Copyright © Cengage Learning. All rights reserved. Trigonometric Functions: Right Triangle Approach

Copyright © Cengage Learning. All rights reserved. 6.1 Angle Measure

3 Objectives ► Angle Measure ► Angles in Standard Position ► Length of a Circular Arc ► Area of a Circular Sector ► Circular Motion

4 Angle Measure An angle AOB consists of two rays R 1 and R 2 with a common vertex O (see Figure 1). We often interpret an angle as a rotation of the ray R 1 onto R 2. Figure 1 Positive angle Negative angle

5 Angle Measure In this case, R 1 is called the initial side, and R 2 is called the terminal side of the angle. An angle is said to be in standard position if its vertex is at the origin of the circle and its initial side is along the positive x- axis. If the rotation is counterclockwise, the angle is considered positive, and if the rotation is clockwise, the angle is considered negative.

6 Angle Measure The measure of an angle is the amount of rotation about the vertex required to move R 1 onto R 2. Intuitively, this is how much the angle “opens.” One unit of measurement for angles is the degree. An angle of measure 1 degree is formed by rotating the initial side of a complete revolution. In calculus and other branches of mathematics, a more natural method of measuring angles is used—radian measure.

7 WHAT IS RADIAN MEASURE? In some applications, it is easier to associate the measure of an angle with the length of an arc of a circle centered at the vertex of the angle. This is the idea of radian measure, a radian is the measure of an angle that, when drawn as a central angle of a circle, intercepts an arc whose length is equal to the length of the radius of the circle.arc

8 Angle Measure hjdf Figure 2

9 Angle Measure Now since the circumference of the circle of radius 1 is 2  a complete revolution of a circle has measure 2  rad, as does a straight angle have a measure of  rad, and a right angle the measure  /2 rad. Other angle measures are determined by the length of the arc intersected by the central angle: An angle that is subtended by an arc of length 2 along the unit circle has radian measure 2 (see Figure 3). Figure 3 Radian measure

10 Angle Measure

11 Example 1 – Converting Between Radians and Degrees (a) Express 60  in radians. (b) Express rad in degrees. Solution: The relationship between degrees and radians gives (a) 60  (b) = 30 

12 Angle Measure A note on terminology: We often use a phrase such as “a 30  angle” to mean an angle whose measure is 30 . Also, for an angle , we write  = 30  or  =  /6 to mean the measure of  is 30  or  /6 rad. When no unit is given, the angle is assumed to be measured in radians.

13 Angles in Standard Position

14 Angles in Standard Position An angle is in standard position if it is drawn in the xy-plane with its vertex at the origin and its initial side on the positive x-axis. Figure 5 gives examples of angles in standard position. Figure 5 Angles in standard position (a) (b) (c)(d)

15 Angles in Standard Position Two angles in standard position are coterminal if their sides coincide. In Figure 5 the angles in (a) and (c) are coterminal. Figure 5 Angles in standard position (a) (b) (c)(d)

16 Example 2 – Coterminal Angles (a) Find angles that are coterminal with the angle  = 30  in standard position. (b) Find angles that are coterminal with the angle  = in standard position. Solution: (a) To find positive angles that are coterminal with , we add any multiple of 360 .

17 Example 2 – Solution Thus 30° + 360° = 390° and 30° + 720° = 750° are coterminal with  = 30 . To find negative angles that are coterminal with , we subtract any multiple of 360°. Thus 30° – 360° = –330° and 30° – 720° = –690° are coterminal with . cont’d

18 Example 2 – Solution See Figure 6. cont’d Figure 6

19 Example 2 – Solution (b) To find positive angles that are coterminal with , we add any multiple of 2 . Thus and are coterminal with  =  /3. To find negative angles that are coterminal with , we subtract any multiple of 2 . cont’d

20 Example 2 – Solution Thus and are coterminal with  =  /3. To find negative angles that are coterminal with , we subtract any multiple of 2 . cont’d

21 Example 2 – Solution Thus and are coterminal with . See Figure 7. cont’d Figure 7

22 Length of a Circular Arc

23 LENGTH OF AN ARC An arc of a circle is a "portion" of the circumference of the circle. The length of an arc is simply the length of its "portion" of the circumference. The length of an arc (or arc length) is traditionally symbolized by s. In the diagram at the right, it can be said that subtends angle and that the measure of θ is the part of the circle determined by the arc rotation.

24 Length of a Circular Arc Recall the formula for length of an arc from Geometry:

25 Length of a Circular Arc Solving for , we get the important formula Now using Radian measure, we obtain the formula: Length of Arc = θ * 2π r 2π

26 Length of a Circular Arc This formula allows us to define radian measure using a circle of any radius r : The radian measure of an angle  is s/r, where s is the length of the circular arc that subtends  in a circle of radius r (see Figure 10). Figure 10 The radian measure of  is the number of “radiuses” that can fit in the arc that subtends  ; hence the term radian.

27 Example 4 – Arc Length and Angle Measure (a) Find the length of an arc of a circle with radius 10 m that subtends a central angle of 30 . (b) A central angle  in a circle of radius 4 m is subtended by an arc of length 6 m. Find the measure of  in radians. Solution: (a) From Example 1(b) we see that 30  =  /6 rad. So the length of the arc is s = r  = =

28 Example 4 – Solution (b) By the formula  = s/r, we have cont’d

29 Area of a Circular Sector

30 Area of a Circular Sector

31 Example 5 – Area of a Sector Find the area of a sector of a circle with central angle 60  if the radius of the circle is 3 m. Solution: To use the formula for the area of a circular sector, we must find the central angle of the sector in radians: 60° = 60(  /180) rad =  /3 rad. Thus, the area of the sector is

32 Circular Motion

33 Circular Motion Suppose a point moves along a circle as shown in Figure 12. There are two ways to describe the motion of the point: linear speed and angular speed. Linear speed is the rate at which the distance traveled is changing, so linear speed is the distance traveled divided by the time elapsed. Figure 12

34 Circular Motion Angular speed is the rate at which the central angle  is changing, so angular speed is the number of radians this angle changes divided by the time elapsed.

35 CIRCULAR MOTION CONSIDER POINT P TRAVELING THROUGH π/3 RADIANS ON A CIRCLE OF RADIUS 2 CM IN 10 SEC A)Find the linear speed: B)Find the angular speed

36 Example 6 – Finding Linear and Angular Speed A boy rotates a stone in a 3-ft-long sling at the rate of 15 revolutions every 10 seconds. Find the angular and linear velocities of the stone. Solution: In 10 s, the angle  changes by 15  2  = 30  radians. So the angular speed of the stone is

37 Example 6 – Solution The distance traveled by the stone in 10 s is s = 15  2  r = 15  2   3 = 90  ft. So the linear speed of the stone is cont’d

38 Example 7 – Finding Linear Speed from Angular Speed A woman is riding a bicycle whose wheels are 26 inches in diameter. If the wheels rotate at 125 revolutions per minute (rpm), find the speed at which she is traveling, in mi/h. Solution: The angular speed of the wheels is 2   125 = 250  rad/min. Since the wheels have radius 13 in. (half the diameter), the linear speed is v = r  = 13  250   10,210.2 in./min

39 Example 7 – Solution Since there are 12 inches per foot, 5280 feet per mile, and 60 minutes per hour, her speed in miles per hour is  9.7 mi/h cont’d