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Angles in a Unit Circle by CHED on May 09, 2017 lesson duration of 6 minutes under Precalculus generated on May 09, 2017 at 06:47 am Tags: Trigonometry CHED.GOV.PH K-12 Teacher's Resource Community Generated: May 09,2017 02:47 PM Angles in a Unit Circle ( 6 mins ) Written By: CHED on July 3, 2016 Subjects: Precalculus Tags: Trigonometry Resources N/A N/A Content Standard Key concepts of circular functions, trigonometric identities, inverse trigonometric functions, and the polar coordinate system Performance Standard Formulate and solve accurately situational problems involving circular functions Apply appropriate trigonometric identities in solving situational problems Formulate and solve accurately situational problems involving appropriate trigonometric functions Formulate and solve accurately situational problems involving the polar coordinate system Learning Competencies Illustrate the unit circle and the relationship between the linear and angular measures of a central angle in a unit circle Introduction 1 mins There are many problems involving angles in several fields like engineering, medical imaging, electronics, astronomy, geography and many more. Surveyors, pilots, landscapers, designers, soldiers, and people in many other professions heavily use angles and trigonometry to accomplish a variety of practical tasks. In this lesson, we will deal with the basics of angle measures together with arc length and sectors. Angle Measure 1 mins 1 / 19 CHED.GOV.PH K-12 Teacher's Resource Community An angle is formed by rotating a ray about its endpoint. In the figure shown below, the initial side of angle /AOB is OA, OA, An angle angle is is said said to to be be positive ifif the the ray ray rotates rotates inin aa counterclockwise counterclockwise direction, direction, and and the the while is terminal side is OB. OB . An angle is negative if it rotates in a clockwise direction. Teaching Notes Angles in trigonometry differ angles in Euclidean geometry in the sense of motion. angle in geometry Angles in trigonometry differ fromfrom angles in Euclidean geometry in the sense of motion. An An angle in geometry is is defined a union of rays is, static) and measure has measure between 0 degrees and degrees. 180 degrees. An angle defined as a as union of rays (that (that is, static) and has between 0 degrees and 180 An angle in in trigonometry is a rotation of a ray, and, therefore, has no limit. It has positive and negative directions and measures. An angle angle isis in in standard position ififititisisdrawn drawnininthe the xy-plane withitsitsvertex vertexatatthe theorigin originand anditsitsinitial initialside sideononthe the An xy -planewith positive x-axis. The angles a, B, and ? in the following figure are angles in standard position. To measure angles, we use degrees, minutes, seconds, and radians. For example, in degrees, minutes, and seconds, 2 / 19 CHED.GOV.PH K-12 Teacher's Resource Community 10 degrees 30’18” = 10(30+(18/60)) = 10 degrees 30.3’ = (10+(30.3/60) degrees = 10.505 degrees and 79.251 degrees = 79 degrees (0.251 x 60)’ = 79 degrees15.06’ = 79 degrees 15’(0.06 x 60)” = 79 degrees 15’3.6”. Recall that the unit circle is the circle with center at the origin and radius 1 unit. trigonometry,as asititwas wasstudied studiedininGrade Grade9,9,the thedegree degreemeasure measureisisoften oftenused. used.On Onthe theother otherhand, hand,ininsome somefields fieldsofof InIntrigonometry, mathematics like like calculus, calculus, radian radian measure measure of of angles angles is is preferred. preferred. Radian Radian measure measure allows allows us us to to treat treat the the trigonometric trigonometric mathematics functions as functions with the set of real numbers as domains, rather than angles. Example 3.1.1. In the following figure, identify the terminal side of an angle in standard position with given measure. 3 / 19 CHED.GOV.PH K-12 Teacher's Resource Community Since aa unit unit circle circle has has circumference circumference 22pi pi, ,aacentral centralangle anglethat thatmeasures measures360 360degrees degreeshas hasmeasure measureequivalent equivalenttoto22pi pi Since radians. Thus, we obtain the following conversion rules. Figure3.2 3.2shows showssome somespecial specialangles anglesininstandard standardposition positionwith withthe theindicated indicatedterminal terminalsides. sides.The Thedegree degreeand andradian radian Figure measures are also given. 4 / 19 CHED.GOV.PH K-12 Teacher's Resource Community Example 3.1.2. Express 75 degrees and 240 degrees in radians. Solution. 75 (pi/180) = 5pi/12 => 75 degrees = 5pi/12 rad 240 (pi/180) = 4pi/3 => 240 degrees = 4pi/3 rad Example 3.1.3. Express pi/8 rad and 11pi/6 rad in degrees. Solution. pi/8 (180/pi) = 22.5 => pi/8 rad = 22.5 degrees 11pi/6 (180/pi) = 330 => 11pi/6 rad = 330 degrees Seatwork 1 mins Seatwork/Homework 1. Convert the following degree measures to radian measure. 5 / 19 CHED.GOV.PH K-12 Teacher's Resource Community 2. Convert the following radian measures to degree measure. Coterminal Angles 1 mins Two angles in standard position that have a common terminal side are called coterminal angles. angles. Observe that the degree measures of coterminal angles differ by multiples of 360 degrees. As a quick illustration, to find one coterminal angle with an angle that measures 410 degrees, just subtract 360 degrees, resulting in 50 degrees. See Figure 3.3. Example 3.1.4. Find the angle coterminal with ?380 degrees that has measure (1) between 0 degrees and 360 degrees, and (2) between ?360 degrees and 0 degrees. Solution. A negative angle moves in a clockwise direction, and the angle ?380 degrees lies in Quadrant IV. (1) ?380 degrees + 2 360 degrees = 340 degrees 6 / 19 CHED.GOV.PH K-12 Teacher's Resource Community (2) ?380 degrees + 360 degrees = ?20 degrees Seatwork/Homework 1. Find the angle between 0 degrees and 360 degrees (if in degrees) or between 0 rad and 2pi rad (if in radians) that is coterminal with the given angle. 2. Find the angle between ?360 degrees and 0 degrees (if in degrees) or between ?2pi rad and 0 rad (if in radians) that is coterminal with the given angle. Seatwork 0 mins Seatwork/Homework 1. Find the angle between 0 degrees and 360 degrees (if in degrees) or between 0 rad and 2pi rad (if in radians) that is coterminal with the given angle. 2. Find the angle between ?360 degrees and 0 degrees (if in degrees) or between ?2pi rad and 0 rad (if in radians) that is coterminal with the given angle. Arc Length and Area of a Sector 1 mins In a circle, a central angle whose radian measure is ? subtends an arc that is the fraction ?/2pi of the circumference of the circle. Thus, in a circle of radius r (see Figure 3.4), the length s of an arc that subtends the angle ? is 7 / 19 CHED.GOV.PH K-12 Teacher's Resource Community Teaching Notes Review how arcs were measured in Grade 10. What unit of measure was used pi for two circles with different radii, do equalcentral centralangles anglesintercept interceptarcs arcsofofthe thesame samemeasure measurepi.pi.Conclude Concludethat thatprevious previousnotion notionofofarc arcmeasure measureisisnot notthe the equal same as length. Arcs are now measured in terms of length and measure changes with the radius of the circle. Example 3.1.5. Find the length of an arc of a circle with radius 10 m that subtends a central angle of 30 degrees. Solution. Since Since given central angle in degrees, have to convert it into radian measure. Then apply the the given central angle is inisdegrees, we we have to convert it into radian measure. Then apply the the formula for an arc length. Example 3.1.6. A A central central angle angle ?? in in aa circle circle of of radius radius 44 m m is is subtended subtended by by an an arc arc of of length length 66 m. m. Find Find the the measure measure of of ?? in radians. Solution. 8 / 19 CHED.GOV.PH K-12 Teacher's Resource Community ? = s/r = 6/4 = 3/2rad sectorofofa acircle circleisisthe theportion portionofofthe theinterior interiorofofa acircle circlebounded boundedbybythe theinitial initialand andterminal terminalsides sidesofofa acentral centralangle angle AAsector and its its intercepted intercepted arc. arc. ItIt is is like like aa “slice “slice of of pizza.” pizza.” Note Note that that an an angle angle with with measure measure 2pi 2pi radians radians will will define define aa sector sector that that and corresponds to the whole “pizza.” Therefore, a central angle of sector a sector measure ? radians, then sector corresponds to the whole “pizza.” Therefore, if aif central angle of a hashas measure ? radians, then thethe sector makesupupthe thefraction fraction?/2pi ?/2piofofa acomplete completecircle. circle.See SeeFigure Figure3.5. 3.5.Since Sincethe thearea areaofofa acomplete completecircle circlewith withradius radius r is makes pir pir^2, we have Example 3.1.7. Find the area of a sector of a circle with central angle 60 degrees if the radius of the circle is 3 m. Solution. First, we have to convert 60 degrees into radians. Then apply the formula for computing the area of a sector. Example 3.1.8. A sprinkler on a golf course fairway is set to spray water over a distance of 70 feet and rotates through an angle of 120 degrees. Find the area of the fairway watered by the sprinkler. Solution. 9 / 19 CHED.GOV.PH K-12 Teacher's Resource Community Seatwork/Homework 1. In a circle of radius 7 feet, find the length of the arc that subtends a central angle of 5 radians. Answer: 35 ft 2. A central angle pi in a circle of radius 20 m is subtended by an arc of length 15pi m. Find the measure of ? in degrees. Answer: 135 degrees 3. Find the area of a sector of a circle with central angle that measures 75 degrees if the radius of the circle is 6 m. Answer: 7.5 m^2 Exercises 3.1 1. Give the degree/radian measure of the following special angles. 2. Convert each degree measure to radians. Leave answers in terms of pi. 10 / 19 CHED.GOV.PH K-12 Teacher's Resource Community 3. Convert each radian measure to degree-minute-second measure (approximate if necessary). 4. Find the angle between 0 degrees and 360 degrees (if in degrees) or between 0 rad and 2pi rad (if in radians) that is coterminal with the given angle. 5. Find the angle between -360 degrees and 0 degrees (if in degrees) or between ?2pi rad and 0 rad (if in radians) that is coterminal with the given angle. 11 / 19 CHED.GOV.PH K-12 Teacher's Resource Community 6. Find the length of an arc of a circle with radius 21 m that subtends a central angle of 15 degrees. Answer: 7pi/4 m 7. A central angle ? in a circle of radius 9 m is subtended by an arc of length 12 m. Find the measure of ? in radians. Answer: 4/3 rad 8. Find the radius of a circle in which a central angle of pi/6 rad determines a sector of area 64 m^2. Answer: 16 m 9. If the radius of a circle is doubled, how is the length of the arc intercepted by a fixed central angle changed? Answer: The length is doubled. 10. Radian measure simplifies many formulas, such as the formula for arc length, s = r ?. Give the corresponding formula when ? is measured in degrees instead of radians. Answer: s = pir pir ?/180 11. As shown below, find the radius of the pulley if a rotation of 51.6 degrees raises the weight by 11.4 cm. Answer: 12.7 cm 12. How many inches will the weight rise if the pulley whose radius is 9.27 inches is rotated through an angle of 71 degrees 500pi Answer: 11.6 in 13. Continuing with the previous item, through what angle (to the nearest minute) must the pulley be rotated to raise the weight 6 in? 12 / 19 CHED.GOV.PH K-12 Teacher's Resource Community Answer: 37 degrees 5' 14. Given a circle of radius 3 in, find the measure (in radians) of the central angle of a sector of area 16 in^2. Answer: 3.6 rad 15. An automatic lawn sprinkler sprays up to a distance of 20 feet while rotating 30 degrees. What is the area of the sector the sprinkler covers? Answer: 104.72 ft^2 16. A jeepney has a windshield wiper on the driver’s side that has total arm and blade 10 inches long and rotates back and forth through an angle of 95 degrees. The shaded region in the figure is the portion of the windshield cleaned by the 7-inch wiper blade. What is the area of the region cleaned? Answer: 75.4 in^2 17. If the radius of a circle is doubled and the central angle of a sector is unchanged, how is the area of the sector changed? Answer: The area is quadrupled. 18.Give the corresponding formula for the area of a sector when the angle is measured in degrees. Answer: A = pir2? 360 19.A frequent problem in surveying city lots and rural lands adjacent to curves of highways and railways is that of finding the area when one or more of the boundary lines is the arc of a circle. Approximate the total area of the lot shown in the figure. Answer: 1909.0 m^2 20. Two gears of radii 2.5 cm and 4.8 cm are adjusted so that the smaller gear drives the larger one, as shown. If the smaller gear rotates counterclockwise through 225 degrees, through how many degrees will the larger gear rotate? Answer: 117 degrees 13 / 19 CHED.GOV.PH K-12 Teacher's Resource Community Seatwork 1 mins Seatwork/Homework 1. In a circle of radius 7 feet, find the length of the arc that subtends a central angle of 5 radians. Answer: 35 ft 2. A central angle pi in a circle of radius 20 m is subtended by an arc of length 15pi m. Find the measure of ? in degrees. Answer: 135 degrees 3. Find the area of a sector of a circle with central angle that measures 75 degrees if the radius of the circle is 6 m. Answer: 7.5 m^2 Exercises 3.1 1. Give the degree/radian measure of the following special angles. 14 / 19 CHED.GOV.PH K-12 Teacher's Resource Community 2. Convert each degree measure to radians. Leave answers in terms of pi. 3. Convert each radian measure to degree-minute-second measure (approximate if necessary). 15 / 19 CHED.GOV.PH K-12 Teacher's Resource Community 4. Find the angle between 0 degrees and 360 degrees (if in degrees) or between 0 rad and 2pi rad (if in radians) that is coterminal with the given angle. 5. Find the angle between -360 degrees and 0 degrees (if in degrees) or between ?2pi rad and 0 rad (if in radians) that is coterminal with the given angle. 16 / 19 CHED.GOV.PH K-12 Teacher's Resource Community 6. Find the length of an arc of a circle with radius 21 m that subtends a central angle of 15 degrees. Answer: 7pi/4 m 7. A central angle ? in a circle of radius 9 m is subtended by an arc of length 12 m. Find the measure of ? in radians. Answer: 4/3 rad 8. Find the radius of a circle in which a central angle of pi/6 rad determines a sector of area 64 m^2. Answer: 16 m 9. If the radius of a circle is doubled, how is the length of the arc intercepted by a fixed central angle changed? Answer: The length is doubled. 10. Radian measure simplifies many formulas, such as the formula for arc length, s = r ?. Give the corresponding formula when ? is measured in degrees instead of radians. Answer: s = pir pir ?/180 11. As shown below, find the radius of the pulley if a rotation of 51.6 degrees raises the weight by 11.4 cm. Answer: 12.7 cm 12. How many inches will the weight rise if the pulley whose radius is 9.27 inches is rotated through an angle of 71 degrees 500pi Answer: 11.6 in 13. Continuing with the previous item, through what angle (to the nearest minute) must the pulley be rotated to raise the weight 6 in? Answer: 37 degrees 5' 14. Given a circle of radius 3 in, find the measure (in radians) of the central angle of a sector of area 16 in^2. Answer: 3.6 rad 15. An automatic lawn sprinkler sprays up to a distance of 20 feet while rotating 30 degrees. What is the area of the sector the sprinkler covers? Answer: 104.72 ft^2 16. A jeepney has a windshield wiper on the driver’s side that has total arm and blade 10 inches long and rotates back and forth through an angle of 95 degrees. The shaded region in the figure is the portion of the windshield cleaned by 17 / 19 CHED.GOV.PH K-12 Teacher's Resource Community the 7-inch wiper blade. What is the area of the region cleaned? Answer: 75.4 in^2 17. If the radius of a circle is doubled and the central angle of a sector is unchanged, how is the area of the sector changed? Answer: The area is quadrupled. 18.Give the corresponding formula for the area of a sector when the angle is measured in degrees. Answer: A = pir2? 360 19.A frequent problem in surveying city lots and rural lands adjacent to curves of highways and railways is that of finding the area when one or more of the boundary lines is the arc of a circle. Approximate the total area of the lot shown in the figure. Answer: 1909.0 m^2 20. Two gears of radii 2.5 cm and 4.8 cm are adjusted so that the smaller gear drives the larger one, as shown. If the smaller gear rotates counterclockwise through 225 degrees, through how many degrees will the larger gear rotate? Answer: 117 degrees 18 / 19 CHED.GOV.PH K-12 Teacher's Resource Community Generated: May 09,2017 02:47 PM 19 / 19 Powered Poweredby byTCPDF TCPDF(www.tcpdf.org) (www.tcpdf.org)