Section 54 Trigonometric Functions


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1 5 Trigonometric Functions Section 5 Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form Exct Vlues for Specil Angles nd Rel Numers Summry of Specil Angle Vlues The six circulr functions introduced in Section 5 were defined without ny reference to the concept of ngle. Historiclly, however, ngles nd tringles formed the core suject mtter of trigonometry. In this section we introduce the six trigonometric functions, whose domin vlues re the mesures of ngles. The six trigonometric functions re intimtely relted to the six circulr functions. Definition of the Trigonometric Functions We re now redy to define trigonometric functions with ngle domins. Since we hve lredy defined the circulr functions with rel numer domins, we cn tke dvntge of these results nd define the trigonometric functions with ngle domins in terms of the circulr functions. To ech of the six circulr functions we ssocite trigonometric function of the sme nme. If is n ngle, in either rdin or degree mesure, we ssign vlues to sin, cos, tn, csc, sec, nd cot s given in Definition. DEFINITION TRIGONOMETRIC FUNCTIONS WITH ANGLE DOMAINS If is n ngle with rdin mesure x, then the vlue of ech trigonometric function t is given y its vlue t the rel numer x. Trigonometric Function sin cos tn csc sec cot Circulr Function sin x cos x tn x csc x sec x cot x W(x) (, ) x rd x units rc length (, 0) If is n ngle in degree mesure, convert to rdin mesure nd proceed s ove. [Note: To reduce the numer of different symols in certin figures, the u nd v xes we strted with will often e leled s the nd xes, respectively. Also, n expression such s sin 0 denotes the sine of the ngle whose mesure is 0.]
2 5 TRIGONOMETRIC FUNCTIONS The figure in Definition mkes use of the importnt fct tht in unit circle the rc length s opposite n ngle of x rdins is x units long, nd vice vers: s r x x EXAMPLE Solutions (A) (B) Exct Evlution for Specil Angles Evlute exctly without clcultor. (A) sin rd (B) tn rd (C) cos 80 (D) csc (50 ) sin rd sin tn rd tn (C) cos 80 cos ( rd) cos (D) csc (50 ) csc 5 rd csc 5 MATCHED PROBLEM Evlute exctly without clcultor. (A) tn (/ rd) (B) cos (/ rd) (C) sin 90 (D) sec (0 ) Clcultor Evlution of Trigonometric Functions How do we evlute trigonometric functions for ritrry ngles? Just s clcultor cn e used to pproximte circulr functions for ritrry rel numers, clcultor cn e used to pproximte trigonometric functions for ritrry ngles. Most clcultors hve choice of three trigonometric modes: degree (deciml), rdin, or grd. The mesure of right ngle 90 rdins 00 grds The grd unit is used in certin engineering pplictions nd will not e used in this ook. We repet cution stted erlier: CAUTION Red the instruction ook ccompnying your clcultor to determine how to put your clcultor in degree or rdin mode. Forgetting to set the correct mode efore strting clcultions involving trigonometric functions is frequent cuse of error when using clcultor.
3 5 Trigonometric Functions 5 Using clcultor with degree nd rdin modes, we cn evlute trigonometric functions directly for ngles in either degree or rdin mesure without hving to convert degree mesure to rdin mesure first. (Some clcultors work only with deciml degrees nd others work with either deciml degrees or degree minute second forms. Consult your mnul.) We generlize the reciprocl identities (stted first in Theorem, Section 5) to evlute cosecnt, secnt, nd cotngent. THEOREM RECIPROCAL IDENTITIES For x ny rel numer or ngle in degree or rdin mesure, csc x sin x sin x 0 sec x cos x cot x tn x cos x 0 tn x 0 EXAMPLE Clcultor Evlution Evlute to four significnt digits using clcultor. (A) cos 7. (B) sin ( rd) (C) tn 7.8 (D) cot (0 5) (E) sec (.59 rd) (E) csc (0.) Solutions (A) cos Degree mode (B) sin ( rd) 0. Rdin mode (C) tn Rdin mode (D) cot (0 5) cot (0.85 ) 0.8 Degree mode (Some clcultors require deciml degrees.) (E) sec (.59 rd).000 Rdin mode (F) csc (0.).5 Rdin mode MATCHED PROBLEM Evlute to four significnt digits using clcultor. (A) sin 9. (B) cos (7 rdins) (C) cot 0 (D) tn ( ) (E) sec (8.09 rdins) (F) csc (.5) Definition of the Trigonometric Functions Alternte Form For mny pplictions involving the use of trigonometric functions, including tringle pplictions, it is useful to write Definition in n lternte form form
4 5 TRIGONOMETRIC FUNCTIONS P(, ) Q(, ) P(, ) FIGURE Angle. O FIGURE Similr tringles. x rd O x units (, 0) tht utilizes the coordintes of n ritrry point (, ) (0, 0) on the terminl side of n ngle (see Fig. ). This lternte form of Definition is esily found y inserting unit circle in Figure, drwing perpendiculrs from points P nd Q to the horizontl xis (Fig. ), nd utilizing the fct tht rtios of corresponding sides of similr tringles re proportionl. Letting r d(o, P) nd noting tht d(o, Q), we hve sin sin x r cos cos x r nd lwys hve the sme sign. nd lwys hve the sme sign. The vlues of the other four trigonometric functions cn e otined using sic identities. For exmple, tn sin cos /r /r We now hve the very useful lternte form of Definition given elow. DEFINITION (Alternte Form) TRIGONOMETRIC FUNCTIONS WITH ANGLE DOMAINS If is n ritrry ngle in stndrd position in rectngulr coordinte system nd P(, ) is point r units from the origin on the terminl side of, then P(, ) r r P(, ) r P(, ) sin r cos r csc r sec r 0 0 r 0; P(, ) is n ritrry point on the terminl side of, (, ) (0, 0) tn 0 cot 0
5 5 Trigonometric Functions 7 DEFINITION continued Domins: Sets of ll possile ngles for which the rtios re defined Rnges: Susets of the set of rel numers (Domins nd rnges will e stted more precisely in the next section.) [Note: The right tringle formed y drwing perpendiculr from P(, ) to the horizontl xis is clled the reference tringle ssocited with the ngle. We will often refer to this tringle.] Explore/Discuss Discuss why, for given ngle, the rtios in Definition re independent of the choice of P(, ) on the terminl side of s long s (, ) (0, 0). The lternte form of Definition should e memorized. As memory id, note tht when r, then P(, ) is on the unit circle, nd ll function vlues correspond to the vlues otined using Definition for circulr functions in Section 5. In fct, using the lternte form of Definition in conjunction with the originl sttement of Definition in this section, we hve n lternte wy of evluting circulr functions: CIRCULAR FUNCTIONS AND TRIGONOMETRIC FUNCTIONS For x ny rel numer, sin x sin (x rdins) cos x cos (x rdins) sec x sec (x rdins) csc x csc (x rdins) () tn x tn (x rdins) cot x cot (x rdins) Thus, we re now free to evlute circulr functions in terms of trigonometric functions, using reference tringles where pproprite, or in terms of circulr points nd the wrpping function discussed erlier. Ech pproch hs certin dvntges in prticulr situtions, nd you should ecome fmilir with the uses of oth pproches. It is ecuse of equtions () tht we re le to evlute circulr functions using clcultor set in rdin mode (see Section 5). Generlly, unless certin emphsis is desired, we will not use rd fter rel numer. Tht is, we will interpret expressions such s sin 5.7 s the circulr function vlue sin 5.7 or the trigonometric function vlue sin (5.7 rd) y the context in which the expression occurs or the form we wish to emphsize. We will remin flexile nd often switch ck nd forth etween circulr function emphsis nd trigonometric function emphsis, depending on which pproch provides the most enlightenment for given sitution.
6 8 5 TRIGONOMETRIC FUNCTIONS EXAMPLE FIGURE Evluting Trigonometric Functions Find the vlue of ech of the six trigonometric functions for the illustrted ngle with terminl side tht contins P(, ) (see Fig. ) P(, ) r 5 Solution (, ) (, ) r () () 5 5 sin r 5 5 cos r 5 5 tn csc r 5 5 sec r 5 5 cot MATCHED PROBLEM Find the vlue of ech of the six trigonometric functions if the terminl side of contins the point (, 8). [Note: This point lies on the terminl side of the ngle in Exmple ; hence, the finl results should e the sme s those otined in Exmple.] EXAMPLE Solution FIGURE Evluting Trigonometric Functions Find the vlue of ech of the other five trigonometric functions for n ngle (without finding ) given tht is IV qudrnt ngle nd sin. The informtion given is sufficient for us to locte reference tringle in qudrnt IV for, even though we don t know wht is. We sketch reference tringle, lel wht we know (Fig. ), nd then complete the prolem s indicted P(, ) Since sin /r 5, we cn let nd r 5 (r is never negtive). If we cn find, then we cn determine the vlues of the other five functions. Terminl side of
7 5 Trigonometric Functions 9 Use the Pythgoren theorem to find : () 5 9 cnnot e negtive ecuse is IV qudrnt ngle. Using (, ) (, ) nd r 5, we hve cos r 5 tn sec r 5 csc r 5 5 cot MATCHED PROBLEM FIGURE 5 Qudrntl ngles. Find the vlue of ech of the other five trigonometric functions for n ngle (without finding ) given tht is II qudrnt ngle nd tn. Exct Vlues for Specil Angles nd Rel Numers Assuming trigonometric function is defined, it cn e evluted exctly without the use of clcultor (which is different from finding pproximte vlues using clcultor) for ny integer multiple of 0, 5, 0, 90, /, /, /, or /. With little prctice you will e le to determine these vlues mentlly. Working with exct vlues hs dvntges over working with pproximte vlues in mny situtions. The esiest ngles to del with re qudrntl ngles since these ngles re integer multiples of 90 or /. It is esy to find the coordintes of point on coordinte xis. Since ny nonorigin point will do, we shll, for convenience, choose points unit from the origin, s shown in Figure 5. (0, ) (, 0) (, 0) In ech cse, r, positive numer. (0, ) EXAMPLE 5 Trig Functions of Qudrntl Angles Find (A) sin 90 (B) cos (C) tn () (D) cot (80 )
8 70 5 TRIGONOMETRIC FUNCTIONS Solutions For ech, visulize the loction of the terminl side of the ngle reltive to Figure 5. With little prctice, you should e le to do most of the following mentlly. (A) sin 90 r (, ) (0, ), r (B) cos r (, ) (, 0), r (C) tn () 0 0 (, ) (, 0), r (D) cot (80 ) 0 (, ) (, 0), r Not defined MATCHED PROBLEM 5 Find (A) sin (/) (B) sec () (C) tn 90 (D) cot (70 ) Explore/Discuss Notice in Exmple 5, prt D, cot (80 ) is not defined. Discuss other ngles in degree mesure for which the cotngent is not defined. For wht ngles in degree mesure is the cosecnt function not defined? Becuse the concept of reference tringle introduced in Definition (lternte form) plys n importnt role in much of the mteril tht follows, we restte its definition here nd define the relted concept of reference ngle. REFERENCE TRIANGLE AND REFERENCE ANGLE. To form reference tringle for, drw perpendiculr from point P(, ) on the terminl side of to the horizontl xis.. The reference ngle is the cute ngle (lwys tken positive) etween the terminl side of nd the horizontl xis. (, ) (0, 0) is lwys positive P(, )
9 5 Trigonometric Functions 7 Figure shows severl reference tringles nd reference ngles corresponding to prticulr ngles. FIGURE Reference tringles nd reference ngles. Reference ngle 80 Reference tringle () / Reference ngle / 90 () 5 Reference tringle 5 / (c) (d) / (e) (f) If reference tringle of given ngle is 0 0 right tringle or 5 right tringle, then we cn find exct coordintes, other thn (0, 0), on the terminl side of the given ngle. To this end, we first note tht 0 0 tringle forms hlf of n equilterl tringle, s indicted in Figure 7. Becuse ll sides re equl in n equilterl tringle, we cn pply the Pythgoren theorem to otin useful reltionship mong the three sides of the originl tringle:
10 7 5 TRIGONOMETRIC FUNCTIONS FIGURE right tringle. c 0 0 c c c () 0 (/) (/) c Similrly, using the Pythgoren theorem on 5 right tringle, we otin the result shown in Figure 8. FIGURE 8 5 right tringle. c 5 c 5 (/) 5 5 (/) Figure 9 illustrtes the results shown in Figures 7 nd 8 for the cse. This cse is the esiest to rememer. All other cses cn e otined from this specil cse y multiplying or dividing the length of ech side of tringle in Figure 9 y the sme nonzero quntity. For exmple, if we wnted the hypotenuse of specil 5 right tringle to e, we would simply divide ech side of the 5 tringle in Figure 9 y. FIGURE AND 5 SPECIAL TRIANGLES 0 (/) 5 (/) 0 (/) 5 (/) If n ngle or rel numer hs 0 0 or 5 reference tringle, then we cn use Figure 9 to find exct coordintes of nonorigin point on the termi
11 5 Trigonometric Functions 7 nl side of the ngle. Using the definition of the trigonometric functions, Definition lternte form, we will then e le to find the exct vlue of ny of the six functions for the indicted ngle or rel numer. EXAMPLE Exct Evlution Evlute exctly using pproprite reference tringles. (A) cos 0, sin (/), tn (/) (B) sin 5, cot (/), sec (/) Solutions (A) Use the specil 0 0 tringle with sides,, nd s the reference tringle, nd use 0 or / s the reference ngle (Fig. 0). Use the sides of the reference tringle to determine P(, ) nd r; then use the pproprite definitions. FIGURE 0 0 (/) (, ) (, ) r cos 0 r sin tn r (B) Use the specil 5 tringle with sides,, nd s the reference tringle, nd use 5 or / s the reference ngle (Fig. ). Use the sides of the reference tringle to determine P(, ) nd r; then use the pproprite definitions. FIGURE 5 (/) (, ) (, ) r sin 5 r cot sec or r MATCHED PROBLEM Evlute exctly using pproprite reference tringles. (A) cos 5, tn (/), csc (/) (B) sin 0, cos (/), cot (/) Before proceeding, it is useful to oserve from geometric point of view multiples of / (0 ), / (0 ), nd / (5 ). These re illustrted in Figure.
12 7 5 TRIGONOMETRIC FUNCTIONS 5 5 () Multiples of / (0) FIGURE Multiples of specil ngles. 0 () Multiples of / (0) (c) Multiples of / (5) 0 8 EXAMPLE 7 Solutions Exct Evlution Evlute exctly using pproprite reference tringles. (A) cos (7/) (B) sin (/) (C) tn 0 (D) sec (0 ) Ech ngle (or rel numer) hs 0 0 or 5 reference tringle. Locte it, determine (, ) nd r, s in Exmple, nd then evlute. (A) cos 7 or (B) sin 7 (, ) (, ) r (, ) (, ) r (C) tn 0 or (D) sec (0 ) 0 0 (, ) (, ) r (, ) (, ) r 0 0
13 5 Trigonometric Functions 75 MATCHED PROBLEM 7 Evlute exctly using pproprite tringles. (A) tn (/) (B) sin 0 (C) cos (/) (D) csc (0 ) Now we reverse the prolem; tht is, we let the exct vlue of one of the six trigonometric functions e given nd ssume this vlue corresponds to one of the specil reference tringles. Cn we find smllest positive for which the trigonometric function hs tht vlue? Exmple 8 shows how. EXAMPLE 8 Finding Find the smllest positive in degree nd rdin mesure for which ech is true. (A) tn / (B) sec Solutions (A) tn FIGURE We cn let (, ) (, ) or (, ). The smllest positive for which this is true is qudrnt I ngle with reference tringle s drwn in Figure. 0 or (, ) 0 (B) sec r Becuse r 0 In qudrnts II nd III, is negtive. The smllest positive is ssocited with 5 reference tringle in qudrnt II, s drwn in Figure. FIGURE 5 or 5
14 7 5 TRIGONOMETRIC FUNCTIONS MATCHED PROBLEM 8 Find the smllest positive in degree nd rdin mesure for which ech is true. (A) sin / (B) cos / Remrk After quite it of prctice, the reference tringle figures in Exmples 7 nd 8 cn e visulized mentlly; however, when in dout, drw figure. Summry of Specil Angle Vlues Tle includes summry of the exct vlues of the sine, cosine, nd tngent for the specil ngle vlues from 0 to 90. Some people like to memorize these vlues, while others prefer to memorize the tringles in Figure 9. Do whichever is esier for you. TABLE Specil Angle Vlues sin cos tn / / or / 5 / or / / or / 0 / 90 0 Not defined These specil ngle vlues re esily rememered for sine nd cosine if you note the unexpected pttern fter completing Tle in Explore/Discuss. Explore/Discuss Fill in the cosine column in Tle with pttern of vlues tht is similr to those in the sine column. Discuss how the two columns of vlues re relted. TABLE Specil Angle Vlues Memory Aid sin cos / 0 / / / / Cosecnt, secnt, nd cotngent cn e found for these specil ngles y using the vlues in Tles or nd the reciprocl identities from Theorem.
15 5 Trigonometric Functions 77 Answers to Mtched Prolems. (A) (B) (C) (D). (A) (B) (C).5 (D) 0.8 (E).77 (F).7. sin. sin 5, 5, cos cos 5, 5, tn csc 5,, csc 5 sec 5,, sec 5 cot, cot 5. (A) (B) (C) Not defined (D) 0. (A) cos 5 /, tn (/), csc (/) (B) sin 0 7. (A) (B) (C), cos (/) /, cot (/) (D) / 8. (A) 0 or / (B) 5 or / EXERCISE 5 A Find the vlue of ech of the six trigonometric functions for n ngle tht hs terminl side contining the point indicted in Prolems.. (, 8). (, ). (, ). (, ) Evlute Prolems 5 to four significnt digits using clcultor. Mke sure your clcultor is in the correct mode (degree or rdin) for ech prolem. 5. sin 5. tn cot 8. csc 9. sin.7 0. tn.7. cot (. ). sec (7.9 ). sin 7. cos 5 7 In Prolems 5, evlute exctly, using reference tringles where pproprite, without using clcultor. 5. sin 0. cos 0 7. tn 0 8. cos 0 9. sin 5 0. csc 0. sec 5. cot 5. cot 0. cot tn 90. sec 0 Find the reference ngle for ech ngle in Prolems B In Prolems 8, evlute exctly, using reference ngles where pproprite, without using clcultor.. cos 0. sin cos (/). sin (/) 7. cot (0 ) 8. sec (0 ) 9. cos (/) 0. cot (/). sin (/). cos (/). csc 50. cot 5 5. tn (/). sec (/) 7. cos tn 90 For which vlues of 0 0, is ech of Prolems 9 5 not defined? Explin why. 9. cos 50. sec 5. tn 5. cot 5. csc 5. sin In Prolems 55 0, find the smllest positive in degree nd rdin mesure for which 55. cos 5. sin tn 59. csc 0. sec Find the vlue of ech of the other five trigonometric functions for n ngle, without finding, given the informtion indicted in Prolems. Sketching reference tringle should e helpful.. sin 5 nd cos 0. tn nd sin 0. cos 5/ nd cot 0 sin
16 78 5 TRIGONOMETRIC FUNCTIONS. 5. Which trigonometric functions re not defined when the terminl side of n ngle lies long the verticl xis. Why?. Which trigonometric functions re not defined when the terminl side of n ngle lies long the horizontl xis? Why? 7. Find exctly, ll, 0 0, for which cos /. 8. Find exctly, ll, 0 0, for which cot /. 9. Find exctly, ll, 0, for which tn. 70. Find exctly, ll, 0, for which sec. C cos 5/ nd tn 0 P(, ) s A Find light intensity I in terms of k for 0, 0, nd Solr Energy. Refer to Prolem 75. Find light intensity I in terms of k for 0, 50, nd Physics Engineering. The figure illustrtes piston connected to wheel tht turns revolutions per second; hence, the ngle is eing generted t () rdins per second, or t, where t is time in seconds. If P is t (, 0) when t 0, show tht for t 0. Solr cell y sin t (cos t) Sun 7. If the coordintes of A re (, 0) nd rc length s is 7 units, find y (A) The exct rdin mesure of (B) The coordintes of P to three deciml plces 7. If the coordintes of A re (, 0) nd rc length s is 8 units, find (A) The exct rdin mesure of (B) The coordintes of P to three deciml plces 7. In rectngulr coordinte system, circle with center t the origin psses through the point (, ). Wht is the length of the rc on the circle in qudrnt I etween the positive horizontl xis nd the point (, )? 7. In rectngulr coordinte system, circle with center t the origin psses through the point (, ). Wht is the length of the rc on the circle in qudrnt I etween the positive horizontl xis nd the point (, )? revolutions per second y inches P(, ) (, 0) t x APPLICATIONS 75. Solr Energy. The intensity of light I on solr cell chnges with the ngle of the sun nd is given y the formul I k cos, where k is constnt (see the figure). 78. Physics Engineering. In Prolem 77, find the position of the piston y when t 0. second (to three significnt digits).
17 55 Solving Right Tringles Geometry. The re of rectngulr nsided polygon circumscried out circle of rdius is given y A n tn 80 n (A) Find A for n 8, n 00, n,000, nd n 0,000. Compute ech to five deciml plces. (B) Wht numer does A seem to pproch s n? (Wht is the re of circle with rdius?) 80. Geometry. The re of regulr nsided polygon inscried in circle of rdius is given y A n 0 sin n n 8 r (A) Find A for n 8, n 00, n,000, nd n 0,000. Compute ech to five deciml plces. (B) Wht numer does A seem to pproch s n? (Wht is the re of circle with rdius?) 8. Angle of Inclintion. Recll (Section ) the slope of nonverticl line pssing through points P (x, y ) nd P (x, y ) is given y slope m (y y )/(x x ). The ngle tht the line L mkes with the x xis, 0 80, is clled the ngle of inclintion of the line L (see figure). Thus, Slope m tn, 0 80 (A) Compute the slopes to two deciml plces of the lines with ngles of inclintion 88.7 nd.. (B) Find the eqution of line pssing through (, 5) with n ngle of inclintion 7. Write the nswer in the form y mx, with m nd to two deciml plces. L y 8. Angle of Inclintion Refer to Prolem 8. (A) Compute the slopes to two deciml plces of the lines with ngles of inclintion 5. nd 9.. (B) Find the eqution of line pssing through (, ) with n ngle of inclintion 0. Write the nswer in the form y mx, with m nd to two deciml plces. L x Section 55 Solving Right Tringles* FIGURE c In the previous sections we hve pplied trigonometric nd circulr functions in the solutions of vriety of significnt prolems. In this section we re interested in the prticulr clss of prolems involving right tringles. A right tringle is tringle with one 90 ngle. Referring to Figure, our ojective is to find ll unknown prts of right tringle, given the mesure of two sides or the mesure of one cute ngle nd side. This is clled solving right tringle. Trigonometric functions ply centrl role in this process. To strt, we locte right tringle in the first qudrnt of rectngulr coordinte system nd oserve, from the definition of the trigonometric functions, six trigonometric rtios involving the sides of the tringle. [Note tht the right tringle is the reference tringle for the ngle.] *This section provides significnt ppliction of trigonometric functions to relworld prolems. However, it my e postponed or omitted without loss of continuity, if desired. Some my wnt to cover the section just efore Sections 7 nd 7.
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