Question

determine the measure of each given angle in radians. note that all circles circumferences are broken up into arcs that are each 1 radius length.
try again. can you identify the subtended arc? for a circle centered at the angles vertex, how long is the subtended arc in comparison to the circles radius? how does this help us determine the angle measure in radians?

Explanation:

Step1: Recall the definition of radian measure

The radian measure of an angle \(\theta\) is given by \(\theta=\frac{s}{r}\), where \(s\) is the length of the arc subtended by the angle and \(r\) is the radius of the circle. When \(r = 1\) (since each arc is 1 radius length), \(\theta=s\).

Step2: Count the number of arc - length segments

In the first circle:
The angle \(\theta\) subtends \(5\) arc - length segments. Since \(r = 1\) (each arc is 1 radius length), using the formula \(\theta=\frac{s}{r}\) with \(s = 5\) and \(r=1\), \(\theta = 5\) radians.
In the second circle:
The angle \(\theta\) subtends \(3\) arc - length segments. Since \(r = 1\) (each arc is 1 radius length), using the formula \(\theta=\frac{s}{r}\) with \(s = 3\) and \(r = 1\), \(\theta=3\) radians.

Answer:

The measures of the angles are \(5\) radians and \(3\) radians.