Pre Cal Notes 6-1 Angles in Radian Measure

Most of the angles that we have been working with have been in degree measure. We are going to start a more in depth study of angles in radian measure. Some problems are easier to solve when the angle is in radian measure. We have already converted the angle measures on the unit circle to degrees and used this information to find the trig values at a given angle measure.

Recall that radian is just another unit of measure for angles. Just as distances can be measured in feet or inches, angles can be measured in radian or degree. Recall that 1 radian = and that 1 degree =

Warm Up:

  1. Convert 330 degrees to radian in terms of
  1. Convert radian to degree.
  1. Use the unit circle to find .

Do you know why one full rotation on the unit circle is equal to 2π radian?

Remember that the definition of a radian is based on the unit circle (circle with r=1, centered at the origin.) Consider an angle in standard position, α. Let P(x,y) be on the intersection of ti’s terminal side with the unit circle. The radian measure of the angle in standard position is defined as the length of the corresponding arc on the unit circle. Thus, the measure of angle α is s radians. Since , a full revolution corresponds to an angle of So since 360⁰= it follows that 180⁰=and that 90⁰= From this we gain the conversation for degrees to radian.

1 radian = 1 degree =

Radian measure can be used to find the length of a circular arc (part of a circle.) An arc is defined by a central angle that intercepts it. A central angle is an angle whose vertex lies on the center of the circle.

Theorem – If 2 central angles in different circles are congruent, then the ratio of the lengths of their intercepted arcs is equal to the ratio of the measures of their redii.

Because of this theorem, we can write a rule for finding the length of any arc.

If two concentric circles share a central angle, and the radius of the larger circle is r and the smaller circle is a unit circle, then the arc length on the unit circle is So by the above theorem:

So to find the arc length, s, we multiply the radius of the circle by the measure of the central angle (in radian.)

You try:

  1. Given a central angle of 125 degrees, find the length of the intercepted arc if r= 7 cm.
  1. The Swiss have been highly regarded as makers of watches for years. The central angle formed by the hand of the watch on “12” and “5” is 150 degrees. The radius of the watch is ¾ cm. Find the distance traveled by the end of the minute hand from “12” to “5”.
  1. An arc has a length of 24 inches and lies on a circle with a radius of 3 inches. Find the measure of the central angle in degrees.
  1. Mikayla rides her bike for 3.5 km. If the radius of the tire on her bike is 32 cm, determine the number of radians that a spot on the tire will travel during the trip.

A sector of a circle is the region bounded by a central angle and the intercepted arc. It looks like a slice of pizza. The ratio of the area of the sector to the area of the circle is equal to the ratio of the arc length to the circumference.

So the area of the sector is ½ the radius squared times the measure of the central angle in RADIAN!

You Try

  1. Find the area of the sector whose central angle is radian and the radius is 11 cm.
  1. A sector has an area of 127.2 square inches and a central angle of 100 degrees. Find the length of the radius of the circle.