \frac{\text{Arc Length}}{\text{Radius}} \] This formula states that the radian measure of an angle is equal to the ratio of the arc length formed by the angle (with its vertex at the center of the circle) to the radius of the circle.
03

Conversion between Radians and Degrees

To relate radians and degrees, we can use the following conversion formula: \[ \text{Radians} = \frac{\pi}{180} \cdot \text{Degrees} \] This formula allows us to convert between degrees and radians by multiplying the degrees by the constant factor (pi/180).
04

Example

Let's say we have an angle of 60 degrees, and we want to find its radian measure. Step 1: Convert the angle from degrees to radians using the conversion formula. \[ \text{Radians} = \frac{\pi}{180} \cdot 60\text{ degrees} \] Step 2: Simplify the expression. \[ \text{Radians} = \frac{\pi}{3} \] So, an angle of 60 degrees is equal to pi/3 radians.

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