4.2. Co-terminal and Related Angles. Investigate

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1 .2 Co-teminal and Related Angles <CATCH photo P-3: lage electic moto with someone standing beside it. One of the 3-stoe GE motos would be good.> Tigonometic atios can be used to model quantities such as the altenating-cuent electicit that powes electic motos, among othe electical devices. When solving poblems involving tigonometic quantities, thee is almost alwas moe than one solution sometimes an infinite numbe of solutions. It is impotant to find all possible solutions and then select which solutions ae appopiate fo the poblem. In this section, ou will lean how to identif diffeent angles that have the same tigonometic atio, as well as lean how the ae elated. Tools gid pape Investigate How can ou find diffeent angles with the same tigonometic atios? 1. a) In the fist quadant, has a sine of. Eplain wh. 2 b) Daw a unit cicle. Add a teminal am to epesent. Label the angle. 1_ 2. Label the - and -coodinates of the point whee the teminal am intesects the unit cicle. 3. Find anothe point on the unit cicle that has the same -coodinate. What is the -coodinate of this point?. a) Daw a second angle in standad position with its teminal am though the point ou found in step 3. b) What is the measue of this second angle?. Which of the tigonometic atios fo the two angles ae the same? Which ae diffeent? 6. Reflect Fo an chosen point on the unit cicle, how man othe points will have the same -coodinate? Eplain wh. 7. a) Which angle in the fist quadant has a cosine of 1_ 2? b) Daw a unit cicle and plot the point whee the teminal am of the angle intesects the cicle. Then, daw the teminal am of this angle. 232 MHR Functions 11 Chapte

2 8. Label the - and -coodinates of the point in step Find anothe point on the unit cicle that has the same -coodinate. What is the -coodinate of this point? 1. a) Daw a second angle in standad position with its teminal am though the point ou found in step 9. b) What is the measue of this second angle? 11. Which of the tigonometic atios fo the two angles ae the same? Which ae diffeent? 12. Reflect Fo an chosen point on the unit cicle, how man othe points will have the same -coodinate? Eplain how the ae elated. 13. Suppose that the cicle being used is not a unit cicle, but athe a cicle with adius. The coodinates (, ) of a point on the cicle ae no longe the cosine and sine of the angle, but ae elated to the cosine and sine. Stud the cicle shown. Wite epessions fo sin, cos, and tan in tems of,, and. (, ) Eample 1 Find Pima Tigonometic Ratios and Angles Using An Cicle and that /A lies in the fist quadant, detemine eact values fo cos A and tan A. a) Given that sin A 3_ b) Detemine the pima tigonometic atios fo anothe angle between and 36 that has the same sine value. c) Daw a diagam showing the locations of the two angles. How ae the two angles elated? d) Use a calculato to help ou find the two angles, to the neaest degee. Solution, possible values of and ae 3 and. Theefoe, let 3 and Use the Pthagoean theoem to find the value of a) Since sin A 3_.2 Co-teminal and Related Angles MHR 233

3 Since /A _ lies in the fist quadant,. cos A tan A 3_ b) The sine atio is positive in the fist and second quadants. The point that defines an angle with the same sine is (, 3). Let /B epesent the angle. tan B _ sin B _ 3_ cos B _ c) Fom the diagam, /B 18 /A. 3_ _ 3 Technolog Tip Kestokes va. On some calculatos, pess 2nd SIN to access the sin 1 function. Tpe 3 ENTER, and then pess ENTER. d) Ensue that the calculato is set to degee measue. 3_ sin A /A sin 1 ( _ 3 ) 37 /B 18 /A (, 3) B A 2 (, 3) Eample 2 Relations fo the Cosine and Tangent Ratios a) Detemine anothe angle between and 36 that has the same cosine as /A in Eample 1. What is the elationship between this angle and /A? b) Detemine anothe angle between and 36 that has the same tangent as /A in Eample 1. What is the elationship between this angle and /A? Solution a) Detemine anothe _ angle, α, such that cos α. Since the cosine atio is positive in the fouth quadant, the coodinates of the equied point ae (, 3). Fom the diagam, /α 36 /A A β α (, 3) (, 3) 23 MHR Functions 11 Chapte

4 . Since the tangent atio is positive in the thid quadant, the coodinates of the equied point ae (, 3). b) Detemine anothe angle, β, such that tan β 3_ Fom the diagam, β 18 /A Eample 3 Solve a Map Poblem The cit plan of Pot Foghon uses a Catesian gid, with each gid mak epesenting a distance of 1 km. The plan places Cit Hall at the oigin of the gid. Ted s house is at gid point ( 6, 2.). a) The angle of otation fo Suzette s house has the same tangent atio as the angle of otation fo Ted s house. Whee is Suzette s house? b) Find the angles in standad position if the lines dawn fom Cit Hall to each of the two houses ae teminal ams. Round ou answes to the neaest degee. Solution a) Ted s house is in the second quadant, so the tangent is negative. If Suzette s house has the same tangent atio, it must be in the fouth quadant at (6, 2.). b) Ted s house is at a eflection of (6, 2.) in the -ais. Suzette s house a eflection of (6, 2.) in the -ais. The efeence angle associated with these points is, whee tan _ Ted ( 6, 2.) Ted s house is at appoimatel Suzette s house is at appoimatel (6, 2.) (6, 2.) W N S E Suzette.2 Co-teminal and Related Angles MHR 23

5 Eample co-teminal angles angles in standad position that have the same teminal am Co-teminal Angles a) Find thee othe positive angles that have the same teminal am as 3. b) Find thee negative angles that have the same teminal am as 3. Solution a) Tun in a counteclockwise diection fom 3. If ou continue fo 36, ou aive back at the same teminal am , so 3 and 39 ae co-teminal angles. Two othe positive angles that ae co-teminal with 3 ae as follows: 3 2(36 ) 7 3 3(36 ) Connections A positive angle is geneated when the teminal am moves in a counteclockwise diection. A negative angle is geneated when the teminal am moves in a clockwise diection. b) Stating at the positive -ais and poceeding in a clockwise diection defines a negative angle. You will each the teminal am of 3 afte a otation of 33. Hence, 33 is co-teminal with 3. Two othe negative angles that ae co-teminal with 3 ae as follows: 3 2(36 ) (36 ) MHR Functions 11 Chapte

6 Ke Concepts The pima tigonometic atios fo the angle in standad position that has a point (, ) on its teminal am can be calculated as, and tan _, whee sin _, cos 2 2. Fo an given sine atio, two distinct angles between and 36 have this sine atio. (, ) Fo an given cosine atio, two distinct angles between and 36 have this cosine atio. Fo an given tangent atio, two distinct angles between and 36 have this tangent atio. Pais of elated angles can be found using the coodinates of the endpoints of thei teminal ams. Use a efeence angle in the fist quadant. Co-teminal angles ae angles with the same teminal am. The can be positive o negative. Communicate You Undestanding C1 Eplain wh thee ae eactl two angles between and 36 that have a given sine atio. C2 The teminal am of an angle is in the fist quadant. What kind of eflection will give the teminal am of an angle that has the same sine atio? the same cosine atio? the same tangent atio? C3 How man co-teminal angles can ou find fo an angle of 3? Eplain. C How do the tigonometic atios fo 3 elate to the tigonometic atios fo 39? Is this tue fo all co-teminal angles? Eplain. A Pactise Note: Unless othewise specified, assume that all angles ae between and 36. a) A(, 12) 8 Fo help with questions 1 and 2, efe to Eample The coodinates of a point on the teminal am of an angle ae shown. Detemine the eact pima tigonometic atios fo. A(, 12) Co-teminal and Related Angles MHR 237 Functions 11 CH.indd 237 6/1/9 :8:3 PM

7 b) B( 3, ) B( 3, ) 2. The coodinates of a point on the teminal am of an angle ae given. Detemine the eact pima tigonometic atios fo a) G( 8, 6) b) H(3, ) c) I( 1, 8) d) J(3, ) e) K(1, 2) f) L(6, 2) Fo help with questions 3 and, efe to Eamples 1 and 2. c) C( 6, 8) d) D(2, ) C( 6, 8) 8 D(2, ) One of the pima tigonometic atios fo an angle is given, as well as the quadant in which the teminal am lies. Find the othe two pima tigonometic atios. a) sin A 8_, fist quadant 17 b) cos B 3_, fouth quadant c) tan C _, second quadant 12 d) sin D 2_, thid quadant 3 e) cos E _, second quadant 6 f) tan F 12_, fist quadant 7. Detemine anothe angle that has the same tigonometic atio as each given angle. Daw a sketch with both angles labelled. a) cos b) sin 1 c) tan 3 d) sin 1 2 e) cos 23 f) tan 3 Fo help with questions and 6, efe to Eample. e) E( 1, 3) a) Detemine an thee positive angles that ae co-teminal with 12. b) Detemine an thee negative angles that ae co-teminal with Detemine the eact pima tigonometic atios fo each angle. You ma wish to use a unit cicle to help ou. E( 1, 3) 3 a) /A b) /B 12 c) /C d) /D 31 e) /E 2 f) /F MHR Functions 11 Chapte

8 B Connect and Appl 7. Without using a calculato, detemine two angles between and 36 that have a cosine of _ Two angles between and 36 have a tangent of 1. Without using a calculato, detemine the angles. 9. The cosine of each of two angles between and 36 is 1. Without using a 2 calculato, detemine the angles. 1. Two angles between and 36 have a tangent that is undefined. What ae the angles? Wh is the tangent undefined fo each of these? 11. The point P(, 9) is on the teminal am of /A. a) Detemine the pima tigonometic atios fo /A and /B, such that /B has the same sine as /A. b) Use a calculato and a diagam to detemine the measues of /A and /B, to the neaest degee. 12. The point R( 3, ) is on the teminal am of /E. Repesenting Reasoning and Poving Poblem Solving a) Detemine Connecting Reflecting the pima Communicating tigonometic atios fo /E and /F such that /F has the same tangent as /E. b) Use a calculato and a diagam to detemine the measues of /E and /F, to the neaest degee. 13. Use Technolog Open The Geomete s Sketchpad. Selecting Tools a) Daw a cicle with a adius of gid units. Plot a point A on the cicle in the fist quadant. b) Measue the coodinate distance between point A and the oigin. Change the label to. Measue the - and -coodinates of point A. c) Constuct fomulas to calculate the sine, cosine, and tangent of /A defined b teminal am OA, using the measues of,, and. d) Right-click on point A and select Animate Point. Obseve the values of the tigonometic atios as A moves aound the cicle. Pause the animation at selected points at which ou know the atios, and compae the values on the sceen to ou knowledge. e) T the contols in the Motion Contolle. Detemine what each of them does. You can also contol the animation fom the Displa menu. f) As the sine inceases in the fist quadant, what happens to the cosine? What happens to the tangent? What happens in the second quadant? 1. An acute angle has the point A(p, q) on its teminal am. Repesenting Connecting Reasoning and Poving Poblem Solving Selecting Tools Reflecting a) Find an Communicating epession fo the distance OA in tems of p and q. b) Wite eact epessions fo the pima tigonometic atios fo. c) Locate the angle 9. Sketch the teminal am fo this angle. Detemine the coodinates of a point B on the teminal am of 9 in tems of p and q. d) Wite eact epessions fo the pima tigonometic atios fo 9. e) Compae the epessions that ou found in pats b) and d)..2 Co-teminal and Related Angles MHR 239

9 1. Chapte Poblem Fo the second leg of ou oienteeing couse, calculate the diection and distance. Caefull daw the leg on ou map, and label all distances and angles. Note: You ma not use a calculato. Diection: Face south. Tun left though an angle with a cosine of and a sine of 1. Distance: Find two angles between and 36 with a sine of 1. Subtact the 2 smalle angle fom the lage angle. Find the sine of the esulting angle. Multipl b to obtain the distance. Achievement Check 16. Conside /C such that sin C 7_ 2. a) What ae the possible quadants in which /C ma lie? b) If ou know that cos C is negative, how does ou answe to pat a) change? c) Sketch a diagam to epesent /C in standad position, given that the condition in pat b) is tue. d) Find the coodinates of a point P on the teminal am of /C. e) Wite eact epessions fo the othe two pima tigonometic atios fo /C. C Etend 17. The side length of a hombus is s. One of its diagonals has the same length. Detemine an eact epession fo the length of the othe diagonal. 18. A egula octagon has side length. A line segment is dawn joining two of its vetices to fom a tiangle and an iegula heptagon. Detemine an epession fo the eact length of this line segment. 19. Use Technolog How can ou use a gaphing calculato to check answes when detemining two angles with the same tigonometic atio? Fo eample, suppose ou ae asked to detemine the values of such that sin.. a) Set the window vaiables as shown. Ente the left side as Y1 and the ight side as Y2. b) Pess GRAPH. Pess 2nd [CALC]. Select :intesect. Use the Intesect opeation twice to detemine the values that satisf the condition. 2. Math Contest A clock face is in the fom of a coodinate sstem with the oigin at the cente of the face. The minute hand passes though the point ( 1, 3 ). The hou hand is somewhee between the and the. The time is A : B :1 C :1 D :7 E : Math Contest Given A(, ), B(3, 3 3 ), and C(2 3, 2), what is the aea, in squae units, of nabc? 3 A 12 B C 6 2 D 3 3 E Math Contest If 7, what is the value of 12 1? A 3 B 7 C 21 D 27 E 31 2 MHR Functions 11 Chapte

10 Use Technolog Use a Compute Algeba Sstem to Find Eact Tigonometic Ratios and Angles A compute algeba sstem (CAS) found on calculatos such as the TI-Nspie CAS gaphing calculato can displa eact values of the tigonometic atios fo the special angles ou leaned about in Section.1. With its equation-solving powe, a CAS can also find all angles that satisf a given function, unlike an odina calculato that gives onl one answe. This activit is witten fo the CAS on a TI-Nspie CAS gaphing calculato. Othe sstems can be used. Tools TI-Nspie CAS gaphing calculato A: Displa Eact Values 1. Tun on the TI-Nspie CAS gaphing calculato. Pess c and select 8:Sstem Info. Select 2:Sstem Settings. Use the e ke to scoll down to Angle, and ensue that it is set to Degee. Continue on to Auto o Appo and ensue that it is set to Auto. Continue down to OK, and pess twice. 2. Pess c and select 6:New Document. Select 1:Add Calculato. 3. Ente sin(3) and pess. Notice how the esult is displaed. Investigate cos(3) and tan(3). Do ou see anthing unusual about the displa? Eplain wh this is equivalent to the value that ou detemined using a unit cicle.. You can use a CAS to obtain eact answes to tigonometic poblems. Fo eample, in nabc, b 1 cm, c 1 cm, and /A. Find the eact measue of c using the cosine law. Use Technolog: Use a Compute Algeba Sstem to Find Eact Tigonometic Ratios and Angles MHR 21

11 B: Find Angles 1. When ou use an odina calculato to find an angle such that sin., ou onl get one answe. If ou use the sin 1 opeation on the TI-Nspie CAS gaphing calculato, the same thing happens. T it. Technolog Tip To obtain the sign, pess / and then <. Technolog Tip You can foce the calculato to give ou an appoimate answe without esetting the mode b pessing / befoe pessing. 2. Now, use the equation-solving powe of the CAS. Pess b. Select 3:Algeba. Select 1:solve. Tpe sin()., ) 36. Pess. Notice that the CAS has found the angles between and 36 that have a sine of.. The domain of the solution needs to be esticted to answes between and 36. This is done with the epession following the with opeato ( ). 3. T the same thing with the equation cos.. Befoe using the CAS, pedict the answes. Then, use the CAS to see if ou pedictions ae coect. Notice that, unlike an odina calculato, the CAS does not use a negative angle fo the fouth quadant.. The CAS can also solve moe comple tigonometic equations. Ae thee an angles between and 36 that satisf the equation sin cos 1? Wite down ou pediction, and justif wh ou think ou ae coect. Then, use the CAS to check ou pediction.. Wite a tigonometic equation that ou think has a solution. Tade equations with a classmate and solve each othe s equation. Discuss the esults. 22 MHR Functions 11 Chapte