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Chapter 3

Chapter 3. Radian Measure and Circular Functions. 3.1. Radian Measure. Measuring Angles. Thus far we have measured angles in degrees For most practical applications of trigonometry this the preferred measure

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Chapter 3

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  1. Chapter 3 Radian Measure and Circular Functions

  2. 3.1 Radian Measure

  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

  4. 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 1radian. (1 rad) Radian Measure

  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: • 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.

  6. More Comments on Radian Measure • As with measures given in degrees, a counterclockwiserotation 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 , there are radians in a complete rotation • Based on the reasoning just discussed:

  7. Converting Between Degrees and Radians • From the preceding discussion these ratios both equal “1”: • To convert between degrees and radians: • Multiply a degree measure by and simplify to convert to radians. • Multiply a radian measure by and simplify to convert to degrees.

  8. Example: Degrees to Radians • Convert each degree measure to radians. • a) 60 • b) 221.7

  9. Example: Radians to Degrees • Convert each radian measure to degrees. • a) • b) 3.25 rad

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

  11. Equivalent Angles in Degrees and Radians continued

  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”

  13. Find exact function value: a) Convert radians to degrees. b) Example: Finding Function Values of Angles in Radian Measure

  14. Homework • 3.1 Page 97 • All: 1 – 4, 7 – 14, 25 – 32, 35 – 42, 47 – 52, 61 – 72 • 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

  15. 3.2 Applications of Radian Measure

  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, and , with corresponding arc lengths and :

  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: • Let one angle be with corresponding arc length and let the other central angle be a whole rotation, with arc length

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

  19. For the same circle with r = 18.2 cm and  = 144, find the arc length convert 144 to radians Example: Finding Arc Length continued

  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

  21. 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 39.72 to radian measure. Example: Finding a Length

  22. 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 Example: Finding an Angle Measure

  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. • This same arc length will occur on the larger gear.

  24. Solution continued • An arc with this length on the larger gear corresponds to an angle measure , in radians where • Convert back to degrees.

  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, and , with corresponding sector areas and :

  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: • Let one angle be with corresponding sector area and let the other central angle be a whole rotation, with sector area

  27. Area of a Sector • The area of a sector of a circle of radius r and central angle  is given by

  28. Example: Area • Find the area of a sector with radius 12.7 cm and angle  = 74. • Convert 74 to radians. • Use the formula to find the area of the sector of a circle.

  29. Homework • 3.2 Page 103 • All: 1 – 10, 17 – 23, 27 – 42 • 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

  30. 3.3 The Unit Circle and Circular Functions

  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 • The value of circular function of a real number “x” is the same as the corresponding trig function of “x radians” • Example:

  32. 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 be the central angle for the arc measured in radians Since s=r , and r = 1, Define circular functions of s to be equal to trig functions of Circular Functions Defined

  33. Circular Functions

  34. 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) Observations About Circular Functions

  35. 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 Further Observations About Circular Functions

  36. Unit Circle with Key Arc Lengths, Angles and Ordered Pairs Shown

  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: • Cotangent and Cosecant Functions:

  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.

  39. Example: Finding Exact Circular Function Values • Find the exact values of • Evaluating a circular function of the real number is equivalent to evaluating a trig function for radians. • Convert radian measure to degrees: • What is the reference angle? • Using our knowledge of relationships between trig functions of angles and trig functions of reference angles:

  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 2.01  b) cos .6207  • For the cotangent, secant, and cosecant functions values, we must use the appropriate reciprocal functions. • c) cot 1.2071

  41. Finding an Approximate Number Given its Circular Function Value • Approximate the value of s in the interval given that: • With calculator set in radian mode use the inverse cosine key to get:

  42. Finding an Exact Number Given its Circular Function Value • Find the exact value of s in the interval given that: • What known reference angle has this exact tangent value? • Based on the interval specified, in what quadrant must the reference angle be placed? • The exact real number we seek for “s” is:

  43. Homework • 3.3 Page 113 • All: 3 – 6, 11 – 18, 23 – 32, 49 – 60 • 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

  44. 3.4 Linear and Angular Speed

  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)

  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) • Angular Speed: the amount of rotation per unit of time, where  is the angle of rotation measured in radians and t is the time.

  47. Angular Speed Linear Speed ( in radians per unit time,  in radians) Formulas for Angular and Linear Speed

  48. Example: Using the Formulas • Suppose that point P is on a circle with radius 20 cm, and ray OP is rotating with angular speed 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.

  49. Solution: Find the angle generated by P in 6 seconds. • Which formula includes the unknown angle and other things that are known? • Substitute for to find

  50. Solution: Find the distance traveled by P in 6 seconds • The distance traveled is along an arc. What is the formula for calculating arc length? • The distance traveled by P along the circle is

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