1. Chapter 5 Trigonometric Functions Section 5.1 Angles and Arcs

2. Definition of an Angle An angle is formed by rotating a given ray about its endpoint to some terminal position. The original ray is the initial side of the angle, and the second ray is the terminal side of the angle. The common endpoint is the vertex of the angle.

3. Definition of an Angle There are several methods that can be used to name an angle. Figure 5.2 below shows and angle that is represented using the Greek letter, , and is designated as . It could also be named  O,  AOB, or  BOA. It is traditional to list the vertex between the other two points.

4. Definition of an Angle Angles formed by a counterclockwise rotation are considered positive angles. Angles formed by a clockwise rotation are considered negative angles.

5. Degree Measure The measure of an angle is determined by the amount of rotation of the initial ray. – The concept of measuring angles in degrees grew out of the belief of the early Sumerians and Babylonians that the seasons repeated every 360 days.

6. Definition of Degree One degree is the measure of an angle formed by rotating a ray 1/360 of a complete revolution. The symbol for degree is 0.

7. Classification of Angles 180 0 angles are straight angles. 90 0 angles are right angles. Angles that have a measure greater than 0 0 but less than 90 0 are acute angles. Angles that have a measure greater than 90 0 but less than 180 0 are obtuse angles.

8. Classification of Angles An angle superimposed in a Cartesian coordinate system is in standard position if its vertex is at the origin and its initial side is on the positive x-axis.

9. Classification of Angles Two positive angles are complementary angles if the sum of the measures of the angles is 90 0. – Each angle is the complement of the other angle.

10. Classification of Angles Two positive angles are supplementary angles if the sum of the measures of the angles is 180 0. – Each angle is the supplement of the other angle.

11. Example 1 For each angle, find the measure (if possible) of its complement and of its supplement. a.  = 40 0 b.  = 125 0

12. Classification of Angles Are the two acute angles of any right triangle complementary angles? Explain. Yes. The sum of the measures of the angles of any triangle is 180 0. The right angle has a measure of 90 0. Thus the measure of the sum of the two acute angles must be 180 0 – 90 0 = 90 0.

13. Classification of Angles It is possible for an angle to have a measure that is greater than 360 0. In this scenario an angle was formed by rotating its terminal side more than one revolution of the initial ray.

14. Classification of Angles If the terminal side of an angle in standard position lies on a coordinate axis, then the angle is classified as a quadrantal angle. – The 90 0 angle, the 180 0 angle, and the 270 0 angle are all examples of quadrantal angles.

15. Classification of Angles Angles in standard position that have the same terminal sides are co-terminal angles. Every angle has an unlimited number of co- terminal angles.

16. Classification of Angles Given   in standard position with measure x 0, then the measures of the angles that are co- terminal with   are given by: x 0 + k  360 0 where k is an integer.

17. Example 2 Assume the following angles are in standard position. Classify each angle by quadrant, and then determine the measure of the positive angle with measure less than 360 0 that is co- terminal with the given angle.  = 550 0  = -225 0  = 1105 0

18. Conversion Between Units There are two popular methods for representing a fractional part of a degree. Decimal Degrees – 29.76 0 DMS (Degree, Minute, Second) – 126 0 12’ 27”

19. Conversion Between Units In the DMS method, a degree is subdivided into 60 equal parts, each of which is called a minute, denoted by ‘. Therefore 1 0 = 60’ Furthermore, a minute is divided into 60 equal parts, each of which is called a second, denoted by “. Thus 1’ = 60” and 1 0 = 3600”. 1 0 /60’ = 11’/60” = 11 0 /3600” = 1

20. Example Convert 126 0 12’ 27” to decimal degrees.

21. Example Convert 31.57 0 to DMS.

22. Radian Measure Another commonly used angle measurement is the radian. To define a radian, first consider a circle of radius, r, and two radii OA and OB. The angle, , formed by the two radii is a central angle. The portion of the circle between A and B is an arc. We say that arc AB subtends the angle . The length of arc AB is s.

23. Definition of Radian One radian is the measure of the central angle subtended by an arc of length r on a circle of radius r. Given an arc of length s on a circle of radius r, the measure of the central angle subtended by the are is  = s / r radians.

24. Radian Measure As an example, consider than an arc of length 15 centimeters on a circle with a radius of 5 centimeters subtends and angle of 3 radians, as shown in Figure 5.22. The same result can be found by 15 centimeters by 5 centimeters.

25. Radian-Degree Conversion To convert from radians to degrees, multiply by 180 0 /  To convert from degrees to radians, multiply by  /180 0.

26. Radian-Degree Conversion

27. Example 3 Convert 300 0 to radians.

28. Example 4

29. Arcs and Arc Length Let r be the length of the radius of a circle and  the nonnegative radian measure of a central angle of the circle. Then the length of the arc, s, that subtends the central angle is s = r 

30. Example 5 Find the length of an arc that subtends a central angle of 120 0 in a circle of radius 10 centimeters.

31. Example 6 A pulley with a radius of 10 inches uses a belt to drive a pulley with a radius of 6 inches. Find the angle through which the smaller pulley turns as the 10-inch pulley makes on revolution. State your answer in radians and also degrees.

32. Linear and Angular Speed Linear speed,, is distance traveled per unit time. Angular speed, , is the angle through which a point on the circle moves per unit time.

33. Linear and Angular Speed

34. Example 7 A hard disk in a computer rotates at 3600 revolutions per minute. Find the angular speed of the disk in radians per second. 3600 rev/minute = 3600 rev ∙ 2  radians ∙ 1 minute 1 minute 1 rev 60 sec = 120p radians/second ≈ 377 radians/second

35. Linear and Angular Speed A wheel has both linear and angular speed. As the wheel moves a distance, s, point A moves through an angle, . the arc length subtending angle  is also s, the distance traveled by the wheel.

36. Linear and Angular Speed The formula from the previous slide,  r  gives the linear speed of a point on a rotating body in terms of distance r from the axis of rotation and the angular speed , provided that  is in radians per unit of time.

37. Example 8 A wind machine is used to generate electricity. The wind machine has propeller blades that are 12 feet in length. If the propeller is rotating at 3 revolutions per second, what is the linear speed, in feet per second, of the tips of the blades?

38. Assignments Day #1 Pgs. 472-473 (1-57 odd) Day #2 Pg. 473 (59-79 odd)