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1 A APPENDIX D TRIGONOMETRY Licensed to: jsamuels@bmcc.cun.edu PDF Ceated with deskpdf PDF Wite - Tial :: D T i g o n o m e t FIGURE a A n g l e s Angles can be measued in degees o in adians (abbeviated as ad). The angle given b a complete evolution contains, which is the same as ad. Theefoe and EXAMPLE (a) Find the adian measue of. (b) Epess ad in degees. SOLUTION (a) Fom Equation o we see that to convet fom degees to adians we multipl b. Theefoe ad (b) To convet fom adians to degees we multipl b. Thus ad In calculus we use adians to measue angles ecept when othewise indicated. The following table gives the coespondence between degee and adian measues of some common angles. Figue shows a secto of a cicle with cental angle and adius subtending an ac with length a. Since the length of the ac is popotional to the size of the angle, and ce the entie cicle has cicumfeence and cental angle, we have Solving this equation fo ad 7. and fo a, we obtain a Remembe that Equations ae valid onl when ad ad.7 ad a Degees 9 7 Radians a is measued in adians. Licensed to: jsamuels@bmcc.cun.edu ad FIGURE FIGURE Angles in standad position APPENDIX D TRIGONOMETRY A In paticula, putting a in Equation, we see that an angle of ad is the angle subtended at the cente of a cicle b an ac equal in length to the adius of the cicle (see Figue ). EXAMPLE (a) If the adius of a cicle is cm, what angle is subtended b an ac of cm? (b) If a cicle has adius cm, what is the length of an ac subtended b a cental angle of ad? SOLUTION (a) Ug Equation with a and, we see that the angle is (b) With cm and ad, the ac length is The standad position of an angle occus when we place its vete at the oigin of a coodinate sstem and its initial side on the positive -ais as in Figue. A positive angle is obtained b otating the initial side counteclockwise until it coincides with the teminal side. Likewise, negative angles ae obtained b clockwise otation as in Figue. teminal side FIGURE Figue shows seveal eamples of angles in standad position. Notice that diffeent angles can have the same teminal side. Fo instance, the angles,, and have the same initial and teminal sides because and ad epesents a complete evolution. = initial side =_. ad a 9 cm = FIGURE < initial side teminal side =_ = Copight Thomson Leaning, Inc. All Rights Reseved. Ma not be copied, scanned, o duplicated, in whole o in pat. Copight Thomson Leaning, Inc. All Rights Reseved. Ma not be copied, scanned, o duplicated, in whole o in pat.

2 A APPENDIX D TRIGONOMETRY APPENDIX D TRIGONOMETRY A7 Licensed to: jsamuels@bmcc.cun.edu PDF Ceated with deskpdf PDF Wite - Tial :: hpotenuse FIGURE P(,) FIGURE 7 œ FIGURE adjacent O opposite œ T h e T i g o n o m e t i c F u n c t i o n s Fo an acute angle the si tigonometic functions ae dened as atios of lengths of sides of a ight tiangle as follows (see Figue ). This denition doesn t appl to obtuse o negative angles, so fo a geneal angle in standad position we let P, be an point on the teminal side of and we let be the distance as in Figue 7. Then we dene Since division b is not dened, tan and sec ae undened when and csc and cot ae undened when. Notice that the denitions in () and () ae consistent when is an acute angle. If is a numbe, the convention is that means the e of the angle whose adian measue is. Fo eample, the epession implies that we ae dealing with an angle of ad. When nding a calculato appoimation to this numbe we must emembe to set ou calculato in adian mode, and then we obtain OP If we want to know the e of the angle we would wite and, with ou calculato in degee mode, we nd that. The eact tigonometic atios fo cetain angles can be ead fom the tiangles in Figue. Fo instance, s s opp hp adj hp tan opp adj tan tan tan. s s csc hp opp sec hp adj cot adj opp csc sec cot s tan s Licensed to: jsamuels@bmcc.cun.edu > all atios> tan> FIGURE 9 P {_, œ } FIGURE FIGURE œ FIGURE = œ > The signs of the tigonometic functions fo angles in each of the fou quadants can be emembeed b means of the ule All Students Take Calculus shown in Figue 9. EXAMPLE Find the eact tigonometic atios fo. SOLUTION Fom Figue we see that a point on the teminal line fo is P(, s). Theefoe, taking in the denitions of the tigonometic atios, we have The following table gives some values of and found b the method of Eample. EXAMPLE If and, nd the othe v e tigonometic functions of. s csc s SOLUTION Since, we can label the hpotenuse as having length and the adjacent side as having length in Figue. If the opposite side has length, then the Pthagoean Theoem gives and so, s. We can now use the diagam to wite the othe v e tigonometic functions: csc s sec s EXAMPLE Use a calculato to appoimate the value of in Figue. SOLUTION Fom the diagam we see that s sec tan tan s Theefoe 9.7 tan tan s cot s cot s s s s s s s s s Copight Thomson Leaning, Inc. All Rights Reseved. Ma not be copied, scanned, o duplicated, in whole o in pat. Copight Thomson Leaning, Inc. All Rights Reseved. Ma not be copied, scanned, o duplicated, in whole o in pat.

3 A APPENDIX D TRIGONOMETRY APPENDIX D TRIGONOMETRY A9 Licensed to: jsamuels@bmcc.cun.edu PDF Ceated with deskpdf PDF Wite - Tial :: Odd functions and even functions ae discussed in Section.. T i g o n o m e t i c I d e n t i t i e s A tigonometic identit is a elationship among the tigonometic functions. The most elementa ae the following, which ae immediate consequences of the denitions of the tigonometic functions. Fo the net identit we efe back to Figue 7. The distance fomula (o, equivalentl, the Pthagoean Theoem) tells us that. Theefoe We have theefoe poved one of the most useful of all tigonometic identities: 7 If we now divide both sides of Equation 7 b and use Equations, we get Similal, if we divide both sides of Equation 7 b, we get 9 The identities a b show that is an odd function and is an even function. The ae easil poved b dawing a diagam showing and in standad position (see Eecise 9). Since the angles and have the same teminal side, we have csc tan sec tan sec cot csc cot cot tan These identities show that the e and ine functions ae peiodic with peiod. The emaining tigonometic identities ae all consequences of two basic identities called the addition fomulas: Licensed to: jsamuels@bmcc.cun.edu a b The poofs of these addition fomulas ae outlined in Eecises,, and 7. B substituting fo in Equations a and b and ug Equations a and b, we obtain the following subtaction fomulas: a b Then, b dividing the fomulas in Equations o Equations, we obtain the coesponding fomulas fo tan : a b If we put in the addition fomulas (), we get the double-angle fomulas: a b Then, b ug the identit, we obtain the following altenate foms of the double-angle fomulas fo : a b If we now solve these equations fo and, we get the following half-angle fomulas, which ae useful in integal calculus: 7a 7b tan tan tan tan tan tan tan tan tan tan Finall, we state the poduct fomulas, which can be deduced fom Equations and : Copight Thomson Leaning, Inc. All Rights Reseved. Ma not be copied, scanned, o duplicated, in whole o in pat. Copight Thomson Leaning, Inc. All Rights Reseved. Ma not be copied, scanned, o duplicated, in whole o in pat.

4 A APPENDIX D TRIGONOMETRY APPENDIX D TRIGONOMETRY A Licensed to: jsamuels@bmcc.cun.edu PDF Ceated with deskpdf PDF Wite - Tial :: FIGURE a b c Thee ae man othe tigonometic identities, but those we have stated ae the ones used most often in calculus. If ou foget an of them, emembe that the can all be deduced fom Equations a and b. EXAMPLE Find all values of in the inteval, such that. SOLUTION Ug the double-angle fomula (a), we ewite the given equation as Theefoe, thee ae two possibilities: o,, o The given equation has v e solutions:,,,, and. G a p h s o f T i g o n o m e t i c F u n c t i o n s The gaph of the function f, shown in Figue (a), is obtained b plotting points fo and then ug the peiodic natue of the function (fom Equation ) to complete the gaph. Notice that the zeos of the e function occu at the o o (a) ƒ= (b) =, Licensed to: jsamuels@bmcc.cun.edu FIGURE intege multiples of, that is, Because of the identit (which can be veied ug Equation a), the gaph of ine is obtained b shifting the gaph of e b an amount to the left [see Figue (b)]. Note that fo both the e and ine functions the domain is, and the ange is the closed inteval,. Thus, fo all values of, we have The gaphs of the emaining fou tigonometic functions ae shown in Figue and thei domains ae indicated thee. Notice that tangent and cotangent have ange,, wheeas ecant and secant have ange,,. All fou functions ae peiodic: tangent and cotangent have peiod, wheeas ecant and secant have peiod. (a) =tan _ (c) =csc = wheneve n, n an intege (b) =cot _ (d) =sec = Copight Thomson Leaning, Inc. All Rights Reseved. Ma not be copied, scanned, o duplicated, in whole o in pat. Copight Thomson Leaning, Inc. All Rights Reseved. Ma not be copied, scanned, o duplicated, in whole o in pat.

5 A APPENDIX D TRIGONOMETRY APPENDIX D TRIGONOMETRY A Licensed to: jsamuels@bmcc.cun.edu PDF Ceated with deskpdf PDF Wite - Tial :: D Eecises Convet fom degees to adians Convet fom adians to degees Find the length of a cicula ac subtended b an angle of ad if the adius of the cicle is cm.. If a cicle has adius cm, nd the length of the ac subtended b a cental angle of 7.. A cicle has adius. m. What angle is subtended at the cente of the cicle b an ac m long?. Find the adius of a cicula secto with angle and ac length cm. 7 Daw, in standad position, the angle whose measue is given ad 7. ad. ad. ad Find the eact tigonometic atios fo the angle whose adian measue is given Find the emaining tigonometic atios. 9.,. tan,. sec.,.,. cot, 9... Find, coect to v e decimal places, the length of the side labeled cm 9 Pove each equation. 9. (a) Equation a (b) Equation b. (a) Equation a (b) Equation b. (a) Equation a (b) Equation b (c) Equation c Pove the identit..... cot. 7. sec tan. 9. cot sec tan csc. csc t sec t csc t..... cm cm. csc, tan tan sec csc cot cm tan tan tan. tan tan If and sec, whee and lie between and, evaluate the epession Find all values of in the inteval, that satisf the equation... cot tan Find all values of in the inteval, that satisf the inequalit tan Gaph the function b stating with the gaphs in Figues and and appling the tansfomations of Section. whee appopiate tan 79.. sec tan... Pove the Law of Coes: If a tiangle has sides with lengths a, b, and c, and is the angle between the sides with lengths a and b, then c a b ab b P(,) tan c (a,) [Hint: Intoduce a coodinate sstem so that is in standad position as in the gue. Epess and in tems of and then use the distance fomula to compute c.]. In ode to nd the distance as a small inlet, a point C is located as in the gue and the follo wing measuements wee ecoded: C m 9 m Use the Law of Coes fom Eecise to nd the equied distance.. Use the gue to po ve the subtaction fomula [Hint: Compute c in two was (ug the Law of Coes fom Eecise and also ug the distance fomula) and compae the two epessions.] AC C A å. Use the fomula in Eecise to pove the addition fomula fo ine (b). 7. Use the addition fomula fo ine and the identities to pove the subtaction fomula fo the e function.. Show that the aea of a tiangle with sides of lengths a and b and with included angle is AB A ab A(å,å) c B(, ) 9. Find the aea of tiangle ABC, coect to v e decimal places, if AB cm BC cm ABC 7 BC B Copight Thomson Leaning, Inc. All Rights Reseved. Ma not be copied, scanned, o duplicated, in whole o in pat. Copight Thomson Leaning, Inc. All Rights Reseved. Ma not be copied, scanned, o duplicated, in whole o in pat.