1. Warm - up

2. Angles and Degree Measure
Chapter 4 Sec 1
Angles and Degree Measure

3. How do you describe angles and angular movement?
Essential Question
How do you describe angles and angular movement?
Key Vocabulary:
Initial side
Terminal side
Linear speed
Angular speed

4. Standard Position
An angle in standard position has its vertex at the origin and initial side on the positive x–axis.
terminal side
initial side

5. Positively Counterclockwise
Angles that have a counterclockwise rotation have a positive measure.
90º or π/2
positive
0º or 2π
180º or π
270º or 3π/2

6. Clockwise means negative
Angles that have a clockwise rotation have a negative measure.
– 270º or –3π/2
–180º or –π
0º or –2π
Negative
–90º or – π/2

7. Coterminal Angles
Coterminal angles are angles that have the same initial and terminal side, but differ by the number of rotations.
Since one rotation equals 2π, the measures of coterminal angles differ by multiples of 2π.

8. Radian Measure
The measure of an angle is determined by the amount of rotation from the initial side to the terminal side.
One way to measure angles is in radians.

9. Since the circumference of a circle is 2πr units.
Radian…still
Since the circumference of a circle is 2πr units.

10. For the positive angle subtract 2π to obtain a coterminal angle.
Example 1
For the positive angle subtract 2π to obtain a coterminal angle.
For the negative angle add 2π to obtain a coterminal angle.

11. Complementary and Supplementary Angles
Two positive angles α and β are complementary if their sum is π/2 or 90°. Two positive angles are supplementary if their sum is π or 180°.
Find the complement and supplement of the following
complement supplement
complement supplement
Because 5π/6 is greater than π/2 it has no complement. (Remember complements are positive angles.)

12. Find (if possible) the complement and supplement of the angle.
Guided Practice
Determine two coterminal angles in radian measure (one positive and one negative) for each angle.
Find (if possible) the complement and supplement of the angle.
complement
supplement

13. A second way to measure angle is in the terms of degree, denoted by °.
One degree = 1/360
To measure angles it is convenient to mark degrees on a circle. So a full revolution is 360°, a half is 180° a quarter is 90°…
We see 360° is one revolution so 360°= 2π rad and 180° = π rad
Thus

14. Degree/Radian Conversion
30°
45°
60°
90°
180°
360°

15. Convert from degrees to radians.
Example 2
Convert from degrees to radians.
135o
540°
– 270°
Convert from radians to degrees.

16. Arc Length
Arc Length
For a circle of radius r, a central angle θ intercepts an arc of length s is given by
s = r θ Length of circular arc
where θ is measured in radians. Note that if r = 1, then s = θ, and the radian measurement of θ equal the arc length
A circle has a radius of 4 inches. Find the length of the arc intercepted by central angle of 240°.

17. Linear and Angular Speed
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 of the particle is:
Linear speed =
Moreover, if θ is the angle (in radian measure) corresponding to the arc length s, then the angular speed of the particle is
Angular speed =
Linear speed measures how fast the particle moves and angular speed measures how fast the angle changes.

18. In one revolution, the arc length traveled is
Example 4
The second hand of a clock is 10.2 centimeters long. Find the linear speed of the tip of this second hand.
In one revolution, the arc length traveled is
The time required for one revolution of the second hand is t = 1 minute or 60 seconds
So the linear speed of the tip of the second hand is

19. A 15-inch diameter tire on a car makes 9.3 revolutions per second.
Example 5
A 15-inch diameter tire on a car makes 9.3 revolutions per second.
Find the angular speed of the tire in radians per second.
Find the linear speed of the car.
Because each revolution generates 2π radians, it follows that the tire turns (9.3)(2π)= 18.6π radians per second. So the angular speed is length traveled is
The linear speed of the tire is

20. How do you describe angles and angular movement?
Essential Question
How do you describe angles and angular movement?

21. Chapter 4 Section 1 Text Book Pg 265 – 267 Show all work for credit.
Daily Assignment
Chapter 4 Section 1
Text Book
Pg 265 – 267
#5 – 21 Mod4
#31 – 55 Mod4
#71 – 87 Mod4, #76
Show all work for credit.