Measurement of Angles

1
Section 7-1

• Measurement of Angles

2
Trigonometry

• The word trigonometry comes two Greek words,
  trigon and metron, meaning triangle measurement.

3
Trigonometry

• In trigonometry, an angle often represents a
  rotation about a point. Thus, the angle T shown
  is the result of rotating its initial ray to its
  terminal ray.

4
Revolutions and Degrees

• A common unit for measuring very large angles is
  the revolution, a complete circular motion.
• A common unit for measuring smaller angles is the
  degree, of which there are 360 in one revolution

5
Degrees, Minutes, and Seconds

• Angles can be measured more precisely by dividing
  one degree into 60 minutes and by dividing one
  minute into 60 seconds. For example, an angle of
  25 degrees, 20 minutes, and 6 seconds is written
  25206.

6
Decimal Degrees

• Angles can also be measured in decimal degrees.
  To convert between decimal degrees and degrees,
  minutes, and seconds, you can reason as follows
• 12.312 0.3(60) 1218
• 2520625

7
Radians

• Relatively recently in mathematical history,
  another unit of angle measurement, the radian,
  has come into use. When an arc of a circle has
  the same length as the radius of the circle, the
  measure of the central angle , is by
  definition 1 radian.

8
Radians

• Likewise, a central angle has a measure of 1.5
  radians when the length of the intercepted arc is
  1.5 times the radius.

9
Radian Measures

• In general, the radian measure of the central
  angle is the number of radius units in
  the length of arc AB. This accounts for the name
  radian. In the diagram at the right, the measure
  (Greek Theta) of the central angle is

10
Arc Length

• Arc length s r

11
Radians

• Let us use this equation to see how many radians
  correspond to 1 revolution. Since the arc length
  of 1 revolution is the circumference of the
  circle, 2?r, we have
  
• . Thus, 1 revolution
  measured in radians is 2? and measured in degrees
  is 360. We have 2? radians 360 degrees or ?
  radians 180 degrees.

12
Conversion Formulas

• This gives us the following conversion formulas
• 1 radian degrees 57.2958 degrees

13
Conversion Formulas

• This gives us the following conversion formulas
• 1 degree radians 0.0174533 radians

14
Radians

• Angle measures that can be expressed evenly in
  degrees cannot be expressed evenly in radians,
  and vice versa. That is why angles measured in
  radians are often given as fractional multiples
  of ?.

15
Degrees vs. Radians

• Angles whose measures are multiples of
  
• appear often in trigonometry.
  The diagrams below will help you keep the degree
  conversions for these special angles in mind.
  Note that a degree measure, such as 45, is
  usually written with a degree symbol, while a
  radian measure such as is usually written
• without any symbol.

16
45

• Multiples of 45,

17
60

• Multiples of 60,

18
30

• Multiples of 30,

19
Standard Position

• When an angle is shown in a coordinate plane, it
  usually appears in standard position, with its
  vertex at the origin and its initial ray along
  the positive x-axis. We consider a
  counterclockwise rotation to be positive and a
  clockwise rotation to be negative. By positive
  and negative angles we mean angles with positive
  and negative measures.

20
Positive Angle

• An angle of 380

21
Negative Angle

• An angle of

22
Quadrantal Angles

• If the terminal ray of an angle in standard
  position lies in the first quadrant, as shown at
  the left above, the angle is said to be a
  first-quadrant angle. Second-, third-, and
  fourth-quadrant angles are similarly defined. If
  the terminal ray of an angle in standard position
  lies along an axis, as shown at the right above,
  the angle is called a quadrantal angle. The
  measure of a quadrantal angle is always a
  multiple of 90 or

23
Coterminal Angles

• Two angles in standard position are called
  coterminal angles if they have the same terminal
  ray. For any given angle there are infinitely
  many coterminal angles.

24
Example

• Convert the degree measures to radians.
• 270 -23.6

25
Example

• Convert the radian measures to degrees
• 12.3

26
Example

• Find two angles, one positive and one negative,
  that are coterminal with each given angle.
• A. 60
• B. -210
• C.
• D.

27
Example

• Find two angles, one positive and one negative,
  that are coterminal with each given angle.
• 2415 -2337

28
Additional Example

• Give the degree measure and the radian measure of
  the angle formed by the hour hand and the minute
  hand of a clock at 230 a.m.

29
Additional Example

• 2. A gear revolves at 40 rpm. Find the number of
  degrees per minute through which the gear turns
  and the approximate number of radians per minute.