1. Welcome Back to Precalculus

2. 4.1 Radian and Degree Measure
Objective: Students will use radian angle measures to model real life problems.
2015

3. What You Should Learn Describe angles Use radian measure
Use degree measure and convert between degree and radian measure
Use angles to model and solve real-life problems

4. Angles
As derived from the Greek language, the word trigonometry means “measurement of triangles.” Initially, trigonometry dealt with relationships among the sides and angles of triangles and was used in the development of astronomy, navigation, and surveying.
With the development of calculus and the physical sciences in the 17th century, a different perspective arose—one that viewed the classic trigonometric relationships as functions with the set of real numbers as their domains.

5. Angles
Consequently, the applications of trigonometry expanded to include a vast number of physical phenomena involving rotations and vibrations, including the following.
sound waves
light rays
planetary orbits
vibrating strings
pendulums
orbits of atomic particles

6. Angles
• An angle is determined by rotating a ray about its endpoint. • The measure of an angle is determined by the amount of rotation from the initial side, to the terminal side. • The shared initial point of the two rays is called the vertex of the angle.

7. Angles in standard position
• An angle is in standard position if its vertex is at the origin of the rectangular coordinate system and the initial side lies along the positive x-axis. • If the rotation of the angle is in the counterclockwise direction, then the angle is said to be positive. If the rotation is clockwise, then the angle is negative.

8. Angles
Angles are labeled with Greek letters such as  (alpha),  (beta), and (theta), as well as uppercase letters such as A,B, and C. In Figure 4.4, note that angles  and  have the same initial and terminal sides. Such angles are coterminal.
Figure 4.4

9. Radian Measure

10. 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.
This type of measure is especially useful in calculus. To define a radian, you can use a central angle of a circle, one whose vertex is the center of the circle, as shown in Figure 4.5.
Figure 4.5

40. How many radii fit around a circle?

41. Is there a more accurate way to calculate the number of radii?

42. Circumference measures the distance around a circle

43. Radians
One radian is the measure of a central angle that intercepts an arc “s” equal in length to the radius of the circle. Since the circumference of a circle is , it takes radians to get completely around the circle once. Therefore, it takes radians to get halfway around the circle.

44. Radians
The four quadrants in a coordinate system are numbered I, II, III, and IV. Figure 4.8 shows which angles between 0 and 2 lie in each of the four quadrants. Note that angles between 0 and 2 and are acute and that angles between 2 and  are obtuse.
Figure 4.8

45. Radian vs Degree Measurey x

46. Coterminal Angles
Two angles in standard position that have the same terminal side are said to be coterminal.

47. Draw an angle with the given measure in standard position
Draw an angle with the given measure in standard position. Tell which quadrant the terminal side lies in. Find one positive angle, and one negative co-terminal angle. y
Coterminal Angles
Quadrant II x

48. Coterminal Angles
Find one positive and one negative angle coterminal with the given angles:
a) b)

49. Coterminal Angles
Find one positive and one negative angle coterminal with the given angles:
a) b)

50. Complimentary and Supplementary Angles
Two positive angles  and  are complementary (complements of each other) when their sum is 90  (or 2) Two positive angles are supplementary (supplements of each other) when their sum is 180  (or  ).
Complementary angles
Supplementary angles
Figure 4.14

51. Example
Find the complement and supplement of

52. Degree Measure

53. Degree Measure
Another way to measure angles is in terms of degrees, denoted by the symbol . A measure of one degree (1) is equivalent to a rotation of of a complete revolution about the vertex.
Figure 4.12

54. Degree Measure
Figure 4.13

55. Degree Measure
So, a full revolution (counterclockwise) corresponds to 360 a half revolution to 180, a quarter revolution to 90  and so on.
Because 2 radians corresponds to one complete revolution, degrees and radians are related by the equations
360 = 2 rad and 180 =  rad.

56. Converting between degrees and radians
Since 180 =  radians, it follows that:
1  = radians and radian =
which lead to the following conversion rules.
To convert degrees to radians, multiply degrees by
To convert radians to degrees, multiply radians by

57. Converting Degrees to Radian Measure

58. Converting Radians to Degree Measure

59. You try:

60. Arc Length and Area of a Sector
central
angle

61. Find the arc length and area of the sector.
10 cm

62. You Try:
Find the length of the arc that subtends a central angle with measure 120° in a circle with radius 5 inches.

63. 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 is the angle (in radians measure) corresponding to the arc length, s, then the angular speed of the particle is Angular speed =

64. Example
A lawn roller with a 10-inch radius makes 1.2 revolutions per second. Find the angular speed of the roller in radians per second. Find the speed of the tractor that is pulling the roller.

65. Student Example – Finding Linear Speed
The second hand of a clock is 10.2 centimeters long, as shown in Figure Find the linear speed of the tip of this second hand.
Figure 4.16

66. Solution

67. Think/Pair/Share
The minute hand on the clock at Gilbert City Hall measures 2.2 meters from the tip to the axle.
a) Through what angle (in radians) does the minute hand pass between 7:05 A.M. and 7:40 A.M.?
b) What distance does the tip of the minute hand travel during this period?

68. HOMEWORK
4.1 pg ,odd odd

69. HWQ
Find one positive and one negative angle co-terminal with the given angle: