Appendix: Trigonometry Basics

Transcription

1 HM LaTorre APP pg NO_ANNO 8/5/0 :6 AM Page 78 Appendi: Trigonmetr Baic 78 Appendi: Trigonometr Baic Thi Appendi give upplementar material on degree meaure and right triangle trigonometr. It can be ued in connection with Section 8.. Degree Meaure One complete revolution of a circle i divided into 60 equal part, and each part i called a degree. Talking about mall part of a rotation i awkward, o we call partial rotation angle. The meaure of an angle i decribed b the amount of rotation in the turn. For eample, a 90-degree angle i one-fourth of a complete counterclockwie revolution. (See Figure A..) A mall circle a a upercript i ued a a mbol to indicate degree (90 ). 90 FIGURE A. A 90 angle i _ 4 of a complete revolution Degree Meaurement 60 degree 60 complete revolution degree of a complete revolution 60 The election of 60 a the number of diviion i rooted in hitor and ma reflect the fact that the Earth complete one revolution about the un in approimatel 65 da. Becaue 65 doe not have a large number of divior (onl 5 and 7), the inventor of the tem probabl picked 60 with it multitude of divior (,, 4, 5, 6, 8, 9, 0,, 5, 8, 0, 4, 0, 6, 40, 45, 60, 7, 90, 0, and 80) a being cloe to the number of da of the ear and ea to ue in computing. That fortunate choice give u a large number of even-degree angle that are imple fraction of a full revolution. EXAMPLE Meauring Angle in Degree a. Epre 8 of a complete rotation in degree. b. Epre of a complete rotation in degree.. It i poible to divide one revolution into 400 equal part. Incidentall, ome do. Thee part are called gradian or grad, and thi angle meaure i available on man calculator. Of coure, ou might prefer to be epeciall patriotic and divide one revolution into 776 equal part. In thi cae, ou would be inventing a new angle meaure.

2 HM LaTorre APP pg NO_ANNO 8/5/0 :7 AM Page APPENDIX Solution Keeping in mind that one complete rotation i 60, we have a. 8(60 ) 45 (See Figure A.a) b. (60 ) 0 (See Figure A.b) 45 0 (a) _ 8 of a revolution i a 45 angle FIGURE A. (b) _ of a revolution i a 0 angle In the tud of trigonometr, we conider revolution around a particular circle. Specificall, we conider the unit circle that i, the circle with radiu, centered at the origin. We a that an angle i in tandard poition when the verte of the angle i at the origin and one of it ide i drawn along the poitive -ai. The ide that i drawn on the poitive -ai i called the initial ide. The other ide of the angle i called the terminal ide. Refer to Figure A.. Figure A.4 how a 90 angle, a 60 angle, and a 50 angle. Terminal ide Radiu = Initial ide FIGURE A. FIGURE A.4 Becaue a 90 angle i one-quarter of a rotation (one quarter of the wa around the full circle), a 80 angle would be one-half of a rotation, a 70 angle would be threequarter of a rotation, and a 60 angle i one full rotation. Note that in Figure A.4, the 60 angle i drawn between a quarter rotation and a half rotation, and the 50 angle i drawn between a half rotation and a three-quarter rotation. You hould alo note that all of the rotation are drawn in the counterclockwie direction. We conider a poitive angle to define counterclockwie rotation and a negative angle to define clockwie rotation. For eample, Figure A.5 illutrate a 90 angle and a 00 angle.

3 HM LaTorre APP pg NO_ANNO 8/5/0 :7 AM Page 78 Appendi: Trigonmetr Baic When angle meaure are larger than 60, the decribe more than one rotation around the circle. For eample, an 80 angle decribe two full rotation plu onequarter of a rotation in the counterclockwie direction ( ). A 505 angle decribe one full rotation plu one-quarter of a rotation plu an etra 55 angle in the clockwie direction ( ). Thee two angle are hown in Figure A.6. FIGURE A FIGURE A.6 Before defining trigonometric function for angle on a unit circle, we take a brief look at the trigonometric function defined in term of a right triangle. Hpotenue Leg adjacent to FIGURE A.7 Leg oppoite Right-Triangle Trigonometr From the earl ue of trigonometr (meaning triangle meaurement ) through preent-da application, right triangle have provided a method for olving problem that involve indirect meaurement. A right triangle i a triangle with one 90 angle. In application, we generall label one of the remaining angle of the right triangle a angle. (See Figure A.7.) We define the ine of the angle a the ratio of the length of the leg oppoite the angle to the length of the hpotenue (the ide oppoite the right angle). The coine of the angle i the ratio of the length of the adjacent leg to the length of the hpotenue. The tangent of the angle i the ratio of the length of the leg oppoite the angle to the length of the adjacent leg. We abbreviate coine b co, ine b in, and tangent b tan. Right-Triangle Trigonometric Definition For a right triangle with one of the non-90 angle having meaure, the ine, coine, and tangent of are defined in term of the ide of the triangle. in co tan length of the leg oppoite the angle length of the hpotenue length of the leg adjacent to the angle length of the hpotenue length of the leg oppoite the angle length of the leg adjacent to the angle

4 HM LaTorre APP pg NO_ANNO 8/5/0 :7 AM Page APPENDIX Thee three function are illutrated in Figure A.8. Hpotenue oppoite leg in = hpotenue FIGURE A.8 Oppoite leg Hpotenue Adjacent leg adjacent leg co = hpotenue Adjacent leg oppoite leg tan = adjacent leg Oppoite leg Three other function that are commonl tudied in trigonometr are the ecant, coecant, and cotangent function. The are defined in term of the firt three trigonometric function a follow: ecant : ec length of the hpotenue co length of the adjacent leg coecant : cc length of the hpotenue in length of the oppoite leg cotangent : cot length of the adjacent leg tan length of the oppoite leg We tud primaril in, co, and tan becaue the other three trigonometric function are defined in term of them. If we are given an right triangle for which we know the angle and know the length of one ide of the triangle, we can ue the trigonometric function in, co, and tan to obtain the length of the other two ide of the triangle. For eample, if we have a right triangle with a 5 angle and hpotenue of length inche, we can ue in 5 and co 5 to find the length of the two leg. (See Figure A.9.) inche b 5 FIGURE A.9 a A B C To find the length of the leg adjacent to the 5 angle, we ue the fact that co 5 i the ratio of the length of the adjacent leg to the length of the hpotenue. We will ue a a to repreent the length of the adjacent leg, o co 5. Solving for a, we have a co 5 (0.906).86 inche That i, the leg adjacent to the 5 angle i approimatel.8 inche long. Similarl, the length of ide b can be found a b in 5 (0.46) inch Trigonometr i often ued in application where indirect meaurement of the length of one or more leg of a triangle i necear. Land urveing often relie on the ue of an intrument called a etant to meaure angle and on trigonometr to determine ditance uing right triangle.

5 HM LaTorre APP pg NO_ANNO 8/5/0 :7 AM Page 785 Appendi: Trigonmetr Baic 785 EXAMPLE Catle Meaurement A oldier in ancient time making ue of triangle ratio i illutrated in Figure A.0. He i uing a etant and ighting it in line with the top of the catle wall. The etant give an angle of 4.6. The oldier then count the brick in the wall and etimate the ditance d without having to enter the battle zone between hi afe pot and the catle wall. h = 4.6 d FIGURE A.0 a. Etimate the height of the wall if each brick i inche tall. b. Find the ditance to the catle. c. Find the ditance from the oldier to the top of the catle wall. Solution a. There are laer of brick in the wall, o the wall i ( brick)( inche per brick) 4 inche 0. feet tall. b. Uing the definition of the tangent of an angle, we know that tan 4.6 o we have d 0. feet d 0. feet 0. feet 77.4 feet tan The oldier i approimatel 77.4 feet awa from the catle. c. The Pthagorean Theorem ield h feet The oldier i approimatel 80 feet awa from the top of the catle wall. Two pecial right triangle that occur in application uing trigonometr are the 0 60 right triangle and the right triangle. The value of ine, coine, and tangent for one of thee triangle are eplored in the net eample. The other triangle i left for an activit. EXAMPLE Ratio for a Special Triangle The 0 60 right triangle i o named becaue in a right triangle with a 0 angle, the remaining angle mut be 60. A reflection of the triangle acro the leg between

6 HM LaTorre APP pg NO_ANNO 8/5/0 :7 AM Page APPENDIX 60 0 h 60 FIGURE A. the 0 angle and the right angle form an equilateral triangle (one whoe three ide are of equal length). The line of reflection i the perpendicular biector of the bae. (See Figure A..) a. Ue the Pthagorean Theorem to find the length h in term of. b. Ue the ide h and to find the ine, coine, and tangent for the 0 angle in a 0 60 right triangle. c. Ue the ide h and to find the ine, coine, and tangent for the 60 angle. Solution algebra a. The ide oppoite the 0 angle ha length that i equal to half of the length of the hpotenue. Appling the Pthagorean Theorem ield h () 60 Solving for h, we find the remaining leg length to be h () 4 b. Referring to Figure A., we find the value of ine, coine, and tangent for the 0 angle in the right triangle. 0 in 0 FIGURE A. co 0 tan 0 c. Similarl, the value of ine, coine, and tangent for the 60 angle follow. in 60 co 60 tan 60 Right triangle aociated with angle FIGURE A. Unit-Circle Trigonometr Right-triangle trigonometr i ueful in man application. However, a we previoul noted, the trigonometric function are not retricted to ue with angle that are le than 90. We can now define the trigonometric function ine, coine, and tangent for all angle b referring to our knowledge of right-triangle trigonometr. When the angle i between 0 and 90, we draw a right triangle in the unit circle uch that the hpotenue of the triangle i along the terminal ide of the angle from the origin to the unit circle, and the other two leg of the right triangle are drawn along the -ai and perpendicular to the -ai, a hown in Figure A.. If we let (, ) repreent the point where the terminal edge of angle interect the unit circle, then the leg drawn along the -ai ha length and the leg drawn perpendicular to the -ai ha length. The hpotenue ha length (the radiu of the

7 HM LaTorre APP pg NO_ANNO 8/5/0 :7 AM Page 787 Appendi: Trigonmetr Baic 787 unit circle). We define the trigonometric function in, co, and tan a we did for the right triangle, o that in co tan We define the trigonometric function imilarl for angle that are not between 0 and 90. Firt, draw the angle on the unit circle. Then draw a right triangle uch that the hpotenue of the triangle i along the terminal ide of the angle from the origin to the unit circle, and the other two leg of the right triangle are drawn along the -ai (poibl in the negative direction) and perpendicular to the -ai (poibl in the negative direction) a hown in Figure A.4a and b. (,)=(-a,b) b -a -d -c (,)=(-c,-d) FIGURE A.4 (a) (b) Once again, call the point where the terminal edge of angle interect the unit circle (, ), and let a repreent the length of the leg drawn along the -ai and let b repreent the length of the leg drawn perpendicular to the -ai. The hpotenue ha length (the radiu of the unit circle). However, we mut indicate whether the leg of the triangle are drawn in the negative and/or the negative direction. We indicate thi b writing the negative of the length in the appropriate direction. For eample, the triangle in Figure A.4a ha one leg along the negative portion of the -ai. Thi leg ha length a in the negative direction, o a. The other leg i drawn up from the -ai; thu it ha poitive direction and ha length b. We define the trigonometric function in, co, and tan a we have done previoul, o that in and tan, co, Thu, for the angle drawn in Figure A.4a, in b, co a, and tan b. a b Similarl, for the angle drawn in Figure A.4b, in d, co c, and tan d a. c d c. In general, for an angle, we define ine, coine, and tangent a follow: Unit-Circle Trigonometric Definition Let (, ) be the point on the unit circle where the terminal ide of the angle interect the circle. Then the ine, coine, and tangent of the angle are in co tan

8 HM LaTorre APP pg NO_ANNO 8/5/0 :7 AM Page APPENDIX Becaue the unit circle i given b the equation, it i true that for an angle,co in. EXAMPLE 4 Ratio for a Specific Angle -40 A 40 angle i hown in Figure A.5. a. Draw the right triangle aociated with a 40 angle. Label the angle in the triangle between the terminal ide and the -ai. b. Find the ine, coine, and tangent for a 40 angle. Solution a. FIGURE A.5 60 Right triangle aociated with the -40 angle FIGURE A.6 b. Thi right triangle i one of the pecial triangle mentioned in Eample. It i a 0 60 right triangle. We know from Eample that in 60 and co 60. Thu the leg on the -ai i poitive and ha length co 60, o. The other leg i drawn down from the -ai, o it i negative and ha length in 60, o. Add thee value to the figure of the 40 angle. See Figure A.7. Now we can read the trigonometric value from the figure: in( 40 ) co( 40 ) - tan( 40 ) FIGURE A.7 We have now defined trigonometric function a the appl to right triangle and, in the much broader ene, a the appl to unit circle. However, our dicuion of unit-circle trigonometr would not be complete if we did not conider another angle meaure called radian meaure. In fact, we cannot ue the trig function in calculu without conidering radian meaure of angle. Radian Meaure We now conider another unit of angle meaurement called a radian. Picture a circular pizza. We can decribe a lice of pizza in term of the angle of the wedge

9 HM LaTorre APP pg NO_ANNO 8/5/0 :7 AM Page 789 Appendi: Trigonmetr Baic of a pizza i a 60 wedge. FIGURE A.8 u r The radian meaure of an angle i the ratio of the arc length to the radiu. FIGURE A.9 r formed b the lice. For intance, in angular term we would decribe one-eighth of a pizza a a 45 wedge. A one-ith lice would be decribed a a 60 wedge (ee Figure A.8). Another wa to think of a lice of pizza i in term of the amount of crut around the edge. If there were omething pecial in the crut, like a cheee filling, then we might want to focu on the length of the crut around the circular edge of the lice. In doing o, we would be talking about an arc of a circle that i, the lice portion of the circumference. (A portion of the circle i called an arc, and the complete ditance around the circle i it circumference.) If the radiu r of the pizza in Figure A.8 were foot, then the circumference of the pizza would be r ( foot) feet 6. feet. Baic geometr tell u that the arc length i the ame fractional part of the circumference of the circle a the angle i of one complete rotation. Thu the wedge would form an arc of 60 ( feet) 0.8 foot of the pecial cheee crut. The wedge would form an arc of 60 ( feet) foot. We define the radian meaure of an angle to be the ratio of the length of the arc cut out of the circumference of a circle b the angle to the radiu r of the circle. (See Figure A.9.) Becaue the total arc length (that i, the circumference) of a circle i time it radiu, a full revolution ha radian meaure of. Note that i a real number approimatel equal to 6.8. It i important to recognize that the radian meaure of an angle i a real number with no unit attached. In the definition of radian meaure, both and r meaure length and mut have the ame unit, which caue to be a unitle quantit. At firt thi might eem to be a trange wa to meaure angle, but we later find that radian meaure of angle give imple calculu reult for the trigonometric function with which we will be dealing. Radian Meaurement radian complete revolution radian of a complete revolution EXAMPLE 5 Meauring Angle in Radian a. Epre 8 of a complete rotation in radian. b. Epre 4 of a complete rotation in radian. c. Epre 4 radian in term of complete rotation. Solution Keeping in mind that complete rotation i radian, we have a. 8( ) 4 radian b. 4 ( ) radian c. Becaue radian of a complete revolution, 4 radian i 4 complete rotation.

10 HM LaTorre APP pg NO_ANNO 8/5/0 :7 AM Page APPENDIX The odometer on an automobile convert turn of the drive haft into ditance o that ou can oberve the number of mile ou have traveled. Although the net eample eplore thi idea, it i not unique to automobile. Whenever wheel rotate to caue movement, the ame tpe of relationhip between rotation of the wheel and ditance traveled hold true. EXAMPLE 6 Rotating Wheel Conider a front-wheel-drive automobile with wheel that are 0 inche in diameter. a. How far will rotation of the wheel caue the automobile to travel? b. How man time will the wheel revolve when the automobile travel mile? Solution a. A an automobile travel, it drive haft and front tire turn at the ame rate; that i, revolution in one caue revolution in the other. If the wheel are 0 inche in diameter (radiu 0 inche), then each rotation caue the automobile to move forward r ( )(0 inche) 6.8 inche 5. feet. Each rotation of the wheel caue the automobile to travel about 6.8 inche or 5. feet. b. There are 580 mile feet inche foot 6,60 inche in a mile. So inche 6,60 mile revolution revolution inche mile The wheel mut revolve approimatel time in each mile traveled. Angle Meaure Converion Whenever ou have two meaurement cale that can be applied to the ame object, it i important to have a converion technique. In thi cae, revolution i both 60 and radian. Thu degree are converted to radian b multipling b 60 80, and 80 radian are converted to degree b multipling b. Our converion formula i Angle Converion Formula radian 80 Up to thi point, we have ued the word radian to pecif our choice of angle meaure, not the unit of the angle. You ma have noted that it i cumberome to write the word radian each time angle meaure i ued. Therefore, we adopt the following convention. Angle Meaure Convention All angle are undertood to be meaured in radian unle the degree mbol i ued to pecif degree meaure of the angle.

11 HM LaTorre APP pg NO_ANNO 8/5/0 :7 AM Page 79 Appendi: Trigonmetr Baic 79 That i, an angle with meaure 60 i quite different than one with meaure 60. An angle of 60 radian i more than 9 full revolution, wherea an angle of 60 i onl one-ith of a revolution. If ou mean an angle of 60 degree, be certain that ou ue the degree mbol with the angle. EXAMPLE 7 Converting Angle A B C 7 6 a. Epre in degree. b. Epre 5 in radian. Solution a. b (5 ) 80 4 Onl radian meaure of angle i ued in Chapter 8 of thi tet becaue our purpoe i to appl calculu to trig function. The definition for the trig function given in thi Appendi coincide with the definition given in Section 8.. Concept A. Inventor Degree meaure of angle Initial and terminal ide and tandard poition of angle Right-triangle trigonometr: ine, coine, and tangent Unit-circle trigonometr: ine, coine, and tangent Radian meaure of angle Angle meaure converion A. Activitie. Becaue the number 60 ha o man integer divior, man fractional revolution have nice epreion when epreed a angle meaured in degree. Complete Table A. of fractional rotation or full rotation and their aociated degree meaure.. There are man eail epreed fraction of a full turn, and the meaure of thee epreion in radian i ea to determine. Complete Table A. of TABLE A. Rotation (in turn) Angle meaure (in degree) Rotation (in turn) Angle meaure (in degree) Rotation (in turn) 60 8 Angle meaure (in degree) 4 5 6

12 HM LaTorre APP pg NO_ANNO 8/5/0 :7 AM Page APPENDIX TABLE A. Rotation (in turn) 4 4 Angle meaure (in radian) 4 Rotation (in turn) Angle meaure (in radian) Rotation (in turn) 60 8 Angle meaure (in radian) fractional rotation or full rotation and their aociated radian meaure.. Convert the following angle in degree into angle in radian, and ketch the angle on a unit circle. a. 0 b. 700 c. 90 d Convert the following angle in radian to angle in degree, and ketch the angle on a unit circle. 7 a. b. c. d Superbike II, the $5,000 bab of the U.S. Ccling Federation (USCF), made it debut in the 996 Olmpic. Superbike II front wheel i.6 inche in diameter, and it rear wheel i 7.56 inche in diameter. Over a ditance of mile, how man turn doe each wheel make? 6. Ue degree meaure to write the angle that the tenth wheel on the odometer turn when the drive haft of the automobile in Eample 6 complete one turn. 7. a. If there are 0 equall paced eat on a Ferri wheel, through what angle doe the wheel move between top to releae paenger in two conecutive eat? b. What i the radian meaurement of thi angle? c. If the Ferri wheel i 00 feet in diameter, what i the ditance traveled b each of the eat a the operator top between conecutive eat to echange paenger? 8. a. If the owner of the automobile in Eample 6 replaced the tire with overized tire that were inche in diameter, what would be the error created on the odometer reading? b. If the tire were replaced with tire that have an 8-inch diameter, what would be the error caued on the odometer? c. If the regular tire (0-inch diameter) lot 0.00 inch of their diameter through wear, how would the odometer reading be affected? 9. For finer meaurement, the degree i further divided into 60 part, and each uch part i called a minute. The mbol ued to indicate minute i the ingle prime; for eample, 5 minute i epreed a 0 5. An angle coniting of 00 minute could alo be epreed a 0. For even finer meaurement, the minute i further divided into 60 equal part called econd. The mbol for a econd i the double prime, o an angle of 0 degree 4 minute 7 econd would be written a You might tr to etimate how mall a econd i to realize how much preciion i ued when ome job tate acceptable tolerance in term of a econd of arc. Convert into decimal degree equivalent the following angle given in degree, minute, and econd. a b Refer to the definition in Activit 9, and convert into meaure uing degree, minute, and econd the following angle given in decimal degree. a. 5.4 b The Saturn miile ued in the moon hot had guidance computer placed on a table gimbaled platform that had to ta within 4 econd of arc for the firt 5 minute of the launch.

13 HM LaTorre APP pg NO_ANNO 8/5/0 :7 AM Page 79 Appendi: Trigonmetr Baic 79. Suppoe ou were to divide a rotation into 776 piece. We will call each piece a patriotic unit and will ue the abbreviation PU. a. How man rotation doe each of the following patriotic unit meaurement repreent? i. 776 PU ii. PU iii. 888 PU iv. 444 PU v. PU vi. PU b. Convert each of the patriotic unit meaurement in part a to radian. c. Find the value of each of the following: i. co(776 PU) ii. in(776 PU) iii. co(666 PU) iv. in(666 PU). Creating different angle meaurement tem uch a degree and grad for partial turn and calculating their converion factor i like making a 5-hour clock. Mr. Morton Rachofk ha built a 5-hour clock for the Circadian Clock Compan. Thi clock divide the 86,400 econd in the tandard da into 5 equal-length period called hour. Noon on the clock i the ame a noon on our regular time cale, but the other hour mark are different. Scientit conducting eperiment in the 90 oberved people in cave where the could not ee the un. Thee people developed activit ccle that lated 5 hour. Becaue we cannot change the olar da, Mr. Rachofk aid, Wh not change the clock? a. How long are Mr. Rachofk hour in our regular minute? b. What time on Mr. Rachofk clock i the regular clock time :00 p.m.? c. What i the regular clock time when it i 6:00 p.m. on Mr. Rachofk clock? For Activitie through 8, identif the trigonometric function ou would ue to find the length of the indicated ide. Aume that the triangle given i a right triangle with one of the other two angle identified a angle.. Given the length of the hpotenue, find the length of the leg adjacent to angle. 4. Given the length of the leg oppoite angle, find the length of the leg adjacent to angle. 5. Find the length of the leg oppoite angle, given the length of the leg adjacent to angle.. New York Time, October 7, 996, page Find the length of the hpotenue, given the length of the leg oppoite angle. 7. Given the length of the leg adjacent to angle, find the length of the hpotenue. 8. Given the length of the hpotenue, find the length of the leg oppoite angle. For Activitie 9 through 4, olve for the length of the indicated ide. Aume that the triangle given i a right triangle. 9. One angle of the triangle i 0. The leg oppoite thi angle i 5 inche long. a. Find the length of the leg adjacent to the given angle. b. Find the length of the hpotenue. 0. One angle of the triangle i 78. The leg adjacent to thi angle i meter long. a. Find the length of the hpotenue. b. Find the length of the leg oppoite the given angle.. One angle of the triangle i 5.. The hpotenue i centimeter long. a. Find the length of the leg adjacent to the given angle. b. Find the length of the leg oppoite the given angle.. One angle of the triangle i The leg oppoite thi angle i 5 inche long. a. Find the length of the leg adjacent to the given angle. b. Find the length of the hpotenue.. One angle of the triangle i The leg adjacent to thi angle i.5 mile long. a. Find the length of the hpotenue. b. Find the length of the leg oppoite the given angle. 4. One angle of the triangle i 0.. The hpotenue i centimeter long. a. Find the length of the leg adjacent to the given angle. b. Find the length of the leg oppoite the given angle. 5. A gable i the triangular egment of a wall created b the roof. The pitch of a gable i it height divided b it width. See Figure A.0. (That i, pitch i equal to half the lope of the roof.) Conider a gable 40 feet wide with an angle of 0 at it peak.

14 HM LaTorre APP pg NO_ANNO 8/5/0 :7 AM Page APPENDIX Height FIGURE A.0 Width a. How tall i the attic at it center? b. What i the pitch of the gable? c. How much board would be needed to cover the roof (not including the overhanging portion of the roof) if the houe i 40 feet long? 6. Find the diameter of the mallet iron rod from which a heagonal nut with ide 4mm can be cut. (Hint: The angle between two adjacent ide of the nut i 0.) 7. A tairwa i to be contructed on a hill with a 4 incline. a. If each tep i to have a 7-inch rie, what mut be it tread (horizontal depth)? b. How man 7-inch tep will be needed if the length of the hill (meaured up the lope) i 4 feet inche? 8. A boat that i ailing S 47 E i ailing on a trajector that i along an angle 47 eat of true outh. A hip ail.8 nautical mile S 47 E from it tarting poition. a. How far outh ha the hip ailed? b. How far eat ha it ailed? 9. Air Force pilot mark their bearing a the clockwie angle meaured from the north. For eample, a bearing of 90 i eat and a bearing of 80 i outh. There i a landing trip 5 mile awa from a plane at bearing. a. What bearing i wet? b. In what direction mut the plane fl to reach the landing trip? c. If the plane were to fl directl north or outh and then eat or wet to reach the landing trip, how far in each direction would the plane have to fl? 0. Becaue the um of the three angle of an triangle i 80, a right triangle with one 45 angle mut contain another 45 angle. A hown in Figure A., thi tpe of triangle i formed b two ide and a diagonal of a quare. Thu the two leg of the triangle are of equal length. B the Pthagorean Theorem, the hpotenue mut have length FIGURE A. 45 Uing the length and, determine the eact value of the following trigonometric ratio. a. in 45 b. co 45 c. tan 45. Ue the value for in 45 and co 45 to determine the ine, coine, and tangent value of each of the following angle. a. 5 b. 5 c. 5 For Activitie through 9, ketch the given angle on the unit circle, and draw the appropriate right triangle correponding to thi angle. Calculate the ine and coine of the angle, and indicate thee value appropriatel on the ketch of the triangle Calculate in 80 and co 80 and eplain our anwer in term of the unit circle. Do the ame thing for in 90 and co