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# Class Note for MATH 1432 at UH

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Date Created: 02/06/15
Lecture 6 Section 77 Inverse Trigonometric Functions Section 78 Hyperbolic Sine and Cosine Jiwen He Department of Mathematics University of Houston jiwenhe mathuhedu http math uh eduNj iwenheMath1432 1 x2 Jiwen He University of Houston Math 1432 Section 26626 Lecture 6 January 31 2008 1 18 Inverse Trig Functions Jiwen He University of Houston Math 1432 Section 26626 Lecture 6 January 31 2008 2 18 Inverse Trig Functions II Inverse Sine Inverse Tanent Inverse Secant Other Tri Invers Inverse Since sin 1 X or arcsin X VII gt K11 x x I I I l I I I L I c 5 3 1 1 3 5 x r 7r JZ39 Jr zr Jr zr 27139 7r 2 27f 2 2 2 2 2 u nquot I n a a o a u a o u a o I a a u no a o u o u a a a a a o o u a o n you d 0 a u a a a n a u u I u on yzsinx M 1 I I 1 x 7Z39 377 ysinxxe 7r7r He University of Houston Math 1432 Section 26626 Lecture 6 January 31 2008 I Inverse Sine Inverse Tanent Inverse Secant Other Tri Invers Inverse Trig Functions Inverse Since sin 1 X or arcsin X VA i i i1 quot39 lx 39I I I I II 27139 7r 7239 Ezr 7T 7T 2JZ39 g7 X anquot i i i 12 1 325in 39 1 1 domain 7t 577 range 1 1 x sinx 1 ylk 7T 1 l 1 7T 3 1 1 l 1 5 2 A 3 1 1 2 2 67139 5 1 0 O 71 ysinxx 39 1 1 47139 22 1 1 37139 2 H 1 57139 1 Jiwen He University of Houston Math 1432 Section 26626 Lecture 6 January 31 2008 Inverse Trig Functions Inverse Since sin 1 X or arcsin X Inverse Sine Inverse Tanent Inverse Secant Other Tri Invers I III I I 1 IR I I I I I L I 5 g u a 2 1 yzsinx 1 1 M domaIn 7T 5W 1 range 1 1 x smx 2 2 1 1 T 1 3 ysinxxe 7r7r l7 1 2 39 39 1 4 2 SInSIn X X 21 21 m 6H 2 1 I 0 0 3 1 1 6 2 I I ff 1 1 x 1 1 37f 2 H 7T 1 7r ysin 1x xe 11 Math 1432 Section 26626 Lecture 6 January 31 2008 Jiwen He University of Houston Inverse Trig Functions Inverse Since sin 1 X or arcsin X ow Inverse Sine Inverse Tanent Jill I I I I I I I L f 7r x 27r 7r 7r 2r 7r 7f 27Ir i 1 ISinx 1 1 M dOWBHVI 5W35WI donmhrI 11 1 1 rangeII 171 I I rangeI n 4 3C 115195 2 1 2 x SiIl l x 1 1 271 1 I 7T l l Sinxx 7r7r I I 3 2 3 y I I 2 3 i 1 1 1 1 1 17 2 2 SInSIn X X 7 2 2 1 1 y l 1 677 2 in 2 6 0 0 2 0 O 1 1 1 1 E 2 2 6 I I i7 1 1 x in 1 1 1 1 3 2 2 i 39 1 7r 57139 ysin 1x xe 11 Math 1432 Section 26626 Lecture 6 January 31 2008 Inverse Secant Other Tri Invers 2m Jiwen He University of Houston quot4i Inverse Sine Inverse Tanent Inverse Secant Other Tri Invers Inverse Trig Functions Trigonometric Properties M WI 1 2 1 c 1 1 X 1 1 A 2775 27139 1 1 2 1 V1 x2 1 1 ysmxxe 7T 7T yzsin lx xe 1 1 sinsin 1 X X Jiwen He University of Houston Math 1432 Section 26626 Lecture 6 January 31 2008 3 18 Inverse Sine Inverse Tanent Inverse Secant Other Tri Invers i i39 ii Iwi 39 iquot Inverse Trig Functions Trigonometric Properties Ui WI 1 2 1 c 1 1 X 1 1 A 2775 27139 1 1 2 I V1 x2 1 1 ysmxxe 7T 7T yzsin lx xe 11 sinsin1 x X cossin1 X V 1 X2 Jiwen He University of Houston Math 1432 Section 26626 Lecture 6 January 31 2008 3 18 Inverse Sine Inverse Tanent Inverse Secant Other Tri Inverse Trig Functions Trigonometric Properties M WI 173 2 1 i l l x 1 1 A 2 27 1 i 2 V1 x2 1 1 ysmxxe 7T 7T yzsin lx xe 1 1 sinsin1 x X cossin1 X V 1 X2 tansin1 x Jiwen He University of Houston Math 1432 Section 26626 Lecture 6 January 31 2008 3 18 Inverse Trig Functions 7 mst r Inverse Sine Inverse Tanent Inverse Secant Other Tri Trigonometric Properties NH gt1 NV 39 or V1 x2 ysin 1x xe 11 1 1 y Sln X XE E 39 sinsin1 x X cossin1 x 1 2 tansin1x cotsin1x X Jiwen He University of Houston Math 1432 Section 26626 Lecture 6 January 31 2008 Inverse Sine Inverse Tanent Inverse Secant Other Tri Invers Inverse Trig Functions a Trigonometric Properties M yll 1 I I I I x in x 1 1 A 1 1 1 6 Vl x2 y539nx xei 2 2 i ySIn 1xxe 11 sinsin1 x X cossin1 X V1 X2 1 X 1 V X2 tan SIn X cot SIn X secsin1 x Jiwen He University of Houston Math 1432 Section 26626 Lecture 6 January 31 2008 Inverse Sine Inverse Tanent Inverse Secant Other Tri Invers Inverse Trig Functions a Trigonometric Properties ylk y i 2 1 gt 39 39 gt 1 l x 1 1 A 2 27 1 i 2 1 1 12 ysmxxe 7T 7T yzsin lx xe 11 sinsin1 x X cossin1 X V 1 X2 tansin 1 X cotsin 1x 1 secsin1 x cscsin1 X X Jiwen He University of Houston Math 1432 Section 26626 Lecture 6 January 31 2008 Inverse Trig Functions Inverse Sine Inverse Tanent Inverse Secant Other Tri Invers Differentiation ylk Theorem I 1 1 x 1 sm 1 X 7 1x2 dX 1X2 y8in 1x M 11 Jiwen He University of Houston Math 1432 Section 26626 Lecture 6 January 31 2008 4 18 Inverse Trig Functions 1LT Inverse Sine Inverse Tanent Inverse Secant Other Tri Invers Differentiation ylk Theorem i I x x 1 sm 1 X 1 I 3 W dX V 1 X2 y8in 1x M 11 Let y sin 1 X Then X sin y d1 1 1 1 1 sm X dX 0 sin y cosy cossm 1 X V1 X2 Jiwen He University of Houston Math 1432 Section 26626 Lecture 6 January 31 2008 4 18 Inverse Trig Functions 3 Inverse Sine Inverse Tanent Inverse Secant Other Tri Invers Differentiation ylk Theorem 1 1 1 x 1 sm 1 X 7 1x2 dX 1X2 y8in 1x M 11 Proof Let y sin 1 X Then X sin y d 1 1 1 1 1 d Sln X I I I 1 I X d ysmy cosy cossm X 1 X 1 1 dU Sln U 3 2mg I Jiwen He University of Houston Math 1432 Section 26626 Lecture 6 January 31 2008 4 18 Inverse Trig Functions 3 Inverse Sine Inverse Tanent Inverse Secant Other Tri Invers Differentiation ylk Theorem 1 1 1 x 1 sm 1 X 7 1x2 dX 1X2 y8in 1x M 11 Proof Let y sin 1 X Then X sin y d 1 1 1 1 1 d Sln X I I I 1 I X d ysmy cosy cossm X 1 X d 1 d sin 1 u u du sin 1 u C 1 a 1 u27 1 u2 H Jiwen He University of Houston Math 1432 Section 26626 Lecture 6 January 31 2008 4 18 Inverse Trig Functions 1 Integration u Substitution gX dxsin1 X C TWW g gtgt Jiwen He University of Houston Math 1432 Section 26626 Lecture 6 January 31 2008 5 18 Inverse Trig Functions i ijv 39rt5 irPiii this Integration u Substitution gX dx sin1gx C 1 155M2 Let u gX Then du g x dX gX X 1 u sin1 u Sin1 X f 1 gX2 d H d c g Jiwen He University of Houston Inverse Trig Functions i ijv 39rt5 irPiii this Integration u Substitution gX dx sin1gx C 1 155M2 Let u gX Then du g x dX gX X 1 u sin1 u Sin1 X f 1 gX2 d H d c g 1 dX 4 X2 Jiwen He University of Houston Inverse Trig Functions i i5v jr iquotii Piii Elim Integration u Substitution gX dx sin1gx C 1 155M2 Let u gX Then du g x dX gX X 1 u sin1 u Sin1 X f 1 gX2 d H d c g x d X 4 X2 Note that 4 X2 4 1 Let u Then du dx H Jiwen He University of Houston Inverse Trig Functions i i5v jr iquotii Piii Elim Integration u Substitution gX dx sin1gx C 1 155M2 Let u gX Then du g x dX gX X 1 u sin1 u Sin1 X f 1 gX2 d H d c g usin1uCsin1gC d f1 d x 4 X2 1 u2 Note that4 X2 41 Let u Then du dx H Jiwen He University of Houston Inverse Trig Functions i i5v jr iquotii Piii Elim Integration u Substitution g X V 1 155M2 dX sin 1gx C Let u gX Then du g x dX gX 1 u sin 1 u sin 1 X i f 1 gW dx M d c g 1 dX 2X X2 Jiwen He University of Houston Inverse Trig Functions 1713425539iwii Elim Integration u Substitution g X V 1 155M2 dX sin 1gx C Let u gX Then du g x dX g X X usin1u sin1 x 1gX2d Md c g 1 dX 2X X2 Note that 2X X2 1 X2 2X 1 1 X 12 complete H the square Let u X 1 Then du dX Jiwen He University of Houston Inverse Trig Functions i iw2 quotiwiii Elim Integration u Substitution g X V 1 155M2 dX sin 1gx C Let u gX Then du g x dX g X X usin1u sin1 x 1gX2d Md c g u sin 1L uC sin 1X 1C 1 1 dX f d 2X X2 xl u2 Note that 2X X2 1 X2 2X 1 1 X 12 complete H the square Let u X 1 Then du dX Jiwen He University of Houston ytanX domain 7T 7T range oo 00 J l l 39 tan x x6 i J 2 27 Jiwen He University of Houston Math 1432 Section 26626 Lecture 6 January 31 2008 6 18 Inverse Trig Functions r V i 7 Inverse Sine Inverse Tanent Inverse Secant Other Tri Inverse Tangent tan IL X or arctan X y tanX domain 7r gw range oo oo y y y tantlx xreal 1 1 v tan x xe 7r 7139 39 2 2 Jiwen He University of Houston Math 1432 Section 26626 Lecture 6 January 31 2008 6 18 Inverse Sine Inverse Tan ent Inverse Secant Other Tri Inverse Trig Functions ieran Inverse Tangent tan IL X or arctan X y y y tanX 39 1 1 x domain 7r 5W 1 range oooo 2 yr I y tantlx xreal Trigonometric Properties 0 tantan1 X X 1 1 v tan x xe 7r 2 727139 Jiwen He University of Houston Math 1432 Section 26626 Lecture 6 January 31 2008 6 18 Inverse Sine Inverse Tanent Inverse Secant Other Tri Inverse Trig Functions ieran Inverse Tangent tan IL X or arctan X ytanX x domain 7r w g y range oo oo y tantlx xreal Trigonometric Properties tantan1 X X 1 1 r tan x 3 6 i Ir 39I 2 27 Jiwen He University of Houston Math 1432 Section 26626 Lecture 6 January 31 2008 6 18 Inverse Sine Inverse Tanent Inverse Secant Other Tri Inverse Trig Functions ieran Inverse Tangent tan IL X or arctan X y y 1 E71quot ytanX x 1 1 domain 7r 7r g y range oooo y tantlx xreal Trigonometric Properties tantan1 X X cottan1 X 1 1 r tan x 3 6 i Ir 39I 2 27 Jiwen He University of Houston Math 1432 Section 26626 Lecture 6 January 31 2008 6 18 Inverse Trig Functions rjsga V Inverse Sine Inverse Tanent Inverse Secant Other Tri Invers Inverse Tangent tan IL X or arctan X y y ytanX x 1 1 domain 7r 7r x 7 yz i range oooo y tantlx xreal 7 T 39 7 T Trigonometric Properties 0 tantan1 X X cottan1 X sintan1 X X Jtanxxei 7rrr 1 I X2 Jiwen He University of Houston Math 1432 Section 26626 Lecture 6 January 31 2008 6 18 Inverse Trig Functions rjsga V Inverse Sine Inverse Tanent Inverse Secant Other Tri Invers Inverse Tangent tan IL X or arctan X y y y tanX x domain 7r w r y range oooo y tantlx xreal g Xg Trigonometric Properties 0 r 1 1 1 tantan X X cottan X X 1 SIntan 1X costan 1X were 1 72 1 x2 Jiwen He University of Houston Math 1432 Section 26626 Lecture 6 January 31 2008 6 18 Inverse Trig Functions rjsga V Inverse Sine Inverse Tanent Inverse Secant Other Tri Invers Inverse Tangent tan IL X or arctan X ytanX domain 7r w range oo oo tantan1 X X cottan1 X gtlt sintan1X L COStan1x 1IX2 Jiwen He University of Houston Math 1432 Section 26626 Lecture 6 January 31 2008 6 18 Other Tri Invers Inverse Sine Inverse Tan ent Inverse Secant Inverse Trig Functions ieran Inverse Tangent tan IL X or arctan X y tanX domain 7r w range oo oo 139 7 1 tantan1X X cottan1X X 1 X 1 1 SI tan X cos tan X I I m I I m 1 2 sectan1 X V 1 IX2 csctan1 X A I X Math 1432 Section 26626 Lecture 6 January 31 2008 6 18 Jiwen He University of Houston Inverse Trig Functions Iti Inverse Sine Inverse Tanent Inverse Secant Other Tri Invers Differentiation tan1x 1 dX 1X2 Jiwen He University of Houston Math 1432 Section 26626 Lecture 6 January 31 2008 7 18 Inverse Trig Functions H1 Inverse Sine Inverse Tanent Inverse Secant Other Tri Invers Differentiation t 1 1 7r X y 2 1 dX 1 X2 y tant1x xreal Let y tan 1 X Then X tan y tan1 1 1 1 1 dX 0 tan y 59C W2 sectan 1 gtlt2 1 X2 Jiwen He University of Houston I Math 1432 Section 26626 Lecture 6 January 31 2008 7 18 Inverse Trig Functions H1 Inverse Sine Inverse Tanent Inverse Secant Other Tri Invers Differentiation t 1 1 7r X y 2 1 dX 1 X2 y tant1x xreal Let y tan 1 X Then X tan y tan1 1 1 1 1 dX 0 tan y 59C W2 sectan 1 gtlt2 1 X2 Theorem I dX 1u2amp7 Fl Jiwen He University of Houston I Math 1432 Section 26626 Lecture 6 January 31 2008 7 18 Inverse Trig Functions H1 Inverse Sine Inverse Tanent Inverse Secant Other Tri Invers Differentiation t 1 1 7r X y 2 1 dX 1 X2 y tant1x xreal Let y tan 1 X Then X tan y tan1 1 1 1 1 dX 0 tan y 59C W2 sectan 1 gtlt2 1 X2 Theorem I d 1 1 du d t 1 C dX 1u20 x7 1u2 u an u H Jiwen He University of Houston I Math 1432 Section 26626 Lecture 6 January 31 2008 7 18 Inverse Sine Inverse Tan39ent Inverse Secant Other Tri Invers Inverse Trig Functions Integration u Substitutlon g X 1 2quotgtlt2 dX tan 1gX C Jiwen He University of Houston Math 1432 Section 26626 Lecture 6 January 31 2008 8 18 Inverse Trig Functions i Ci f39ijhjiiif we gu Integration u Su stltution g X 1 1 i gX2 Let u gX Then du g X dX dX tan 1gX C tan 1x gX 1 1 1 f1 dX1 u2dutan uC tan gXC 002 Jiwen He University of Houston Inverse Trig Functions i Ci f39ijhjiiif we gu Integration u Su stltution g X 1 1 i gX2 Let u gX Then du g X dX dX tan 1gX C tan 1x gX 1 1 1 f1 dX1 u2dutan uC tan gXC 002 Jiwen He University of Houston Inverse Trig Functions 1quot lavisgalit Tia we gervl Integration u Substitution ER 1 1 i gX2 i lllllllllllllllllllll Let u gX Then du g X dX dX tan 1gX C tan 1x gX 1 u an1u an 1 X wmyw A d t Ct g c 1 4x2 Note that 4X2 4 1 Let u Then du dx H Jiwen He University of Houston Inverse Trig Functions giieriq ir we ml Integration u Substitution t 1 g X an x 1 1 i gX2 i lllllllllllllllllllll Let u gX Then du g X dX dX tan 1gX C gX 1 u an1u an 1 X wmyw A d t Ct g c 1 1 1 1 1 x d d t 1 C t 1 C 4x2 X 21u2 2 a 2 a 2 Note that 4X2 4 1 Let u Then du dx H Jiwen He University of Houston Inverse Trig Functions i wE 39isi i we ml Integration u Substitution g X 1 2quotgtlt2 i ailIIIIIIIIIIIIIIIIIII Let u gX Then du g X dX dX tan 1gX C gX 1 u an1u an 1 X wmyw A d t Ct g c J Examples 1 dX 22X i X2 Jiwen He University of Houston Inverse Trig Functions g a riq ir we ml Integration u Substitution g X 1 2quotgtlt2 i ailIIIIIIIIIIIIIIIIIII Let u gX Then du g X dX dX tan 1gX C gX 1 u an1u an 1 X wmyw A d t Ct g c J Examples 1 dX 2 2Xl X2 Note that 22xx2 1x22x1 1x12 complete the square Let u X 1 Then du dX H Jiwen He University of Houston Inverse Trig Functions g a riq ir we ml Integration u Substitution g X 1 2quotgtlt2 i ailIIIIIIIIIIIIIIIIIII Let u gX Then du g X dX dX tan 1gX C gX 1 u an1u an 1 X wmyw A d t Ct g c J Examples 1 1 22XX2 X 1u2 u an x Notethat2 l 2X l X21 l X2 l 2X l 11 X 12 complete the square Let u X 1 Then du dX H Jiwen He University of Houston Inverse Trig Functions 1quot lavisgalit Tia we gervl Integration u Substitution g X 1 2quotgtlt2 e i lllllllllllllllllllll Let u gX Then du g X dX dX tan 1gX C gX 1 u an1u an 1 X wmyw A d t Ct g c Examples LJL w 1 e 2X Jiwen He University of Houston Inverse Trig Functions g a riq ir we ml Integration u Substitution g X 1 gX2 dX tan gX C i ailIIIIIIIIIIIIIIIIIII Let u gX Then du g X dX gX 1 u an1u an 1 X wmyw A d t Ct g c Examples LJL w 1 e 2X Note that 1 e 2X 1 eX2 complete the square Let u eX Then du eXdX Jiwen He University of Houston Inverse Trig Functions g a riq ir we ml Integration u Substitution g X 1 2quotgtlt2 dX tan 1gX C i ailIIIIIIIIIIIIIIIIIII Let u gX Then du g X dX gX 1 u an1u an 1 X wmyw A d t Ct g c Examples eX 1 d d t 1 X 1e2x x 1u2 u an e Note that 1 e 2X 1 eX2 complete the square Let u eX Then du eXdX Jiwen He University of Houston m m t rmm m anu mm awn a m m 2 Fm k gt u doub e ume T b T t 7 Inverse Trig Functions u Vi v Inverse Sine Inverse Tanent Inverse Secant Other Tri Inverse Secant sec 1L X y secx domain0 7TU me range oo 1U 1700 Jiwen He University of Houston Math 1432 Section 26626 Lecture 6 January 31 2008 10 18 Inverse Trig Functions Ni x Inverse Sine Inverse Tanent Inverse Secant Other Tri 1 Inverse Secant sec X y secx domain0 7TU m range oo 1U 1709 1 y sec 1 X e 0 1U1 39 Jiwen He University of Houston Math 1432 Section 26626 Lecture 6 January 31 2008 10 18 Inverse Trig Functions i 39 1 Inverse Secant sec X yseCX l domain0 WU 1 I m 7T range oo 1U 17 e1U1 Trigonometric Properties secsec1 X X Jiwen He University of Houston Math 1432 Section 26626 Lecture 6 January 31 2008 Inverse Trig Functions i 39 1 Inverse Secant sec X y secx domain0 WU J wvwi x2 1 range oo 1U 1 e1U1 1 Jiwen He University of Houston Math 1432 Section 26626 Lecture 6 January 31 2008 Inverse Trig Functions i 39 1 Inverse Secant sec X y seCX domain0 WU J Gmw m range oo 1U 17 00 ysec 1xe 1u 1 secsec1 X X cscsec1 x Jiwen He University of Houston Math 1432 Section 26626 Lecture 6 January 31 2008 10 18 Inverse Trig Functions i 39 1 Inverse Secant sec X y seCX domain0 WU J Gmw m range oo 1U 17 00 ysec 1xe 1u 1 secsec1 X X cscsec1 x 1 X2 1 2 1 sinsec1 X X ysecxxe0U X Jiwen He University of Houston Math 1432 Section 26626 Lecture 6 January 31 2008 10 18 Inverse Trig Functions 1 39 1 Inverse Secant sec X y seCX domain0 WU J Gmw m range oo 1U 17 00 ysec 1xe 1u 1 secsec1 X X cscsec1 x ysecxxe0U X 1 X2 1 2 1 1 sinsec1 X X cossec1 X X Jiwen He University of Houston Math 1432 Section 26626 Lecture 6 January 31 2008 10 18 Inverse Trig Functions 1 5vi quot a 1 Inverse Secant sec X y secx domain0 WU J wvwi x2 1 range oo 1U 1 e1U1 1 quot secltsec 1x x cscsec 1X X 4 X2 1 2 1 X 1 1 l ysecxxeo3ug smsec X X COSSeC x X tansec1 x X2 1 Jiwen He University of Houston Math 1432 Section 26626 Lecture 6 January 31 2008 10 18 Inverse Trig Functions i Inverse Secant y seCX E domain0 WU J Gm range oo 1U 17 ysec 1xe 1u 1 1 secsec1 X X cscsec1 x X 1 X2 1 2 1 X 1 1 l ysecxxeogug smsec X X COSSeC X X 1 tansec1 X X2 1 cotsec1 x I X2 1 Jiwen He University of Houston Math 1432 Section 26626 Lecture 6 Ja nuary 31 2008 10 18 Inverse Trig Functions HI Inverse Sine Inverse Tanent Inverse Secant Other Tri Invers Differentiation x 1 1 SEC 1x X 1 dX XX2 1 1 Ix2 1 y tant1x xrea Let y sec 1 X Then X sec y 1 1 1 1 dX d ysecy secytany XX2 1 Jiwen He University of Houston I Math 1432 Section 26626 Lecture 6 January 31 2008 11 18 Inverse Trig Functions Hi Inverse Sine Inverse Tan ent Inverse Secant Other Tri Invers Differentiation ix2 1 x 1 1 r SEC 1x X 1 dX XX2 1 t ntlx xrea Proof Let y sec 1 X Then X sec y sec1 1 1 1 X dX diysecy secytany2 XX2 1 1 1 dU SEC U dX U u2 17 H Jiwen He University of Houston I Math 1432 Section 26626 Lecture 6 January 31 2008 11 18 Inverse Trig Functions Hi Inverse Sine Inverse Tan ent Inverse Secant Other Tri Invers Differentiation Vx2 1 x 1 1 r SEC 1x X 1 dX XX2 1 t ntlx xrea Proof Let y sec 1 X Then X sec y 1 1 1 1 sec X dX diysecy secytany2 XX2 1 d 1 d 1 sec 1 u u du sec 1uC dX uu2 1 dX UU2 1 H Jiwen He University of Houston I Math 1432 Section 26626 Lecture 6 January 31 2008 11 18 Inverse Sine Inverse Tanent Inverse Secaht Other Tri Invers Inverse Trig Functions Integration u Substitution g X quotgtltWggtlt2 1 dX sec 1igXi C Iii Jiwen He University of Houston Math 1432 Section 26626 Lecture 6 January 31 2008 12 18 Inverse Trig Functions i Ci f39ijhjiiif we gu Integration u Su stltution g X quotgtltWggtlt2 1 Let u gX Then du g X dX dX sec 1igXi C g X dX du sec 1 X C gXgx2 1 Am ig i Examples Jiwen He University of Houston Inverse Trig Functions g 2riq ir we r wl Integration u Su stltution g X quotgtltWggtlt2 1 Let u gX Then du g X dX dX sec 1igXi C g X dX duseC 1 X C gXgx2 1 Am ig i j d xm X Jiwen He University of Houston Inverse Trig Functions if r w W Integration u Substltution g X quotgtltWggtlt2 1 Let u gX Then du g X dX dX sec 1igXi C g X dX du sec 1 X C gXgx2 1 Am ig i j r1 d x X X l Note that x 1 gt lt2 1 Let u Then x u2 H dX 2udu Jiwen He University of Houston Inverse Trig Functions if is w W Integration u Substltution g X quotgtltWggtlt2 1 Let u gX Then du g X dX dX sec 1igXi C g X dX du sec 1 X C gXgx2 1 Am ig i j 1 1 1 dx2 du sec1 xC XX 1 fuxu2 1 2 Note thatX 1 lt2 1 Letugt lt ThenXu2 H dX2udu Jiwen He University of Houston Inverse Trig Functions v A i Other Trigonometric Inverses Other Trigonometric Inverses Jiwen He University of Houston Math 1432 Se ti n 26626 Lemma 6 January 31 21108 13 1 Inverse Trig Functions v A i Other Trigonometric Inverses Other Trigonometric Inverses sin 391L X cos X Mgt1 Jiwen He University of Houston Math 1432 Se ti n 26626 Lemma 6 January 31 21108 13 1 Inverse Trig Functions v A Other Trigonometric Inverses Other Trigonometric Inverses sin 391L X cos X or cos X sin X Mgt1 D Jiwen He University of Houston Math 1432 Segtion 26626 Lemma 6 January 31 21108 Inverse Trig Functions v A Other Trigonometric Inverses Jiwen He University of Houston Math 1432 Se ti n 26626 Lemma 6 January 31 21108 13 1 Inverse Trig Functions Other Trigonometric Inverses 39 n sin 391L X cos 1L X tan 1X cot 1 x MIgt1Ngt1 Jiwen He University of Houston Math 1432 Seetien 26626 Lecture 6 January 31 21008 13 1 Inverse Trig Functions 7 A Other Trigonometric Inverses 39 n Other Trigonometric Inverses i 1 1 1 7T 1 sm Xcos X or cos X sm X 2 1 1 7T 1 tan Xcot X or cot X tan X sec 391L X cscL X mlgt1mgt1wgt1 Jiwen He University of Houston Math 1432 Seetien 26626 Lecture 6 January 31 21008 13 1 Inverse Trig Functions 7 A Other Trigonometric Inverses 39 n sin 391L X cos 1L X or cos X tan 1X cot 1 x mlgt1mgt1wgt1 1 sec 391L X csc X 7r 2 7T or cot X tan X 2 7r or csc X 2 Jiwen He University of Houston Math 1432 Se ti 9n 26626 Lemme 6 January 31 21108 13 1 Inverse Trig Functions Differentiation cos 1 X a Jiwen He University of Houston Math 1432 Se ti 9n 26626 Lemme 6 January 311 2308 14 1 Inverse Trig Functions Differentiation cos 1 X a COS1X sin X dX dX il X2 Jiwen He University of Houston Math 1432 Se ti 9n 26626 Lemme 6 January 311 2308 14 1 Inverse Trig Functions Differentiation X a 1 1 1 dX cos X dX sm X W cot1x tan1x 1 d d 1 X2 Jiwen He University of Houston Math 1432 Se ti 9n 26626 Lemme 6 January 311 2308 14 1 Inverse Trig Functions Differentiation X a cos 391L dX X 11 Jiwen He University of Houston Math 1432 Semen 26526 Lecture 6 January 311 2130 141 m m zttheend mm yam M2 vumw eo stun mmad 210zomvounda 3 WW zwu1um b1m1um z1m1ms a mums 2 mth b ma a z mum um Hyperbolic Sine and Cosine Hyperbolic Sine and Cosine Definition sinhX DII l ex eX coshx DII l eweX T y sinh x d sinhX cosh d coshX sinh X X Jiwen He University of Houston Math 1432 Section 26626 Lecture 6 January 31 2008 16 18 Hyperbolic Sine and Cosine Identities Pcosh r sinh t area of hyperbolic sector 2 gr y x2 2 1 cos 2 sin I 0 10 X x area of circular sector 2 m Math 1432 Section 26626 Lecture 6 January 31 2008 17 18 Jiwen He University of Houston Hyperbolic Sine and Cosine Identities Pcosh r sinh t cosh2x sinh2X 1 area of hyperbolic sector 2 r y x2 322 1 cos 2 sin I 0 10 X area of circular sector 2 Math 1432 Section 26626 Lecture 6 January 31 2008 1718 Jiwen He University of Houston Hyperbolic Sine and Cosine Identities Pcosh r sinh t cosh2x sinh2X 1 area of hyperbolic sector 2 r y x2yz1 COS2X sm2x 1 cos 2 sin I 0 10 X area of circular sector 2 Math 1432 Section 26626 Lecture 6 January 31 2008 1718 Jiwen He University of Houston Hyperbolic Sine and Cosine Identities Pcosh r sinh t cosh2x sinh2X 1 sinhX y sinhxcoshy coshxsinhy Q 0 10 area of hyperbolic sector 2 r y x2yz1 COS2X sm2x 1 cos 2 sin I 0 10 X area of circular sector 2 m Jiwen He University of Houston Math 1432 Section 26626 Lecture 6 January 31 2008 17 18 Hyperbolic Sine and Cosine Identities Pcosh r sinh t 0 10 Q cosh2 X sinh2 X 1 sinhX y sinhxcoshy coshxsinhy 1 area of hyperbolic sector 2 Er y x2 2 1 2 cos2xsin X 1 cos 2 sin I sinX y sinxcosy cosxsiny 0 10 X area of circular sector 2 m Jiwen He University of Houston Math 1432 Section 26626 Lecture 6 January 31 2008 v i Hyperbolic Sine and Cosine Identities Pcosh r sinh r 0 10 Q cosh2 X sinh2 X 1 sinhX y sinhxcoshy coshxsinhy coshX y coshxcoshy sinhxsinhy area of hyperbolic sector 2 r y x2 2 1 2 cos2xsin X 1 cos 2 sin I sinX y sinxcosy cosxsiny 0 10 X area of circular sector 2 Jiwen He University of Houston Math 1432 Section 26626 Lecture 6 January 31 2008 v i Hyperbolic Sine and Cosine Identities Pcosh r sinh r 0 10 Q cosh2 X sinh2 X 1 sinhX y sinhxcoshy coshxsinhy coshX y coshxcoshy sinhxsinhy area of hyperbolic sector 2 r y x2 2 1 2 cos2xsin X 1 cos 2 sin I sinX y sinxcosy cosxsiny 0 10 X cosX y cosxcosy smxsmy area of circular sector 2 Jiwen He University of Houston Math 1432 Section 26626 Lecture 6 January 31 2008 6 Wm THE mqu Wm 5m Wm mm Wm S m 0sz THE Wag a men Sveznd me a demun

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#### "Knowing I can count on the Elite Notetaker in my class allows me to focus on what the professor is saying instead of just scribbling notes the whole time and falling behind."

Kyle Maynard Purdue

#### "When you're taking detailed notes and trying to help everyone else out in the class, it really helps you learn and understand the material...plus I made \$280 on my first study guide!"

Bentley McCaw University of Florida

Forbes

#### "Their 'Elite Notetakers' are making over \$1,200/month in sales by creating high quality content that helps their classmates in a time of need."

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### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com