1. Angles and Their Measurements
MAT 200
Angles and Their Measurements
Prepared By:
Still John F. Reyes

2. Rectangular Coordinate System
Definitions
X-axis – the horizontal axis
Y –axis – the vertical axis
Rectangular coordinate system: where the x-axis and the y-axis intersect
Ordered pair: identifies the location of a point on the rectangular coordinate system
(x coordinate, y coordinate)
Origin: (0,0) – intersection of the x and y axis
Quadrant: a section of the rectangular coordinate system

4. How to graph a point? First Locate the origin
Second, graph your x -coordinate:
IF the value of your x coordinate is positive move to the right, if negative move to the left
Third, graph your y -coordinate:
IF the value of your y- coordinate is positive move up, if negative move down

5. How to identify the quadrant?
First – locate where your point is
Second – check your graph on which quadrant it is in.

6. In this section, we will study the following topics:
MAT 200
In this section, we will study the following topics:
Terminology used to describe angles
Degree measure of an angle
Radian measure of an angle
Converting between radian and degree measure
Coterminal angles
Angles in a triangle

7. Angles
Angle Measure

8. Standard Position Vertex at origin
The initial side of an angle in standard position is always located on the positive x-axis.

9. Positive and negative angles
When sketching angles, always use an arrow to show direction.

10. Measuring Angles
The measure of an angle is determined by the amount of rotation from the initial side to the terminal side.
There are two common ways to measure angles, in degrees and in radians.
We’ll start with degrees, denoted by the symbol º.
One degree (1º) is equivalent to a rotation of of one revolution.

11. Measuring Angles

12. Classifying Angles
Angles are often classified according to the quadrant in which their terminal sides lie.
Ex: Name the quadrant in which each angle lies.
50º
208º II I
-75º III IV
Quadrant 1
Quadrant 3
Quadrant 4

13. Classifying Angles
Standard position angles that have their terminal side on one of the axes are called quadrantal angles.
For example, 0º, 90º, 180º, 270º, 360º, … are quadrantal angles.

14. Angles  and  are coterminal.
Coterminal Angles
Angles that have the same initial and terminal sides are coterminal.
Angles  and  are coterminal.

15. Example of Finding Coterminal Angles
You can find an angle that is coterminal to a given angle  by adding or subtracting multiples of 360º.
Ex: Find one positive and one negative angle that are coterminal to 112º.
For a positive coterminal angle, add 360º : 112º + 360º = 472º
For a negative coterminal angle, subtract 360º: 112º - 360º = -248º

16. Degree System of Angular Measure
1° (one degree) = 60’ (60 minutes) 1’ (one minute) = 60” (60 seconds) 360° (360 degrees) = 1 revolution

17. There are two ways of measuring angles:
Degrees, Minutes, Seconds (DMS)
Decimal Degrees
Examples:
Express each angle measure using DMS
1) 10.5° 2) 42.72°
Express the ff. in decimal degrees
1) 95° 45’ 45”

18. Exercises
Find one positive and one negative angle that is coterminal with the angle  = 30° in standard position.
Find one positive and one negative angle that is coterminal with the angle  = 272 in standard position.
Express 27.3° using DMS.
Express 28° 18’ 36” in decimal degrees.

19. Do Worksheet 1

20. Radian Measure A second way to measure angles is in radians.
Definition of Radian:
One radian is the measure of a central angle  that intercepts arc s equal in length to the radius r of the circle.
In general,

21. Radian Measure

22. Radian Measure

23. To convert degrees to radians, multiply degrees by
Conversions Between Degrees and Radians
To convert degrees to radians, multiply degrees by
To convert radians to degrees, multiply radians by

24. Ex. Convert the degrees to radian measure.
60
30

25. Ex. Convert the radians to degrees.
a) b)
rad
rad

26. Ex. Find one positive and one negative angle that is coterminal with the angle  = in standard position.
Ex. Find one positive and one negative angle that is coterminal with the angle  = in standard position.

27. Interior Angles of a Triangle
The sum of the interior angles of any triangle is 180°.
In a right triangle, the sum of the acute angles is 90°. The two angles are said to be complementary angles.
Two angles with a sum of 180° is said to be supplementary angles.

28. Examples Given: ∆ABC, if angle A=95°, angle B=45°, find angle C.
Given: ∆ABC is a right triangle and angle B=46° 20’, find angle A.

29. Exercises

30. Applications of Angle Measures

31. Arc Length and Central Angles

32. Example
Find the measure of a rotation in radians when a point 2 m from the center of rotation travels 4 m.

33. Example
Find the length of an arc of a circle of radius 5 cm associated with an angle of /3 radians.

34. Linear & Angular Velocity
Things that turn have both a linear velocity and an angular velocity.

35. Linear Velocity is distance/time:
Definition:
Linear Velocity is distance/time:
Distance
Linear Speed
Time

36. Angular Velocity is turn/time:
Definition:
Angular Velocity is turn/time:
Rotation in radians
Angular Speed (omega)
Time

37. Linear & Angular Velocity
Definition of Linear Velocity:
Recall Arc Length Formula
Linear Velocity in terms of Angular Velocity:

38. Let us take 2 pendulums hung on a slim rotating rod for analysis.
If the 2 pendulums (A and B) rotate one full cycle, the time taken by them is the same. They covered the same amount of angular distance (360 degree) within the same amount of time. This showed that they have exactly the SAME angular speed. But is the Linear speed the same?

39. Let us take 2 pendulums hung on a slim rotating rod for analysis.
The length of the 2 circumferences travelled by the individual pendulums are not the same. The linear length or distance is therefore NOT the same. Length = 2 x (pi) x radius = 2πr They took the same time to complete one full cycle, though. The linear speed is thus DIFFERENT, having travelled different length for the same amount of time.

40. Example:
A satellite traveling in a circular orbit approximately 1800 km. above the surface of Earth takes 2.5 hrs. to make an orbit. The radius of the earth is approximately 6400 km.
a) Approximate the linear speed of the satellite in kilometers per hour.
b) Approximate the distance the satellite travels in 3.5 hrs.

41. Example:
r = = 8200
t = 2.5 hrs.
a) Approximate the linear speed of the satellite in kilometers per hour.

42. Example:
r = = 8200
t = 2.5 hrs.
b) Approximate the distance the satellite travels in 3.5 hrs.

43. A small pulley 6 cm in diameter is connected by a belt to a larger pulley 15cm in diameter. The small pulley is turning at 120 revolutions/rotations per minute (rpm).
Find the angular velocity of the small pulley in radians per second.
Find the linear velocity of the rim of the small pulley.

44. A small pulley 6 cm in diameter is connected by a belt to a larger pulley 15cm in diameter. The small pulley is turning at 120 rpm.
Find the angular velocity of the small pulley in radians per second.

45. A small pulley 6 cm in diameter is connected by a belt to a larger pulley 15cm in diameter. The small pulley is turning at 120 rpm.
Find the linear velocity of the rim of the small pulley.

46. Exercises
Find the length of the arc intercepted by a central angle of measure 2.5 radians in a circle whose radius is 15 cm.
A central angle of 38° intercepts an arc of 5 meters. Find the radius of the circle.
A flywheel 6 ft. in diameter makes 40 rpm.
Find its angular velocity in radians per second.
Find the speed of the belt that drives the flywheel.

47. Do Worksheet 2