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1 5 TRtGOhiOAMTRiC WNCTIONS D O E T F F R F l I F U A R N G TO N I l O R C G T N I T Triangle ABC bas a right angle (9Oo) at C and sides of length u, b, c. The trigonometric functions of angle A are defined as follows. 5 sintz. of A = sin A 1 = : = 5 cosine. of A = ~OS A 2 = i = opposite hypotenuse adjacent hypotenuse B opposite 5 tangent. of A = tana 3 = f = ~ adjacent adjacent 5 c. of A = ocet A 4 = k = t c z n g opposite A 5.5 hypotenuse secant of A = sec A = t = ~ adjacent 5 cosecant. of A = csc A 6 = z = hypotenuse opposite Fig. 51 E TX A WO T N M 3 H GE G A TE I RN 9 L Y H C ES0 E A H AI Consider an rg coordinate system [see Fig. 52 and 53 belowl. A point P in the ry plane has coordinates (%,y) where x is eonsidered as positive along OX and negative along OX while y is positive along OY and negative along OY. The distance from origin 0 to point P is positive and denoted by r = dm. The angle A described cozmtwczockwlse from OX is considered pos&ve. If it is described dockhse from OX it is considered negathe. We cal1 X OX and Y OY the x and y axis respectively. The various quadrants are denoted by 1, II, III and IV called the first, second, third and fourth quadrants respectively. In Fig. 52, for example, angle A is in the second quadrant while in Fig. 53 angle A is in the third quadrant. Y Y II 1 II 1 III IV III IV Y Y Fig. 52 Fig f
2 12 TRIGONOMETRIC FUNCTIONS For an angle A in any quadrant the trigonometric functions of A are defined as follows. 5.7 sin A = ylr 5.8 COS A = xl?. 5.9 tan A = ylx 5.10 cet A = xly 5.11 sec A = vlx 5.12 csc A = riy RELAT!ONSHiP BETWEEN DEGREES AN0 RAnIANS A radian is that angle e subtended at tenter 0 of a eircle by an arc MN equal to the radius r. Since 2~ radians = 360 we have radian = 180 /~ = o = ~/180 radians = radians B N 1 r e M 0 r Fig. 54 REkATlONSHlPS AMONG TRtGONOMETRK FUNCTItB4S 5.15 tana = sine A + ~OS~A = COS A &A ~II ~ zz  tan A sin A 1 sec A = ~ COS A 1 csca =  sin A 5.20 sec2a  tane A = csce A  cots A = 1 SIaNS AND VARIATIONS OF FUNCTIONS to 1 1 to 0 0 to m CC to 0 1 to uz m to II to 0 0 to 1 mtoo otom cc to 1 1 to ca III IV to 11 to 0 0 to d Cc to 01tom CO to to 0 0 to 1  too otom uz to 11 to 
3 TRIGONOMETRIC FUNCTIONS 1 3 E V X F AT A O RL FC R IU OUT V GE F NA A OS CN R N TG I Angle A Angle A in degrees in radians sin A COS A tan A cet A sec A csc A w 1 cc 15O riil2 #fi) &(&+fi) 2fi 2+* fifi &+fi 300 ii/6 1 +ti *fi fi $fi zl4 Jfi $fi 1 1 fi fi 60 VI3 Jti r 1 fi.+fi 2 ;G 750 5~112 2+& 2& &+fi fifi 900 z *CU 0 km 1 105O 7~112 *(fi+&) &(&Y% (2+fi) (2&) (&+fi) fifi 120 2~13 *fi * fi $fi fi *fi fi \h 150 5~ ti *fi fi +fi 2 165O llrll2 $(fi fi) &(G+ fi) (2fi) (2+fi) (fifi) Vz+Vc? 180?r Tm 1 *ca ~112 $(fifi) *(&+fi) 2fi 2 + ti (&fi) (&+fi) & 6 l f 3 i  g 2 f i 225O 5z14 Jfi *fi 1 1 fi fi 240 4%J3 # 4 ti &fi O 17~112 &&+&Q &(&fi) 2+fi 26 (&+?cz) (fifi) km 0 Tm O 19?rll2 &(&+fi) *(&fi) (2+6) &+fi (fifi) ïrl3 *fi 2 ti *fi 2 $fi 315O 7?rl44fi *fi 11 fi fi rl6 1 *fi +ti ti $fi O i(fi 6) &(&+ fi) (2  fi) (2+6) fifi (&+fi) 360 2r TJ 1?m For tables involving other angles see pages and f
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9 TRIGONOMETRIC FUNCTIONS y = cet1% 5.90 y = secl% 5.91 y = csclx _/ I T Y / /A , /, //  Fig Fig Fig RElAilONSHfPS BETWEEN SIDES AND ANGtGS OY A PkAtM TRlAF4GlG The following results hold for any plane triangle ABC with sides a, b, c and angles A, B, C. A 5.92 Law of Sines 5.93 Law of Cosines with =Y=a b c sin A sin B sin C cs = a2 + bz  Zab COS C similar relations involving the other sides and angles. C /A 1 f 5.94 Law of Tangents with a+b tan $(A + B)  ab = tan i(a B) similar relations involving the other sides and angles. Fig sina = :ds(s  a)(s  b)(s  c) where s = &a + b + c) is the semiperimeter of the triangle. Similar relations involving B and C cari be obtained. See also formulas 4.5, page 5; 4.15 and 4.16, page 6. angles Spherieal triangle ABC is on the surface of a sphere as shown in Fig Sides a, b, c [which are arcs of great circles] are measured by their angles subtended at tenter 0 of the sphere. A, B, C are the angles opposite sides a, b, c respectively. Then the following results hold Law of Sines zx_ sin a sin b sin c sin A sin B sin C 5.97 Law of Cosines cosa = cosbcosc + sinbsinccosa COSA =  COSB COSC + sinb sinccosa with similar results involving other sides and angles.
10 2 0 T R F I U G N O C N T O 5 L o. T a 9f a w 8 n g e n t s t & + B a t( $ ) + n b aa ( ) n u t &  B a= t( ) n A a n i ( w s r i i i et o mn s sh a t aiv i un h nl o d l d e ag l e t r r l v s s e i w s = & h 1 + c S( e r ) u i h r f e. o m+ ose o sa t a i li r un h n l d ld e ga e t r l r s w S = + h + B + C ( Se r) A i rh f e. o me os o sa t a i li r un h nl d ld e ga e t r lr S a f e 4 l op 1e. s ra 0 4o mg. 4 e u, l a NAPIER S RlJlES FGR RtGHT ANGLED SPHERICAL TRIANGLES E f r x a o C it c na r f, ghp e gor si hae pt le f pv At r t rwe he i Zae t ih ei f t r3s o ai nr h rc r nc a g i F s i wn i b va b i c g B e le,, u,.. 9 n A l, d a C O F 5 i  g 1. 9 F 5 i  g 2. S t uq h a pua e ri p a r c s ae n ionrfi 5e s n wsta ir w a e hi ng ct e 2p t etcg. lh 0r t ri Oeee [ c i ot hn mo c ayd a p A anpi B n l nd oc g. c d ta ml et ee ni n s A o o t n p n of th y ac e f i hce ri a ms ia prt ti ts l ancs hd wl pvle e ad eo c ae i rl da frg a p da t at rj n h wr p ea d a e c to a omc r a ps rp eate lan pt xi hr r l aaos t n eut e psr i nsl di ie 5 T s. o a h m i1 f p n e i n0 t a py q d e o1 ht r rt u d f oe h t oa aa l f e ph dn dl e ae ug js r ce a T s o a h m i f p n e i n t a py q d e o ht r rc u d f o e ht t o a l f pe h dsp l e ae uip s r nco E S x C = 9i a O C n0 = m9a  O wc h0 p, B A  e e a l, B ve e : s a = t b it ( a n a Co n s n = r Ot i a n B n a ) b s ( = Ci Ca C ( n O OoO C ~ A S = r S C Oa Os B A O  S i ) S B n T c o ch ba o f o e a e r fb uts rl i rt 5 rhe o es p oa 1. se n o s a m i 9 e u g n. 7 l e e
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