Name Date. Trigonometric Functions of Any Angle For use with Exploration 5.3

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1 5.3 Tigonometic Functions of An Angle Fo use with Eploation 5.3 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 1 EXPLORATION: 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. b. c. 1, 3 1, 1 0, 1) 175 Copight Big Ideas Leaning, LLC All ights eseved.

2 Name Date 5.3 Tigonometic Functions of An Angle continued) 1 EXPLORATION: Witing Tigonometic Functions continued) d. e. f. 1, 0) 1, 3 1, 1 Communicate You Answe 2. How can ou use the unit cicle to define the tigonometic functions of an angle? 3. Fo which angles ae each function undefined? Eplain ou easoning. a. tangent b. cotangent c. secant d. cosecant Copight Big Ideas Leaning, LLC All ights eseved. 176

3 5.3 Notetaking with Vocabula Fo use afte Lesson 5.3 In ou own wods, wite the meaning of each vocabula tem. unit cicle quadantal angle efeence angle Coe Concepts 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 + = 2. The si tigonometic functions of ae defined as shown. sin = cos = csc =, 0 sec =, 0, ) tan =, 0 cot =, 0 These functions ae sometimes called cicula functions. 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 = = = cos = = = 1 1, ) = 1 Notes: 177 Copight Big Ideas Leaning, LLC All ights eseved.

4 Name Date 5.3 Notetaking with Vocabula continued) 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 π angles such that 90 < < 360 o, in adians, < < 2 π. 2 Quadant II Quadant III Quadant IV Degees: = 180 Degees: = 180 Degees: = 360 Radians: = π Radians: = π Radians: = 2π Notes: Evaluating Tigonometic Functions Use these steps to evaluate a tigonometic function fo an angle : Step 1 Find the efeence angle. Step 2 Evaluate the tigonometic function fo. Step 3 Detemine the sign of the tigonometic function value fom the quadant in which lies. Notes: 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 : Copight Big Ideas Leaning, LLC All ights eseved. 178

5 5.3 Notetaking with Vocabula continued) Pactice A Eta Pactice In Eecises 1 and 2, evaluate the si tigonometic functions of , 6) 4, 3) In Eecises 3 and 4, use the unit cicle to evaluate the si tigonometic functions of. 3. = = 4π In Eecises 5 and 6, sketch the angle. Then find its efeence angle π Evaluate the function csc 150 without using a calculato. 179 Copight Big Ideas Leaning, LLC All ights eseved.

6 Pactice 8.3 BPactice B In Eecises 1 4, evaluate the si tigonometic functions of , 3) 4, 3) , 3) 2, 4) In Eecises 5 7, use the unit cicle to evaluate the si tigonometic functions of. 5. 5π π 2 In Eecises 8 13, find the angle s efeence angle π π π 3 In Eecises 14 16, evaluate the function without using a calculato. 14. cot sin π sec The hoizontal distance d in feet) taveled b a pojectile launched at an angle and 2 with an initial speed v in feet pe second) is given b d = v 32 sin 2. To win a shot-put competition, ou last thow must tavel a hoizontal distance of at least 15 feet. You elease the shot put at a 45 angle with an initial speed of 22 feet pe second. Do ou win the competition? Justif ou answe. Copight Big Ideas Leaning, LLC All ights eseved. 180