1. 4.1 Radian and Degree measure
Changing Degrees to Radians
Linear speed
Angular speed

2. Definition of an angle
An angle is made from two rays with a common initial point.
In standard position the initial side is on the x axis

3. Positive angle vs. Negative angle
Positive angles are Counter clockwise C.C.W.
Negative angles are Clockwise C.W.

4. Angles with the same initial side and terminal side are coterminal.

6. The measure of an angle is from initial side to terminal side
Vertex at the origin (Center)

7. Definition of a Radian
Radian is the measure of the arc of a unit circle.
Unit circle is a circle with a radius of 1.

8. The quadrants in terms of Radians
What is the circumference of a circle with radius 1?

9. The quadrants in terms of Radians
What is the circumference of a circle with radius 1?

10. The quadrants in terms of Radians
The circumference can be cut into parts.

11. The quadrants in terms of Radians
The circumference can be cut into parts.

12. Find the Coterminal Angle
Since equals 0. it can be added or subtracted from any angle to find a coterminal angle.
Given

13. Complementary Angles – two angles are complementary
if their sum is 90 degrees or
Supplementary Angles have a sum of 180 degrees or
Find the complementary and supplementary angles for

14. Radian vs. Degree measurements
360º =
180º =
So or

15. Radian vs. Degree measurements
360º =
180º =
So or
To convert Degrees into Radians multiply by
To convert Radians into Degrees multiply by

16. Conversions: Radians Degrees
To convert degrees to radians, multiply by
To convert radians to degrees, multiply by
Converting an angle from to decimal form.

17. Change 140º to Radians Change to degrees
Use degree to rads.
Use rads to degrees

18. How to use radian to find Arc length
The geometry way was to find the circumference of the circle and multiply by the fraction. Central angle
360º
In degrees Are length called S would be

19. How to use radian to find Arc length
In degrees Are length called S would be
In radian the equation is

20. r = 9, θ = 215º Changing to rads
Are length S

21. Linear speed and Angular speed
Linear speed is
Angular speed is
Assuming “constant speed”

22. Consider a particle moving at a constant speed along a circular arc of
Linear and Angular Speeds
Consider a particle moving at a constant speed along a circular arc of
radius r. If s is the length of the arc traveled in time t, then the linear
speed v of the particle is
Linear speed v
Moreover, if
is the angle (in radian measure) corresponding to the arc
length s, then the angular speed
(the lowercase Greek letter omega)
of the particle is
Angular speed
A relationship between linear speed and angular speed is

23. where is measured in radians
Area of a Sector of a Circle
where is measured in radians
A sprinkler on a golf course fairway is set to spray water over a distance
of 70 feet and rotates through an angle of 120 degrees. Find the area of the
fairway watered by the sprinkler.
120o = how many radians?
70 ft
79-99 odd, 107

24. Tip of the second hand as it passes around the clock face.
Finding Linear Speed
The second hand of a clock is 10.2 cm long. Find the linear speed of the
Tip of the second hand as it passes around the clock face.
Linear speed v
Arc length s
=1.068 cm/sec