1. Chapter 6: Trigonometry 6.3: Angles and Radian Measure
Essential Questions: How many degrees are in a circle? How many radians are in a circle?

2. 6.3: Angles and Radian Measure
In Geometry & Triangle Measurement, angles are formed by two rays that meet at an endpoint
In trigonometry, angles are formed by taking an initial ray (called the initial side) and rotating it around itself, the point after rotation being called the terminal side.
A trigonometry angle can rotate around itself multiple times
terminal
Initial

3. 6.3: Angles and Radian Measure
An angle in the coordinate plane is said to be in standard position if its vertex is at the origin (0,0) and its initial side is on the positive x-axis (going to the right)
Angles formed by rotations that have the same initial and terminal sides are called coterminal. 0º and 360º angles are coterminal, because their ending points are the same as their starting points.
Example 1: Coterminal angles
Find three angles coterminal with an angle of 60º
60º + 360º = 420º
60º - 360º = -300º
60º + 2(360º) = 780º

4. 6.3: Angles and Radian Measure
Arc Length
The length of the arc of a circle is equal to the central angle created:
Arc length can be calculated by considering an arc as a fraction of the entire circle. Since there are 360º in a full circle, the arc is of the circle. Since the circumference is 2r, the length of the arc is
arc 

5. 6.3: Angles and Radian Measure
Arc Length
Arc length:
Example 2: Finding an Angle Given an Arc Length
An arc in a circle has an arc length l which is equal to the radius r. Find the measure of the central angle that the arc intercepts.

6. 6.3: Angles and Radian Measure
There are 360º in a circle
The circumference of a circle = 2r. So if the radius of a circle were 1, then there a circle would contain 2 radians.
This gives us our conversion factor: 360º = 2 radians
Note: Dividing both sides by 2 gives: 180º =  radians
1 revolution around a circle = 2 radians
3/4 revolution: 3/4 * 2 = 3/2 radians
1/2 revolution: 1/2 * 2 =  radians
1/4 revolution: 1/4 * 2 = /2 radians

7. 6.3: Angles and Radian Measure
Radian Measure of Special Angles
We’ve dealt with three special angles so far: 30°, 45°, & 60°
30° = 360/12, so 30° = 1/12 of a circle * 2 radians = /6 radians
45° = 360/8, so 45° = 1/8 of a circle * 2 radians = /4 radians
60° = 360/6, so 60° = 1/6 of a circle * 2 radians = /3 radians
You won’t be required for homework, but it’s advisable you copy Figure on page 437 to have reference to many special angles.

8. 6.3: Angles and Radian Measure
Converting Between Degree and Radians
Use the conversion factor:  = 180°
Convert the following radian measurements to degrees
/5
/5 * 180°/ = 180/5 = 36°
4/9
4/9 * 180°/ = 720/9 = 80° 6
6 * 180°/ = 1080°
Convert the following degree measurements to radians
75°
75° * /180° = 75/180 = 5/12
220°
220° * /180° = 220/180 = 11/9
400°
400° * /180° = 400/180 = 20/9

9. 6.3: Angles and Radian Measure
Assignment
Page 441
1 - 45, odd problems

10. Chapter 6: Trigonometry 6.3: Angles and Radian Measure Day 2
Essential Questions: How many degrees are in a circle? How many radians are in a circle?

11. 6.3: Angles and Radian Measure
Arc Length
An arc with central angle measure θ radians has length: l = r • θ
The arc length is the radius times the radian measure of the central angle of the arc.
Example 5: The second hand on a clock is 6 inches long. How far does the tip of the second hand move in 15 seconds?
Answer
The second hand makes a full revolution every 60 seconds, so 60 seconds = 2π radians. Use conversion factors.
l = r • θ = (6 in)(π/2 rad) = 3π ≈ 9.4 inches

12. 6.3: Angles and Radian Measure
Central Angle Measure
Example 6: Find the central angle measure (in radians) of an arc of length 5 cm on a circle with a radius of 3 cm.
Answer:
Use the formula l = r • θ
l = 5cm, r = 3cm
5cm = 3cm • θ
5/3 = θ

13. 6.3: Angles and Radian Measure
Linear and Angular Speed
Linear speed = arc length / time =
Angular speed = angle / time =
Linear/Angular speed are simply the rate at which it takes to move from one point on the circle to another. Linear speed calculates the time to move from the end of the radius; angular speed calculates the time to change along the angle.
Note that the only difference between the two is the “r”, which means that linear speed totally depends on the distance the point is away from the center of the circle.

14. 6.3: Angles and Radian Measure
Linear and Angular Speed
Linear speed = arc length / time =
Angular speed = angle / time =
Example 7: A merry-go-round makes 8 revolutions per minute
What is the angular speed of the merry go round in radians per minute?
If one revolution = 2π radians, then 8 revolutions = 8 • 2π = 16π radians
How fast is a horse 12 feet from the center traveling?
Use the linear speed formula
r • angular speed = 12 feet • 16π = 192π feet/minute
How fast is a horse 4 feet from the center traveling?
r • angular speed = 4 feet • 16π = 64π feet/minute

15. 6.3: Angles and Radian Measure
Assignment
Page 441
, odd problems