Chapter 3. Radian Measure and Circular Functions. Copyright 2005 Pearson Education, Inc.

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1 Chapter 3 Radian Measure and Circular Functions Copyright 2005 Pearson Education, Inc.

2 3.1 Radian Measure Copyright 2005 Pearson Education, Inc.

3 Measuring Angles Thus far we have measured angles in degrees For most practical applications of trigonometry this the preferred measure For advanced mathematics courses it is more common to measure angles in units called radians In this chapter we will become acquainted with this means of measuring angles and learn to convert from one unit of measure to the other Copyright 2005 Pearson Education, Inc. Slide 3-3

4 Radian Measure An angle with its vertex at the center of a circle that intercepts an arc on the circle equal in length to the radius of the circle has a measure of 1 radian. (1 rad) Copyright 2005 Pearson Education, Inc. Slide 3-4

5 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: q radius of 2 rotated along an arc length of 2 q radius of 1 rotated along an arc length of 1 q radius of 5 rotated along an arc length of 5, etc. r 1 r 2 1rad r 1 r 2 Copyright 2005 Pearson Education, Inc. Slide 3-5

6 More Comments on Radian Measure As with measures given in degrees, a counterclockwise rotation gives a measure expressed in positive radians and a clockwise rotation gives a measure expressed in negative radians Since a complete rotation of a ray back to the initial position generates a circle of radius r, and since the circumference of that circle (arc length) is 2πr, there are radians in a complete rotation 2π Based on the reasoning just discussed: 0 2 π rad = π rad = rad = 57.3 π π rad =180 π rad 0 =1 180 Copyright 2005 Pearson Education, Inc. Slide 3-6 0

7 Converting Between Degrees and Radians From the preceding discussion these ratios both 0 equal 1 : π rad 180 = = π rad To convert between degrees and radians: π rad Multiply a degree measure by and simplify 0 to convert to radians. 180 Multiply a radian measure by to convert to degrees π rad and simplify Copyright 2005 Pearson Education, Inc. Slide 3-7

8 Example: Degrees to Radians Convert each degree measure to radians. a) π rad 60 = 60 = π rad 3 b) π rad = rad Copyright 2005 Pearson Education, Inc. Slide 3-8

9 Example: Radians to Degrees Convert each radian measure to degrees. a) 11π rad 4 11π rad 11π rad = 4 4 b) 3.25 rad π rad = rad rad = 1 π rad Copyright 2005 Pearson Education, Inc. Slide 3-9

10 Equivalent Angles in Degrees and Radians Degrees Radians Degrees Radians Exact Approximate Exact Approximate π π π π π π π Copyright 2005 Pearson Education, Inc. Slide 3-10

11 Equivalent Angles in Degrees and Radians continued Copyright 2005 Pearson Education, Inc. Slide 3-11

12 Finding Trigonometric Function Values of Angles Measured in Radians All previous definitions of trig functions still apply Sometimes it may be useful when trying to find a trig function of an angle measured in radians to first convert the radian measure to degrees When a trig function of a specific angle measure is indicated, but no units are specified on the angle measure, ALWAYS ASSUME THAT UNSPECIFIED ANGLE UNITS ARE RADIANS! When using a calculator to find trig functions of angles measured in radians, be sure to first set the calculator to radian mode Copyright 2005 Pearson Education, Inc. Slide 3-12

13 Example: Finding Function Values of Angles in Radian Measure Find exact function value: 4π a) tan 3 b) 4π sin 3 Convert radians to degrees. sin 4π = 3 sin tan 4π 4 o = tan = tan 240 o = 0 tan 60 = = sin 60 0 = 3 2 = 3 Copyright 2005 Pearson Education, Inc. Slide 3-13

14 Homework 3.1 Page 97 All: 1 4, 7 14, 25 32, 35 42, 47 52, MyMathLab Assignment 3.1 for practice MyMathLab Homework Quiz 3.1 will be due for a grade on the date of our next class meeting Copyright 2005 Pearson Education, Inc. Slide 3-14

15 3.2 Applications of Radian Measure Copyright 2005 Pearson Education, Inc.

16 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, θ1 and θ, with 2 corresponding arc lengths and : s s 1 s 2 s 1 = θ 1 θ 2 2 Copyright 2005 Pearson Education, Inc. Slide 3-16

17 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: s 1 2 s = θ 1 θ 2 Let one angle be with corresponding arc length s and let the other central angle be a whole rotation, 2π rad with arc length 2πr s θ rad = 2πr 2π rad θ rad s = r θ s = rθ in radians! θ Copyright 2005 Pearson Education, Inc. Slide 3-17

18 Example: 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: 3π θ = 8 s = rθ π s = cm π s = cm 21.4cm 8 Copyright 2005 Pearson Education, Inc. Slide 3-18

19 Example: Finding Arc Length continued For the same circle with r = 18.2 cm and θ = 144, find the arc length convert 144 to radians 144 o π = π = radians 5 s s s = rθ π = cm π = cm 45.7cm 5 Copyright 2005 Pearson Education, Inc. Slide 3-19

20 Note Concerning Application Problems Involving Movement Along an Arc When a rope, chain, belt, etc. is attached to a circular object and is pulled by, or pulls, the object so as to rotate it around its center, then the length of the movement of the rope, chain, belt, etc. is the same as the length of the arc s l s = l Copyright 2005 Pearson Education, Inc. Slide 3-20

21 Example: Finding a Length A rope is being wound around a drum with radius.8725 ft. How much rope will be wound around the drum if the drum is rotated through an angle of 39.72? Convert to radian measure. s s = rθ π = ft. 180 Copyright 2005 Pearson Education, Inc. Slide 3-21

22 Example: Finding an Angle Measure Two gears are adjusted so that the smaller gear drives the larger one, as shown. If the smaller gear rotates through 225, through how many degrees will the larger gear rotate? The motion of the small gear will generate an arc length on the small gear and an equal movement on the large gear Copyright 2005 Pearson Education, Inc. Slide 3-22

23 Solution Find the radian measure of the angle and then find the arc length on the smaller gear that determines the motion of the larger gear. o π 5π 225 = 225 = π 12.5π 25π s = rθ = 2.5 = = cm This same arc length will occur on the larger gear. Copyright 2005 Pearson Education, Inc. Slide 3-23

24 Solution continued An arc with this length on the larger gear corresponds to an angle measure θ, in radians where 25π = 4.8θ 8 125π = θ 192 Convert back to degrees. s = rθ (Anglemeasured in radians) 125π π o 117 o Copyright 2005 Pearson Education, Inc. Slide 3-24

25 Sectors and Central Angles of a Circle The pie shaped portion of the interior of circle intercepted by the central angle is called a sector From geometry it is know that in a specific circle the area of a sector is proportional to the measure of its central angle For any two central angles, θ1 and θ2, with corresponding sector areas and : A 1 A 2 A 1 = θ 1 A θ 2 2 Copyright 2005 Pearson Education, Inc. Slide 3-25

26 Development of Formula for Area of Sector Since this relationship is true for any two central angles and corresponding sectors in a circle of radius r: A Let one angle be with corresponding sector area A and let the other central angle be a whole rotation, 2π rad 2 with sector area πr θ A = rad 2 πr π A 1 = θ 1 2 rad θ 2 2 θ rad A r r 2 θ = A = θ 2 2 Copyright 2005 Pearson Education, Inc. Slide in radians! θ

27 Area of a Sector The area of a sector of a circle of radius r and central angle θ is given by A = r 2 θ A= r θ θ, in radians. Copyright 2005 Pearson Education, Inc. Slide 3-27

28 Example: Area Find the area of a sector with radius 12.7 cm and angle θ = 74. Convert 74 to radians. o π 74 = 74 = radians 180 Use the formula to find the area of the sector of a circle. A= ( ) cm 2 r θ = Copyright 2005 Pearson Education, Inc. Slide 3-28

29 Homework 3.2 Page 103 All: 1 10, 17 23, MyMathLab Assignment 3.2 for practice MyMathLab Homework Quiz 3.2 will be due for a grade on the date of our next class meeting Copyright 2005 Pearson Education, Inc. Slide 3-29

30 3.3 The Unit Circle and Circular Functions Copyright 2005 Pearson Education, Inc.

31 Circular Functions Compared with Trigonometric Functions Circular Functions are named the same as trig functions (sine, cosine, tangent, etc.) The domain of trig functions is a set of angles measured either in degrees or radians The domain of circular functions is a set of real numbers The value of a trig function of a specific angle in its domain is a ratio of real numbers π The value of circular function of a real number x is the same as the corresponding trig function of x radians Example: 23 = sin sin 23 rad 0 sin 30 = sin 6 rad = 1 2 Copyright 2005 Pearson Education, Inc. Slide 3-31

32 Circular Functions Defined The definition of circular functions begins with a unit circle, a circle of radius 1 with center at the origin Choose a real number s, and beginning at (1, 0) mark off arc length s counterclockwise if s is positive (clockwise if negative) Let (x, y) be the point on the unit circle at the endpoint of the arc Let θ be the central angle for the arc measured in radians Since s=r, and r = 1, θ Define circular functions of s to be equal to trig functions of ( x, y) s θ s ( 1,0 ) = θ θ Copyright 2005 Pearson Education, Inc. Slide 3-32

33 Circular Functions y y sin s = sinθ = = = y r 1 x x cos s = cosθ = = = x r 1 y tan s = tanθ = ( x 0) x csc s sec s cot s = = = cscθ = secθ = cotθ = r y r x x y = = ( x, y) s 1 y 1 x ( y 0) ( y 0) ( x 0) θ ( 1,0 ) Copyright 2005 Pearson Education, Inc. Slide 3-33

34 Observations About Circular Functions If a real number s is represented in standard position as an arc length on a unit circle, the ordered pair at the endpoint of the arc is: ( cos s, sin s) s (cos s, sin s) ( 1,0 ) Copyright 2005 Pearson Education, Inc. Slide 3-34

35 Further Observations About Circular Functions Draw a vertical line through (1,0) and draw a line segment from the endpoint of s, through the origin, to intersect the vertical line The two triangles formed are similar sin s t tan s = = = t cos s 1 ( cos s, sin s) 1 s ( 1,0 ) t tan s Copyright 2005 Pearson Education, Inc. Slide 3-35

36 Unit Circle with Key Arc Lengths, Angles 2 and Ordered Pairs Shown 5 cos π 3 = cos π tan 2 π = 2 = tan π sin 315 = 2 0 sin Copyright 2005 Pearson Education, Inc. Slide

37 Domains of the Circular Functions Assume that n is any integer and s is a real number. Sine and Cosine Functions: (-, ) Tangent and Secant Functions: s can' t be any odd multiple of π 2 Cotangent and Cosecant Functions: ( s can't be any multipleof π ) π s s ( 2n+ 1) 2 s s nπ { } Copyright 2005 Pearson Education, Inc. Slide 3-37

38 Evaluating a Circular Function Circular function values of real numbers are obtained in the same manner as trigonometric function values of angles measured in radians. This applies both methods of finding exact values (such as reference angle analysis) and to calculator approximations. Calculators must be in radian mode when finding circular function values. Copyright 2005 Pearson Education, Inc. Slide 3-38

39 Example: Finding Exact Circular Function Values 7π 7π 7π Find the exact values of sin, cos, and tan Evaluating a circular function of the real number 7π 4 is equivalent to evaluating a trig function for 4 radians. 7π Convert radian measure to degrees: = π 0 What is the reference angle? 45 Using our knowledge of relationships between trig functions of angles and trig functions of reference angles: 7π o o 2 7π o o cos = cos315 = cos 45 = sin 4 = sin 315 = sin 45 7π tan = tan315 4 Copyright 2005 Pearson Education, Inc. Slide 3-39 o = 2 = tan 45 o = 1 π

40 Example: Approximating Circular Function Values with a Calculator Find a calculator approximation to four decimal places for each circular function. (Make sure the calculator is in radian mode.) a) cos b) cos For the cotangent, secant, and cosecant functions values, we must use the appropriate reciprocal functions. c) cot cot =.3806 tan Copyright 2005 Pearson Education, Inc. Slide 3-40

41 Finding an Approximate Number Given its Circular Function Value Approximate the value of s in the interval given that: cos s = , π 2 With calculator set in radian mode use the inverse cosine key to get: cos ( Is this in theintervalspecified? ) Yes, it's between 0 and π Copyright 2005 Pearson Education, Inc. Slide 3-41

42 Finding an Exact Number Given its Circular Function Value Find the exact value of s in the interval given that: tan s =1 What known reference angle has this exact tangent value? π 45 0 = 4 Based on the interval specified, in what quadrant must the reference angle be placed? III The exact real number we seek for s is: s π = π + = 4 5π 4 π, 3π 2 Copyright 2005 Pearson Education, Inc. Slide 3-42

43 Homework 3.3 Page 113 All: 3 6, 11 18, 23 32, MyMathLab Assignment 3.3 for practice MyMathLab Homework Quiz 3.3 will be due for a grade on the date of our next class meeting Copyright 2005 Pearson Education, Inc. Slide 3-43

44 3.4 Linear and Angular Speed Copyright 2005 Pearson Education, Inc.

45 Circular Motion When an object is traveling in a circular path, there are two ways of describing the speed observed: We can describe the actual speed of the object in terms of the distance it travels per unit of time (linear speed) We can also describe how much the central angle changes per unit of time (angular speed) Copyright 2005 Pearson Education, Inc. Slide 3-45

46 Linear and Angular Speed Linear Speed: distance traveled per unit of time (distance may be measured in a straight line or along a curve for circular motion, distance is an arc length) distance speed = or v = s, time t Angular Speed: the amount of rotation per unit of time, where θ is the angle of rotation measured in radians and t is the time. θ ω = t Copyright 2005 Pearson Education, Inc. Slide 3-46

47 Formulas for Angular and Linear Speed Angular Speed Linear Speed ω θ ω = t ( in radians per unit time, θ in radians) v v v = = = s t rθ t rω Memorize these formulas! Copyright 2005 Pearson Education, Inc. Slide 3-47

48 Example: Using the Formulas Suppose that point P is on a circle with radius 20 π cm, and ray OP is rotating with angular speed 18 radians per second. a) Find the angle generated by P in 6 sec. b) Find the distance traveled by P along the circle in 6 sec. c) Find the linear speed of P. r = 20 cm O P Copyright 2005 Pearson Education, Inc. Slide 3-48

49 Solution: Find the angle generated by P in 6 seconds. π r = 20 cm, ω = rad/sec, t = 6 sec 18 Which formula includes the unknown angle and other things that are known? θ ω = t Substitute for ω and t to find θ θ ω = t π θ = π θ = = 18 π 3 radians. Copyright 2005 Pearson Education, Inc. Slide 3-49

50 Solution: Find the distance traveled by P in 6 seconds π π r = 20 cm, ω = rad/sec, t = 6 sec 18 θ = The distance traveled is along an arc. What is the formula for calculating arc length? s = rθ The distance traveled by P along the circle is 3 rad s = rθ π 20π = 20 = cm. 3 3 Copyright 2005 Pearson Education, Inc. Slide 3-50

51 Solution: Find the linear speed of P π r = 20 cm, ω = rad/sec, t = 6 sec 18 π θ = rad 3 There are three formulas for linear speed. You can use any one that is appropriate for the s v = information that you know: t Better way : rθ v = Linear speed: v = rω t 20π v = rω π 10π s v = = 3 v = 20 = cm/sec 18 9 t 6 20π 20π 1 10π = 6 = = cm per sec π cm 3 Copyright 2005 Pearson Education, Inc. Slide 3-51 s =

52 Observations About Combinations of Objects Moving in Circular Paths When multiple objects, moving in circular paths, are connected by means of being in contact, or by being connected with a belt or chain, the linear speeds of all objects and any connecting devices are all the same In this same situation, angular speeds may be different and will depend on the radius of each circular path Every point in red will be moving at the same linear speed v Angular speeds may be different : ω = r Copyright 2005 Pearson Education, Inc. Slide 3-52

53 Observations About Angular Speed Angular speed is sometimes expressed in units such as revolutions per unit time or rotations per unit time In these situations you should convert to the units of radians per unit time by normal unit conversion methods before using the formulas Example: Express 55 rotations per minute in terms of angular speed units of radians per second 55 rot 2π rad 1min 110π rad 11π = = radians per second 1min 1rot 60 sec 60 sec 6 Copyright 2005 Pearson Education, Inc. Slide 3-53

54 Example: A belt runs a pulley of radius 6 cm at 80 revolutions per min. a) Find the angular speed of the pulley in radians per second. ω = 80 rev 1min 160π 8π ω = = π rad 1rev 1min 60 sec radians per sec v b) Find the linear speed of the belt in centimeters per second. The linear speed of the belt will be the same as that of a point on the circumference of the pulley. = rω 8π = 6 = 16π 50.3 cm pe rs ec. 3 Copyright 2005 Pearson Education, Inc. Slide 3-54

55 Homework 3.4 Page 119 All: 3 43 MyMathLab Assignment 3.4 for practice MyMathLab Homework Quiz 3.4 will be due for a grade on the date of our next class meeting Copyright 2005 Pearson Education, Inc. Slide 3-55