Chapter 4. Trigonometric Functions. 4.1 Angle and Radian Measure /45

Transcription

1 Chapter 4 Trigonometric Functions 4.1 Angle and Radian Measure 1 /45

2 Chapter 4 Homework 4.1 p290 13, 17, 21, 35, 39, 43, 49, 53, 57, 65, 73, 75, 79, 83, 89, 91 2 /45

3 Chapter 4 Objectives Recognize and use the vocabulary of angles. Use degree measure. Use radian measure. Convert between degree and radian measures Draw angles in standard position Find coterminal angles Find the length of a circular arc Use linear and angular speed to describe motion on a circular path Find the area of a Sector 3 /45

4 Angles An angle is formed by two rays that have a common endpoint. One ray is called the initial side and the other the terminal side. θ C Terminal A Initial B Vertex 4 /45

5 Angles An angle is in standard position if its vertex is at the origin of a rectangular coordinate system and its initial side lies along the positive x-axis. Terminal θ Vertex Initial on x-axis Vertex Initial on x-axis Terminal θ Positive Measure Negative Measure 5 /45

6 θ Angles When we see an initial side and a terminal side in place, there are two kinds of rotations that could have generated the angle. Positive angles are generated by counterclockwise rotation. Thus, angle α is positive. Negative angles are generated by clockwise rotation. Thus, angle θ is negative. Vertex Initial on x-axis Terminal α Terminal Vertex Initial on x-axis 6 /45

7 Angles An angle is called a quadrantal angle if its terminal side lies along an axis. quadrant 2 quadrant I Angle β is an example β of a quadrantal angle. Terminal quadrant 3 quadrant 4 7 /45

8 Measuring Angles in Degrees Angles are measured by determining the amount of rotation from the initial side to the terminal side. A complete rotation of the circle is 360 degrees, or 360. An acute angle measures less than 90. A right angle measures 90. An obtuse angle measures more than 90 but less than 180. A straight angle measures 180. acute right obtuse angle straight angle angle angle θ θ θ < 90 θ = 90 0<θ<180 θ=180 8 /45

9 Measuring Angles in Degrees A complete rotation of a circle is 360. One degree (1 ) is 1/360th of a complete rotation. 360 An angle may be measured in degrees, minutes, seconds. One degree = 60 minutes, one minute = 60 seconds. Or one second = 1/3600th degree, one minute = 1/60th degree = 49 degrees 32 minutes 58 seconds. 9 /45

10 Converting to decimal Be cautious when converting degrees and decimal form = 36 degrees, 30 minutes. 48 degrees, 20 minutes = degrees = 39 45/60 degrees = degrees = / /3600 degrees = /3600 degrees degrees 10/45

11 TI-84 You can convert decimal degrees to degrees, minutes, seconds on the TI-84. Make certain the calculator is in Degree Mode. MODE RADIAN DEGREE ENTER Back to home screen and enter degree measure in decimal form, then 2nd APPS 4: DMS ENTER 11 /45

12 TI-84 You can also convert degrees, minutes, seconds to decimal degrees but it takes a few steps. Make certain the calculator is in Degree Mode. Enter the degrees only, then 2nd APPS 1: ᵒ ENTER Enter the minutes next, then Now when you hit ENTER the calculator will convert to decimal. 2nd APPS 2: ENTER Enter the seconds, then ALPHA + ENTER 12/45

13 TI-84 From this point forward you will have to be careful to ensure the calculator is in the correct mode. Get in the habit of checking for degrees or radians before you start working. Most of our work (and calculus) is done in radians so I keep my calculator in radians and try to remember to switch to degrees when needed. To convert degrees to radians. Make certain the calculator is in Radian Mode. Enter the degrees, then, before hitting ENTER 2nd APPS 1: ᵒ ENTER Aaaaaah Radians 13/45

14 Measuring Angles in Radians An angle whose vertex is at the center of the circle is called a central angle. The radian measure of any central angle of a circle is the length of the intercepted arc divided by the α length of the circle s radius. length of intercepted arc (s) s #radians = length of radius (r ) = s r α r 14/45

15 Definition of a Radian One radian is the measure of the central angle of a circle that intercepts an arc equal in length to the radius of the circle. Terminal α = 1 radian r α r Initial 15/45

16 A little Vocabulary An angle intercepts an arc when the initial and terminal sides of an angle intersect a circle. The arc between the angle sides is the intercepted arc. Chords of a circle also intercept an arc An angle is subtended by an arc when the initial and terminal sides of an angle are at ends of the arc. Any object that forms an angle from its extremities is said to subtend an angle. 16/45

17 Radian Measure A central angle that intercepts an arc of length s, on a circle with radius r, has a measure of s r radians. s Terminal s α = r radians 2 radians 1 radian α 3 radians 0 radians r Initial 6 radians 4 radians 5 radians 17/45

18 Example: Computing Radian Measure A central angle θ, in a circle of radius 12 feet intercepts an arc of length 42 feet. What is the radian measure of θ? 42 ft θ = 42ft 12ft = 21ft 6ft = 3.5 radians θ 12 ft Note: radians have no unit of measure other than simply, radians. 18/45

19 Radian Measure Remember the definition of π. π = Circumference Diameter = Circumference 2 x radius A little algebra 2πr = Circumference 2π radians = Circumference A little substitution 2π radians = 360! 19/45

20 Conversion between Degrees and Radians To convert degrees to radians or radians to degrees remember a circle has 360 and a circumference of 2πr (or 2π radians) 2πr = 360, solving for 1r, 1radian = 360 2π = 180 π 2πr = 360, solving for 1, 1degree = 2π 360 = π 180 To convert degrees to radians multiply the degrees by the number of radians for 1 degree: radians = degrees x π 180 To convert radians to degrees multiply the radians by the number of degrees for 1 radian: degrees = radians x 180 π 20/45

21 Example: Converting from Degrees to Radians Convert each angle in degrees to radians: radians = degrees x π 180 a x π 180 = 60 π 180 = π 3 radians b x π 180 = 270 π 180 = 3π 2 radians c x π 180 = 300 π 180 = 5π 3 radians 21/45

22 Example: Converting from Radians to Degrees Convert each angle in radians to degrees: degrees = radians x 180 π a. π 4 radians π 4 radians x 180 π radians = 45 b. 4π 3 radians 4π 3 radians x 180 π radians = = 240 c. 6 radians 6 radians x 180 π radians = π = 1080 π 22/45

23 Sums of Angles Complementary angles. Two positive angles whose sum is 90 complementary angles. π 2 radians are said to be Find the complements of 38, and π 3 radians 38 + x = 90 x = 52 π 3 radians + x = π 2 radians x = π 6 radians Supplementary angles. Two positive angles whose sum is 180 supplementary angles. ( ) π radians are said to be 23/45

24 Drawing Angles in Standard Position The figure illustrates that when the terminal side makes one full revolution, it forms an angle whose radian measure is 2π. The figure shows the quadrantal angles formed by 3/4, 1/2, and 1/4 of a revolution. π 2 radians Initial π radians 2π radians Terminal 3π 2 radians 24/45

25 Example: Drawing Angles in Standard Position Draw and label the angle π 4 radians in standard position: The angle is negative so we rotate the terminal side clockwise. Initial One full rotation is -2π radians Vertex π 4 radians π 4 2π = 1 8 rotations Terminal So we rotate the terminal side 1 clockwise revolution 8 25/45

26 Example: Drawing Angles in Standard Position Draw and label the angle 3π 4 radians in standard position: Terminal The angle is positive so we rotate the terminal side counter-clockwise. 3π Vertex 4 radians 8 Initial One full rotation is 2π radians 3π 4 2π = 3 8 rotations So we rotate the terminal side counter-clockwise 3 8 revolution 26/45

27 Example: Drawing Angles in Standard Position Draw and label the angle 6 7π 4 radians in standard position: The angle is negative so we rotate the 5 8 Terminal terminal side clockwise. 4 8 Vertex 7π 4 radians Initial One full rotation is -2π radians 7π 4 1 2π = 7 8 rotations So we rotate the terminal side 7 clockwise revolution /45

28 Example: Drawing Angles in Standard Position Draw and label the angle 13π 4 radians in standard position: 2 The angle is positive so we rotate the terminal side counter-clockwise. 8 One full rotation is 2π radians 4 8 Vertex Initial 13π 4 1 2π = rotations 13π 4 radians So we rotate the terminal side Terminal counter-clockwise revolution 28/45

29 Degree and Radian Measures of Angles Commonly Seen in Trigonometry π π 120 3π 105 2π 3 7π π 2 5π π 3 60 π 4 45 π 6 π π 0, 2π π 12 23π π 6 5π π 5π 3 17π 3π 19π π 6 7π /45

30 Degree and Radian Measures of Angles Commonly Seen in Trigonometry π π 17π π 2 19π π π π 6 13π 12 11π 6 23π π 0, 2π π π π 6 π π 4 π π 3 7π π π 12 π /45

31 Positive Angles in Revolutions of the Angle s Terminal Around the Origin 31/45

32 Positive Angles in Revolutions of the Angle s Terminal Around the Origin 32/45

33 Co-terminal Angles Two angles with the same initial and terminal sides but possibly different rotations are called co-terminal angles. Increasing or decreasing the degree measure of an angle in standard position by an integer Terminal multiple of 360 results in a co-terminal angle. So, an angle of θ 0 is co-terminal with Initial angles of θ 0 ± 360 k, where k is an integer. Vertex Also, an angle of θ radians is co-terminal with angles of θ radians ± 2π k radians, where k is an integer. 33/45

34 Example: Finding Coterminal Angles Assume the following angles are in standard position. Find a positive angle less than 360 that is co-terminal with each of the following: a. a 400 angle = 40 b. a 135 angle = 225 c. a 430 angle = 70 Vertex d. a -40 angle = /45

35 Example: Finding Coterminal Angles Assume the following angles are in standard position. Find a positive angle less than 2π that is co-terminal with each of the following: 13π a. radians 5 13π 5 13π 2π = 5 10π 5 = 3π 5 b. radians π 15 π π = π π 15 = 29π 15 c. 37π radians 6 37π 6 37π + 8π = π 6 = 11π 6 35/45

36 The Length of a Circular Arc Assume a circle with radius r has positive central angle θ radians Terminal s=rθ θ The length of the arc intercepted by the r central angle is s = rθ 36/45

37 Example: Finding the Length of a Circular Arc A circle has a radius of 6 inches. Find the length of the arc intercepted by a central angle of 45. Express arc length in terms of π. (Then approximate your answer to two decimal places. First convert 45 to radians (We will calculate, but soon you should be able to convert 45 x π 180 = 45 π 180 = π 4 radians 45 automatically): Then find the length of the arc. s = π 4 radians x 6in radian = 3π 2 in 4.71in 37/45

38 Example: Finding the Length of a Circular Arc A circle has a radius of 9 inches. Find the length of the arc intercepted by a central angle of 135. Express arc length in terms of π. (Then approximate your answer to two decimal places. First convert 135 to radians 135 x π 180 = 135 π 180 = 3π 4 radians Find the length of the arc. s = 3π 4 radians x 9in radian = 27π in = 21.21in 4 38/45

39 ω(omega)= θ t Definitions of Linear and Angular Speed If a point is in motion on a circle of radius r through an angle of θ radians in time t, then the point s linear speed (how fast our little man is running around the circle) is: s=rθ v = s t s is the arc length (s = rθ) θ r The angular speed (spin rate in # revolutions/unit of time) is given by 39/45

40 Linear Speed in terms of Angular Speed The linear speed, v (velocity), of a point a distance r from the center of rotation is given by: v = s t = rθ t = r θ t = rω v = rω v is the linear speed of the point and ω is the angular speed of the point. 40/45

41 Example: Finding Linear Speed The largest wind turbine has blades 244 feet long and the diameter of the rotating assembly is 500 feet. If one revolution takes 6 seconds, what is the linear velocity (feet/sec) of a point at the tip of a blade? We are told it takes 6 seconds for one revolution, so the angular speed, ω, is 1/6 revolutions/second. Before applying the formula v = r ω, we must express ω in terms of radians per second: ω = 1revolution 6seconds x 2π radians 1revolution = 2π radians 6seconds = π radians 3seconds 41/45

42 Example: Finding Linear Speed The largest wind turbine has blades 244 feet long and the diameter of the rotating assembly is 500 feet. If one revolution takes 6 seconds, what is the linear velocity (feet/sec) of a point at the tip of a blade? ω = 1revolution 6seconds x 2π radians 1revolution = 2π radians 6seconds = π radians 3seconds The radius of the rotating assembly is 250 feet. v = rω v = 250ft 1radian x π radians 3seconds = 250π ft 3seconds 261.8ft / sec That is about miles/hour. 42/45

43 Example: Finding Linear Speed A typical HDD (hard disk drive in your computer) spins at 7200 rpm (revolutions per minute). In a desktop computer the form factor of an hdd is 3.5 inches. What is the linear speed of a spot 3 inches from the center? The angular speed of the record is 7200 x 2π = 14400π radians per minute. The linear speed at 3 inches is v = rω v = 3 in 1radian x 14400π radians 1minute = 43200π in 1 minute in / min 43/45

44 Sector A sector of a circle is the region formed by two radii of the circle and the intercepted arc For a circle with radius r and central angle θ radians, the area of a sector is given by A = (1/2)θr 2. r θ This is easily confirmed considering the area of circle A = πr 2 = 1 ( 2 2π )r 2 44/45

45 Example: Area of a Sector A sprinkler on a shameful hotel lawn sprays water over a distance of 60 feet through an angle of 180 degrees. What area of the lawn is watered by the sprinkler? First convert 180ᵒ to π radians. A = (1/2)r 22θ A = (1/2)60 2 feet 2 x π radians = 1800π square feet 45/45