1. Radian and Degree Measure
6-1 & 6-2
Radian and Degree Measure

2. Objectives: Describe angles Use radian measure Use degree measure
Use angles to model and solve real-life problems

3. Angles

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. An angle is determined by rotating a ray (half-line) about its endpoint.

5. Angles
The starting position of the ray is the initial side of the angle, and the position after rotation is the terminal side, as shown.
Angle

6. Angles
The endpoint of the ray is the vertex of the angle. This perception of an angle fits a coordinate system in which the origin is the vertex and the initial side coincides with the positive x-axis. Such an angle is in standard position, as shown.
Angle
Angle in standard position

7. Angles
Counterclockwise rotation generates positive angles and clockwise rotation generates negative angles. Angles are labeled with Greek letters such as  (alpha),  (beta), and  (theta), as well as uppercase letters A, B, and C. Note that angles  and  have the same initial and terminal sides. Such angles are coterminal.
Coterminal angles

8. 120° –210° Positive Angle- rotates counter-clockwise (CCW)
Angle describes the amount and direction of rotation
120°
–210°
Positive Angle- rotates counter-clockwise (CCW)
Negative Angle- rotates clockwise (CW)

9. Angles - Review
Angle = determined by rotating a ray (half-line) about its endpoint. Initial Side = the starting point of the ray Terminal Side = the position after rotation Vertex = the endpoint of the ray Positive Angles = generated by counterclockwise rotation Negative Angles = generated by clockwise rotation
Angle
Angle in standard position

10. Radian Measure

11. Radian Measure
Measure of an Angle = determined by the amount of rotation from the initial side to the terminal side (one way to measure angle is in radians) Central Angles = an angle whose vertex is the center of the circle
Central Angle

12. Radian Measure
You determine the measure of an angle 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.
Arc length = radius when  = 1 radian

13. arc length is also r (s=r)
Radian Measure
Given a circle of radius (r) with the vertex of an angle as the center of the circle, if the arc length (s) formed by intercepting the circle with the sides of the angle is the same length as the radius (r), the angle measures one radian.
terminal side
arc length is also r (s=r) r s r
initial side
This angle measures 1 radian
radius of circle is r

14. Radian Measure
The radian measure of a central angle θ is obtained by dividing the arc length s by r (θ = s/r).
Because the circumference of a circle is 2 r units, it follows that a central angle of one full revolution (counterclockwise) corresponds to an arc length of s = 2 r.
Therefore, a full circle measures 2𝜋𝑟 𝑟 , or 2𝜋 radians.

15. Radian Measure
Moreover, because 2  6.28, there are just over six radius lengths in a full circle, as shown. Because the units of measure for s and r are the same, the ratio s / r has no units—it is a real number.

16. Radian Measure
Because the measure of an angle of one full revolution is s/r = 2 r/r = 2 radians, you can obtain the following.
180 ̊
90 ̊
60 ̊

17. Radian Measure These and other common angles are shown below. 30 ̊
45 ̊
60 ̊
90 ̊
180 ̊
360 ̊

18. Radian Measure
We know that the four quadrants in a coordinate system are numbered I, II, III, and IV. The figure to the right shows which angles between 0 and 2 lie in each of the four quadrants. Note that angles between 0 and  / 2 are acute angles and angles between  / 2 and  are obtuse angles.

19. Comments on Radian Measure
A radian is an amount of rotation that is independent of the radius chosen for rotation
For example, all of these give a rotation of 1 radian:
radius of 2 rotated along an arc length of 2
radius of 1 rotated along an arc length of 1
radius of 5 rotated along an arc length of 5, etc.

20. Coterminal Angles
Two angles are coterminal if they have the same initial and terminal sides.
The angles 0 and 2𝜋 are coterminal.
You can find an angle that is coterminal to a given angle θ by adding or subtracting 2𝜋.
For negative angles subtract 2𝜋 and for positive angles add 2𝜋.

21. Coterminal Angles
Two angles are coterminal when they have the same initial and terminal sides. For instance, the angles 0 and 2 are coterminal, as are the angles  / 6 and 13 / 6. You can find an angle that is coterminal to a given angle  by adding or subtracting 2 (one revolution), as demonstrated in Example 1. A given angle  has infinitely many coterminal angles. For instance,  =  / 6 is coterminal with where n is an integer.

22. Example 1 – Finding Coterminal Angles
a. For the positive angle 13 / 6, subtract 2 to obtain a coterminal angle

23. Example 1 – Finding Coterminal Angles
cont’d
b. For the negative angle –2 / 3, add 2 to obtain a coterminal angle

24. Example 2 – Finding Coterminal Angles
Find a positive coterminal angle to 20º
Find a negative coterminal angle to 20º
Find 2 coterminal angles to (one positive and one negative).

25. Your Turn:
Find two Coterminal Angles (+ and -)
positive
negative
25𝜋 6
− 11𝜋 6
11𝜋 4
− 5𝜋 4
4𝜋 3
− 8𝜋 3

26. Complementary & Supplementary Angles
Two positive angles  and  are complementary (complements of each other) when their sum is  / 2. Two positive angles are supplementary (supplements of each other) when their sum is .
Complementary angles
Supplementary angles

27. Example:
Complementary Angles: Two angles whose sum is 90 or (𝜋/2) radians.
Supplementary Angles: Two angles whose sum is 180 or 𝜋 radians.

28. Your Turn: Complement 𝜋 10 Supplement 3𝜋 5 Supplement 𝜋 5
Two positive angles are complementary if their sum is 𝜋/2. Two positive angles are supplementary if their sum is 𝜋. Find the Complement and Supplement
Complement 𝜋 10
Supplement 3𝜋 5
Supplement 𝜋 5
Complement - None

29. Degree Measure
A second way to measure angles is in degrees, denoted by the symbol . A measure of one degree (1) is equivalent to a rotation of of a complete revolution about the vertex. To measure angles, it is convenient to mark degrees on the circumference of a circle.

30. 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.
From the latter equation, you obtain
and
which lead to the conversion rules in the next slide.

31. Degree Measure

32. Degree Measure
When no units of angle measure are specified, radian measure is implied. For instance,  = 2, implies that  = 2 radians.

33. Example – Converting from Degrees to Radians
a. b.
Multiply by  rad / 180.
Multiply by  rad / 180.

34. Example – Converting from Radians to Degrees
Multiply by 180/ rad.
Multiply by 180/ rad.

35. Your Turn: 135° 540° -270° Convert from degrees to radians
Convert from radians to degrees
3𝜋 4 3𝜋
3𝜋 2
−90 °
810 °
360 𝜋 ° ≈ °

36. Equivalent Angles in Degrees and Radians
6.28 2
360
1.05
60
4.71
270
.79
45
3.14 
180
.52
30
1.57
90 0
Approximate
Exact
Radians
Degrees

37. Equivalent Angles in Degrees and Radians cont.

38. A Sense of Angle Sizes
See if you can guess the size of these angles first in degrees and then in radians.
You will be working so much with these angles, you should know them in both degrees and radians.

39. Applications

40. Vocabulary
Arc
Sector
Tangent
Secant
Chord
Segment

41. Arc, Sector, Segment Arc: any unbroken part of the circumference
Measured in radians
Sector: a plane figure bounded by two radii and the included arc
Segment: a part cut off from a circle by a line, as a part of a circular area contained by an arc and its chord

42. Intercepted Arcs
An intercepted arc is the arc that is formed when segments intersect portions of a circle and create arcs. These segments in effect 'intercept' parts of the circle.

43. Major Arcs/Minor Arcs
The bigger one is major (≥π). The smaller is minor(<π).
Arc HN is the minor arc
Arc HKN is the major arc

44. Arc Lengths and Central Angles of a Circle
Given a circle of radius “r”, any angle with vertex at the center of the circle is called a “central angle”.
The portion of the circle intercepted by the central angle is called an “arc” and has a specific length called “arc length” represented by “s”.
From geometry it is know that in a specific circle the length of an arc is proportional to the measure of its central angle.
For any two central angles, and , with corresponding arc lengths and :

45. Development of Formula for Arc Length
Since this relationship is true for any two central angles and corresponding arc lengths in a circle of radius r: Let one angle be with corresponding arc length and let the other central angle be , a whole rotation, with arc length .

46. Applications
The radian measure formula,  = s / r, can be used to measure arc length along a circle.

47. Example – Finding Arc Length
A circle has a radius of 4 inches. Find the length of the arc intercepted by a central angle of 240.

48. Example – Solution
To use the formula s = r, first convert 240 to radian measure.

49. Example – Solution
cont’d
Then, using a radius of r = 4 inches, you can find the arc length to be s = r Note that the units for r determine the units for r because  is given in radian measure, which has no units.

50. Your Turn: Finding Arc Length
A circle has radius 18.2 cm. Find the length of the arc intercepted by a central angle having the following measure:

51. Your Turn: Finding Arc Length cont’d
For the same circle with r = 18.2 cm and  = 144, find the arc length convert 144 to radians

52. Your Turn:
s = rθ (s = arc length, r = radius, θ= central angle measure in radians)
A circle has a radius of 4 inches. Find the length of the arc intercepted by a central angle of 240°.
A circle has a radius of 8 cm. Find the length of the arc intercepted by a central angle of 45°
16/3𝜋 ≈ 16.8 in
2𝜋 ≈ 6.3 cm

53. Applications
A sector of a circle is the region bounded by two radii of the circle and their intercepted arc.

54. Applications

55. Example:
Find the area of the sector of a circle of radius 3 meters formed by an angle of 45 ̊. Round your answer to two decimal places.
WARNING! The angle again must be given in radians
Answer:
𝐴= 1 2 𝑟 2 𝜃
𝜃=45°= 𝜋 4
r = 3 m
𝐴= 𝜋 4
𝐴= 9𝜋 8 =3.53 𝑚 2

56. Your Turn:
Given an arc of length 4 ft and a circle of radius 7 ft, find the exact radian measure of the central angle
subtended by the arc; then find the area of the sector determined by the central angle.
Answer:
𝑠=𝑟𝜃 or 𝜃= 𝑠 𝑟 , 𝜃= 4 7
𝐴= 1 2 𝑟 2 𝜃= =14 𝑓𝑡 2

57. Applications
The formula for the length of a circular arc can help you analyze the motion of a particle moving at a constant speed along a circular path.

58. Linear and Angular Speed
Linear speed measures how fast the particle moves, and angular speed measures how fast the angle changes.
r = radius, s = length of the arc traveled, t = time, θ = angle (in radians)
(distance/time)
Ex mph, 6 ft/sec, 27 cm/min, 4.5 m/sec
(turn/time)
Ex. 6 rev/min, 360°/day, 2π rad/hour

59. Example
A bicycle wheel with a radius of 12 inches is rotating at a constant rate of 3 revolutions every 4 seconds.
a) What is the linear speed of a point on the rim of this wheel?

60. Example
A bicycle wheel with a radius of 12 inches is rotating at a constant rate of 3 revolutions every 4 seconds.
b) What is the angular speed of a point on the rim of this wheel?

61. Your Turn
In 17.5 seconds, a car covers an arc intercepted by a central angle of 120˚ on a circular track with a radius of 300 meters.
a) Find the car’s linear speed in m/sec.
b) Find the car’s angular speed in radians/sec.

62. Picture it…
300 m

63. Solution
(a) Linear speed:
Note: θ must be expressed in radians

64. Solution
(b) Angular speed:
ω ≈ 0.12 radians/sec

65. Example Solution = 1256 radians/s
A race car engine can turn at a maximum rate of rpm. (revolutions per minute).
What is the angular velocity in radians per second.
Solution
a) Convert rpm to radians per second
= 1256 radians/s

66. Your Turn:
The second hand of a clock is 8.4 centimeters long. Find the linear speed of the tip of this second hand as it passes around the clock face.
A lawn roller with a 12-inch radius makes 1.6 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.
1.6 𝑟𝑒𝑣 𝑠𝑒𝑐 = 2𝜋 𝑟𝑎𝑑 𝑟𝑒𝑣 =3.2𝜋 𝑟𝑎𝑑 𝑠𝑒𝑐
𝑟𝜃 𝑡 = (12 𝑖𝑛)(3.2𝜋 𝑟𝑎𝑑) 1 𝑠𝑒𝑐 =38.4𝜋 𝑖𝑛 𝑠𝑒𝑐 =6.85 𝑚𝑖 ℎ𝑟
𝑟𝜃 𝑡 = 8.4𝑐𝑚 2𝜋 1𝑚𝑖𝑛 =16.8 𝑐𝑚 𝑚𝑖𝑛