Angles and their Measure

1
Angles and their Measure

• Geometric Representation of Angles

2
Definition of Angles

• Angles
• Initial Side and Standard Position

3
Angles

• Degrees One degree is 1/360 of a revolution.
• A right angle is an angle that measures 90
  degrees or ¼ revolution
• A straight angle is an angle that measures 180
  degrees or ½ revolution

4
Angles

• Drawing an Angle
• (a) 45 degrees
• (b) -90 degrees
• (c) 225 degrees
• (d) 405 degrees

5
Converting between Degrees, Minutes, Seconds and
Decimal

• 1 degree equals 60 (minutes)
• 1 (minute) equals 60 (seconds)
• Using graphing calculator to convert

6
Radians

• Definition
• Arc Length
• For a circle of radius r, a central angle of q
  radians subtends an arc whose length s is
• sr q

7
Finding the Length of a Circle

• Find the length of the arc of a circle of radius
  2 meters subtended by a central angle of 0.25
  radian.
• sr q with r 2 meters and T 0.25
• 2(0.25) 0.25 meter

8
Relationship Between Degrees and Radians

• One revolution is 2p therefore, 2pr r? (arc
  length formula)
• It follows then that 2p ? and
• 1 revolution 2p radians
• 360 degrees 2p radians
• or 180 degrees p radians
• so . . . 1 degree p/180 radian and
• 1 radian 180/p degrees

9
Converting from Degrees to Radians

• Convert each angle in degrees to radians
• (a) 60 degrees
• (b) 150 degrees
• (c) 45 degrees
• (d) 90 degrees

10
Converting Radians to Degrees

• Convert each angle in radians to degrees
• (a) p/6 radian
• (b) 3p/2 radian
• (c) -3p/4
• (d) 7p/3

11
Common Angles in Degrees and Radians

• Page 375 has common angles in degree and radian
  measures

12
Finding Distance Between two Cities

• Steps
• (1) Find the measure of the central angle between
  the two cities
• (2) Convert angle to radians
• (3) Find the arc length (remember we live on a
  sphere and the distance between two cities on the
  same latitude is actually an arc length)

13
Area of a Sector

• The area A of the sector of a circle of radius r
  formed by a central angle of ? radians is
• A ½ r2?
• Examples

14
Circular Motion

• Linear Speed
• v s/t
• Angular Speed
• ? ?/t

15
Circular Motion

• Angular Speed is usually measured in revolutions
  per minute (rpms).
• Converting to radians per minute
• Linear Speed given an Angular Speed
• v r?
• where r is the radius

16
Finding Linear Speed

• A child is spinning a rock at the end of a 2-ft
  rope at the rate of 180 rpms. Find the linear
  speed of the rock when it is released.

17
Cable Cars of San Francisco

• At the Cable Car Museum you can see four cable
  lines that are used to pull cable cars up and
  down the hills of San Francisco. Each cable
  travels at a speed of 9.55 miles per hour, caused
  by rotating wheel whose diameter is 8.5 feet. How
  fast is the wheel rotating? Express your answer
  in rpms.

18
Circular Motion

• On-line Examples
• On-line Tutorial