9. Tigonometic Functions of An Angle Essential Question How can ou use the unit cicle to define the tigonometic functions of an angle? Let be an angle in standad position with, ) a point on the teminal side of and = + 0. The si tigonometic functions of ae defined as shown. sin = csc =, 0, ) cos = sec =, 0 tan =, 0 cot =, 0 Witing Tigonometic Functions Wok with a patne. Find the sine, cosine, and tangent of the angle in standad position whose teminal side intesects the unit cicle at the point, ) shown. a. 1, b. 1, 1 c. 0, 1) d. e. f. 1, 0) CONSTRUCTING VIABLE ARGUMENTS To be poficient in math, ou need to undestand and use stated assumptions, definitions, and peviousl established esults. 1, 1, 1 Communicate You Answe. How can ou use the unit cicle to define the tigonometic functions of an angle?. Fo which angles ae each function undefined? Eplain ou easoning. a. tangent b. cotangent c. secant d. cosecant Section 9. Tigonometic Functions of An Angle 477
9. Lesson What You Will Lean Coe Vocabula unit cicle, p. 479 quadantal angle, p. 479 efeence angle, p. 480 Pevious cicle adius Pthagoean Theoem Evaluate tigonometic functions of an angle. Find and use efeence angles to evaluate tigonometic functions. Tigonometic Functions of An Angle You can genealize the ight-tiangle definitions of tigonometic functions so that the appl to an angle in standad position. Coe Concept Geneal Definitions of Tigonometic Functions Let be an angle in standad position, and let, ) be the point whee the teminal side of intesects the cicle + =. The si tigonometic functions of ae defined as shown. sin = csc =, 0, ) cos = sec =, 0 tan =, 0 cot =, 0 These functions ae sometimes called cicula functions. Evaluating Tigonometic Functions Given a Point Let 4, ) be a point on the teminal side of an angle in standad position. Evaluate the si tigonometic functions of. 4, ) Use the Pthagoean Theoem to find the length of. = + = 4) + = 5 = 5 Using = 4, =, and = 5, the values of the si tigonometic functions of ae: sin = = 5 cos = = 4 5 tan = = 4 csc = = 5 sec = = 5 4 cot = = 4 478 Chapte 9 Tigonometic Ratios and Functions
ANOTHER WAY The geneal cicle + = can also be used to find the si tigonometic functions of. The teminal side of intesects the cicle at 0, ). So, sin = = = 1. The othe functions can be evaluated similal. Coe Concept The Unit Cicle The cicle + = 1, which has cente 0, 0) and adius 1, is called the unit cicle. The values of sin and cos ae simpl the -coodinate and -coodinate, espectivel, of the point whee the teminal side of intesects the unit cicle. sin = = 1 = cos = = 1 = It is convenient to use the unit cicle to find tigonometic functions of quadantal angles. A quadantal angle is an angle in standad position whose teminal side lies on an ais. The measue of a quadantal angle is alwas a multiple of 90º, o π adians. Using the Unit Cicle, ) Use the unit cicle to evaluate the si tigonometic functions of = 70º. = 1 Step 1 Daw a unit cicle with the angle = 70º in standad position. Step Identif the point whee the teminal side of intesects the unit cicle. The teminal side of intesects the unit cicle at 0, 1). Step Find the values of the si tigonometic functions. Let = 0 and = 1 to evaluate 0, 1) the tigonometic functions. sin = = 1 1 = 1 csc = = 1 1 = 1 cos = = 0 1 = 0 sec = = 1 0 tan = = 1 0 undefined undefined cot = = 0 1 = 0 Monitoing Pogess Help in English and Spanish at BigIdeasMath.com Evaluate the si tigonometic functions of. 1.. 8, 15)., ) 5, 1) 4. Use the unit cicle to evaluate the si tigonometic functions of = 180º. Section 9. Tigonometic Functions of An Angle 479
READING The smbol is ead as theta pime. Refeence Angles Coe Concept Refeence Angle Relationships Let be an angle in standad position. The efeence angle fo is the acute angle fomed b the teminal side of and the -ais. The elationship between and is shown below fo nonquadantal angles such that 90 < < 0 o, in adians, π < < π. Quadant II Quadant III Quadant IV Degees: = 180 Degees: = 180 Degees: = 0 Radians: = π Radians: = π Radians: = π Finding Refeence Angles Find the efeence angle fo a) = 5π and b) = 10º. a. The teminal side of lies in Quadant IV. So, = π 5π = π. The figue at the ight shows = 5π and = π. b. Note that is coteminal with 0º, whose teminal side lies in Quadant III. So, = 0º 180º = 50º. The figue at the left shows = 10º and = 50º. Refeence angles allow ou to evaluate a tigonometic function fo an angle. The sign of the tigonometic function value depends on the quadant in which lies. Coe Concept Evaluating Tigonometic Functions Use these steps to evaluate a tigonometic function fo an angle : Step 1 Find the efeence angle. Step Evaluate the tigonometic function fo. Step Detemine the sign of the tigonometic function value fom the quadant in which lies. Signs of Function Values Quadant II Quadant I sin, csc : + cos, sec : sin, csc : + cos, sec : + tan, cot : tan, cot : + Quadant III Quadant IV sin, csc : cos, sec : tan, cot : + sin, csc : cos, sec : + tan, cot : 480 Chapte 9 Tigonometic Ratios and Functions
Evaluate a) tan 40º) and b) csc 17π. Using Refeence Angles to Evaluate Functions INTERPRETING MODELS This model neglects ai esistance and assumes that the pojectile s stating and ending heights ae the same. a. The angle 40º is coteminal with 10º. The efeence angle is = 180º 10º = 0º. The tangent function is negative in Quadant II, so tan 40º) = tan 0º =. b. The angle 17π is coteminal with 5π. The efeence angle is = π 5π = π. The cosecant function is positive in Quadant II, so csc 17π = csc π =. Solving a Real-Life Poblem The hoizontal distance d in feet) taveled b a pojectile launched at an angle and with an initial speed v in feet pe second) is given b d = v sin. Model fo hoizontal distance Estimate the hoizontal distance taveled b a golf ball that is hit at an angle of 50 with an initial speed of 105 feet pe second. Note that the golf ball is launched at an angle of = 50º with initial speed of v = 105 feet pe second. d = v sin Wite model fo hoizontal distance. = 105 sin 50 ) Substitute 105 fo v and 50º fo. 9 Use a calculato. The golf ball tavels a hoizontal distance of about 9 feet. = 0 = 40 π = 17π = 50 Monitoing Pogess Sketch the angle. Then find its efeence angle. 5. 10. 0 7. 7π 9 Evaluate the function without using a calculato. Help in English and Spanish at BigIdeasMath.com 8. 15π 4 9. cos 10º) 10. sec 11π 4 11. Use the model given in Eample 5 to estimate the hoizontal distance taveled b a tack and field long jumpe who jumps at an angle of 0 and with an initial speed of 7 feet pe second. Section 9. Tigonometic Functions of An Angle 481
9. Eecises Dnamic Solutions available at BigIdeasMath.com Vocabula and Coe Concept Check 1. COMPLETE THE SENTENCE An) is an angle in standad position whose teminal side lies on an ais.. WRITING Given an angle in standad position with its teminal side in Quadant III, eplain how ou can use a efeence angle to find cos. Monitoing Pogess and Modeling with Mathematics In Eecises 8, evaluate the si tigonometic functions of. See Eample 1.). 4. In Eecises 15, sketch the angle. Then find its efeence angle. See Eample.) 15. 100 1. 150 17. 0 18. 70 4, ) 5.. 5, 1) 19. 15π 4 1. 5π 0. 8π. 1π 7., 8) 8., 1). ERROR ANALYSIS Let, ) be a point on the teminal side of an angle in standad position. Descibe and coect the eo in finding tan. tan = = 1, 9) 1, ) In Eecises 9 14, use the unit cicle to evaluate the si tigonometic functions of. See Eample.) 9. = 0 10. = 540 11. = π 1. = 7π 1. = 70 14. = π 4. ERROR ANALYSIS Descibe and coect the eo in finding a efeence angle fo = 50. is coteminal with 90, whose teminal side lies in Quadant IV. So, = 90 70 = 0. In Eecises 5, evaluate the function without using a calculato. See Eample 4.) 5. sec 15. tan 40 7. sin 150 ) 8. csc 40 ) 9. tan π 4 ) 0. cot 8π ) 1. cos 7π 4. sec 11π 48 Chapte 9 Tigonometic Ratios and Functions
In Eecises, use the model fo hoizontal distance given in Eample 5.. You kick a football at an angle of 0 with an initial speed of 49 feet pe second. Estimate the hoizontal distance taveled b the football. See Eample 5.) 4. The fogbot is a obot designed fo eploing ough teain on othe planets. It can jump at a 45 angle with an initial speed of 14 feet pe second. Estimate the hoizontal distance the fogbot can jump on Eath. 8. REASONING A Feis wheel has a adius of 75 feet. You boad a ca at the bottom of the Feis wheel, which is 10 feet above the gound, and otate 55 counteclockwise befoe the ide tempoail stops. How high above the gound ae ou when the ide stops? If the adius of the Feis wheel is doubled, is ou height above the gound doubled? Eplain ou easoning. 9. DRAWING CONCLUSIONS A spinkle at gound level is used to wate a gaden. The wate leaving the spinkle has an initial speed of 5 feet pe second. a. Use the model fo hoizontal distance given in Eample 5 to complete the table. Angle of spinkle, Hoizontal distance wate tavels, d 5. At what speed must the in-line skate launch himself off the amp in ode to land on the othe side of the amp? 0 5 40 45 50 55 18 5 ft. To win a javelin thowing competition, ou last thow must tavel a hoizontal distance of at least 100 feet. You elease the javelin at a 40 angle with an initial speed of 71 feet pe second. Do ou win the competition? Justif ou answe. 7. MODELING WITH MATHEMATICS A ock climbe is using a ock climbing teadmill that is 10 feet long. The climbe begins b ling hoizontall on the teadmill, which is then otated about its midpoint b 110 so that the ock climbe is climbing towad the top. If the midpoint of the teadmill is feet above the gound, how high above the gound is the top of the teadmill? 0 b. Which value of appeas to maimize the hoizontal distance taveled b the wate? Use the model fo hoizontal distance and the unit cicle to eplain wh ou answe makes sense. c. Compae the hoizontal distance taveled b the wate when = 45 k) with the distance when = 45 + k), fo 0 < k < 45. 40. MODELING WITH MATHEMATICS You school s maching band is pefoming at halftime duing a football game. In the last fomation, the band membes fom a cicle 100 feet wide in the cente of the field. You stat at a point on the cicle 100 feet fom the goal line, mach 00 aound the cicle, and then walk towad the goal line to eit the field. How fa fom the goal line ae ou at the point whee ou leave the cicle? ft 5 ft 110? 00, ) stating position 50, 0) 100 ft? goal line Section 9. Tigonometic Functions of An Angle 48
41. ANALYZING RELATIONSHIPS Use smmet and the given infomation to label the coodinates of the othe points coesponding to special angles on the unit cicle. 1, 0, 1), 90 10 0 15 45, 1 150 0 0 180 0 1, 0) 10 0 5 15 40 00 70 4. THOUGHT PROVOKING Use the inteactive unit cicle tool at BigIdeasMath.com to descibe all values of fo each situation. a. sin > 0, cos < 0, and tan > 0 b. sin > 0, cos < 0, and tan < 0 4. CRITICAL THINKING Wite tan as the atio of two othe tigonometic functions. Use this atio to eplain wh tan 90 is undefined but cot 90 = 0. 44. HOW DO YOU SEE IT? Detemine whethe each of the si tigonometic functions of is positive, negative, o zeo. Eplain ou easoning. 4. MAKING AN ARGUMENT You fiend claims that the onl solution to the tigonometic equation tan = is = 0. Is ou fiend coect? Eplain ou easoning. 47. PROBLEM SOLVING When two atoms in a molecule ae bonded to a common atom, chemists ae inteested in both the bond angle and the lengths of the bonds. An ozone molecule is made up of two ogen atoms bonded to a thid ogen atom, as shown., ) d 18 pm 117 0, 0) 18 pm 18, 0) a. In the diagam, coodinates ae given in picometes pm). Note: 1 pm = 10 1 m) Find the coodinates, ) of the cente of the ogen atom in Quadant II. b. Find the distance d in picometes) between the centes of the two unbonded ogen atoms. 48. MATHEMATICAL CONNECTIONS The latitude of a point on Eath is the degee measue of the shotest ac fom that point to the equato. Fo eample, the latitude of point P in the diagam equals the degee measue of ac PE. At what latitude is the cicumfeence of the cicle of latitude at P half the distance aound the equato? cicle of latitude C P 45. USING STRUCTURE A line with slope m passes though the oigin. An angle in standad position has a teminal side that coincides with the line. Use a tigonometic function to elate the slope of the line to the angle. equato O D E Maintaining Mathematical Poficienc Find all eal zeos of the polnomial function. Section 4.) Reviewing what ou leaned in pevious gades and lessons 49. f ) = 4 + + + 8 1 50. f ) = 5 + 4 4 14 14 15 18 Gaph the function. Section 4.8) 51. f ) = + ) 1) 5. f ) = 1 4) + 5) + 9) 5. f ) = + 1) ) 484 Chapte 9 Tigonometic Ratios and Functions