M 408 K Fall 2005 Inverse Trig Functions Important Decimal Approximations and Useful Trig Identities Decimal Approximations: p

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1 M 408 K Fall 005 Inverse Trig Fnctions Imortant Decimal Aroimations an Usefl Trig Ientities Decimal Aroimations: ( æ ö ç ø è ; æ ö ç è ø ( Trigonometric Ientities: When + v i / 90 egrees, sin ( b / c cos (v cos ( i/, v tan ( b / a cot (v cot ( i/, c b sec ( c / a csc (v csc ( i/ a SOHCAHTOA sin ( o / hy tan ( o / aj sec ( hy / aj cos ( aj / hy hy aj o sin ( sin ( tan ( tan ( cos ( cos ( sec ( sec (

2 Inverse Sine Fnction: sin (, Î [, ], sin ( when sin ( an é ù Î ê, ú ë û é ù Î ê, ú ë û For in [, + ], inverse sine ( When is in [ 0, ], inv sin ( is in [ 0, i/ ] Locate on the vertical ais 077 sin ( 088 raians sin ( / / When is in [, 0 ], inv sin ( is in [ i/, 0 ] inverse sine ( 095 sin ( 44 raians / inverse sine ( 0 sin ( 0 raians / sin ( sin ( / /

3 inverse sine ( 06 sin ( 065 raians sin ( / inverse sine ( 097 sin ( 9 raians sin ( / / / Drawing the angle inv sin ( when > 0 : inv sin ( sin ( / o / hy, Make o an hy cos ( aj / hy cos ( inv sin ( Sqrt( ^ a Sqrt( ^ Table of vales: y sin ( y sin ( π / 57 / 0866 π / 047 / 0707 π / / π / / π / / 0707 π / / 0866 π / 047 π / 57 y inv sin ( i/ i/ i/4 i/6 y i/6 i/4 i/ y 00 sin ( 00 i/

4 Inverse Tangent Fnction: Î ç è æ ö Î ç, è ø tan æ ö (, Î ( infinity, infinity,, ø tan ( when tan ( an For in ( inf, inf, inverse tangent ( Locate on the nmber line lace vertically with the origin at oint (,0 4 As > infinity, inv tan ( > i / tan ( 79 raians tan ( As > 0, inv tan ( > As > infinity, inv tan ( > i /

5 inverse tangent ( inverse tangent ( 8 tan ( 66 raians tan ( 7 tan ( 047 raians tan ( Drawing the angle inv tan ( when > 0 : inv tan ( tan ( / o / aj, Make o an aj Sqrt( + ^ c sec ( hy / aj sec ( inv tan ( Sqrt( + ^ Table of vales: y tan ( y tan ( π/ 047 π / / 0577 π / / 0577 π /6 054 π / π / y inv tan ( i 5i/6 i/ i/ i/ i/ i/6 i/ i/ i/ 5i/6 i y 67 y inv tan ( 4

6 Inverse Secant Fnction: sec (, Î (, ] U [,, sec ( when sec ( an é ö é ö Î ê 0,, ë ø U ë ê ø é ö é ö Î ê 0,, ë ø U ë ê ø inverse secant ( 00 raians sec ( 00 sec ( 4 4 For <, inverse secant ( 450 inv sec ( 47 raians sec ( As > +, inv sec ( > 0 As > infinity, inv sec ( > i/ 4 inverse secant ( 40 raians sec ( sec ( 4 inverse secant ( 6 5 raians sec ( sec ( 4 6 When, sec + sec (

7 inverse secant ( 6 inv sec ( 05 raians 4 inverse secant ( sec ( 450 inv sec ( 47 raians sec ( 4 Drawing the angle inv sec ( when > : inv sec ( sec ( / hy / aj Make hy an aj b Sqrt( ^ tan ( o / aj tan ( inv sec ( Sqrt( ^ Table of vales: y sec ( y sec ( π/ π/4 97 / 55 7π/6 665 π 46 0 / 55 π/ π/ π/ i/ y* y sec i 5i/6 i/ i/ y i/ i/6 0 5 * 5 0 i/6

8 Inverse Cosine Fnction: cos (, Î [, ], [ 0, ] Î cos ( when cos ( an [ 0, ] Î For in [, + ], inverse cosine ( When is in [, 0 ], inv cos ( is in [ i/, i ] When is in [ 0, ], inv cos ( is in [ 0, i/ ] Locate on the horizontal ais 08 cos ( 58 raians cos ( / / As > 0, inv cos ( > i / As >, inv cos ( > i As >, inv cos ( > 0 As > 0, inv cos ( > i / inverse cosine ( inverse cosine ( inverse cosine ( / / / / / / cos ( 880 raians cos ( cos ( 94 raians cos ( cos ( 475 raians cos (

9 inverse cosine ( / cos ( 5 raians / cos ( y inv cos ( i 5i/6 i/4 i/ y 04 cos ( 0 i/ y inverse cosine ( i/ i/4 i/ / / i/6 i/4 i/ cos ( 04 raians cos ( The figres below show that cos ( an sin ( are closely relate: cos cot ( ( sin tan ( ( Similar argments show that: an csc ( sec ( By constrction, inv sin ( v Fliing the triangle aron shows that angle v is v inv cos ( Since + v i / 90 egrees, inv sin ( + inv cos ( i / Ths, inv cos ( i / inv sin ( v

10 Differentiation an Integration with Inverse Trig Fnctions The Inverse Sine Fnction : y sin ( ( sin ( ö ç ø è æ ö an ( ç ø sin ( æ ç è For any ositive constant a > 0 : ò sin ( + C an ò a sin æ ö ç è a ø + C The Inverse Tangent Fnction : y tan ( ( tan ( + + ö ç ø è æ ö an ( ç ø tan ( æ ç è For any nonzero constant a : æ ö ò tan ( + C an + ò tan ç a + a è a ø + C The Inverse Secant Fnction : y sec ( ( sec ( For any ositive constant a > 0 : sec ( ò + C an an ( sec ( ò æ ö æ ö ç ç è è ø ø sec æ ö ç + a a èaø C Using the formlas relating the other three inverse trig fnctions with these three, the erivatives of the other three fnctions are easily calclate: ( cos ( ( sin ( ; ( cot ( ( tan ( ; + ( csc ( ( sec (

11 Derivation of erivatives for Inverse Trig fnctions Derivation # : Proof that ( sin ( : Write sin ( We seek ( sin ( Then, sin( sin( sin ( ; so, ( sin( ( ; so, cos(, an ths cos( cos( sin ( Claim: cos( sin ( for all Î[, ] Proof of Claim: The metho of rawing sin ( resente above shows that cos( sin [, ] sch that > 0 ( for all Î If < 0, then, an so sin ( sin ( sin ( This last sin eqality is a irect conseqence of the fact that ( z sin( z for all z Ths, cos( sin ( cos ( sin ( cos( sin ( cos( sin (, an so, cos( sin ( cos( sin ( When 0, the claim is tre becase sin (0 0 an cos(0, an the claim is roven cos( Finally, ( sin ( roof is comlete cos( sin (, an the

12 + Derivation # : Proof that ( tan ( : Write tan ( We seek ( tan ( Then, tan( tan( tan ( ; so, ( tan( ( ; so, sec (, an ths sec ( sec ( tan ( Claim: sec( tan ( + for all Î (, Proof of Claim: The metho of rawing tan ( resente above shows that sec( tan ( + for all sch that > 0 If < 0, then, an so tan ( tan ( tan ( This last eqality is a irect conseqence of the fact that tan( z tan( z for all z Ths, sec( tan ( sec ( tan ( sec( tan ( sec( tan (, an so, sec( tan ( sec( tan ( + + When 0, the claim is tre becase tan (0 0 an sec(0, an the claim is roven Finally, ( ( tan sec ( sec ( tan ( + ( +, an the roof is comlete

13 Derivation # : sec ( for all Î, È, Proof that ( ( ( Write sec ( We seek ( sec ( : Then, sec( sec( sec ( so, sec( tan(, an ths sec( tan( sec( sec ( tan( sec ; so, ( sec( ( ( tan( sec ; ( Claim: tan(sec ( [ ì for all Î, ü ï ï í ý ï ï ïî for all Î (,] ïþ Proof of Claim: If >, then metho of rawing sec ( resente above shows that tan( sec ( If, then, an so sec ( sec ( < + This last eqality is evient from the efinition of sec ( resente above ( ( ( ( ( Ths, ( tan(sec ( tan sec + tan sin an so, tan(sec ( tan sec When, the claim is tre becase sec ( 0 an tan(0 0 ; when, the claim is tre becase sec ( π an tan(π 0, an the claim is roven Ths, when > + or <, tan(sec (, an the roof is comlete

Math 144 Activity #7 Trigonometric Identities

144 p 1 Math 144 Activity #7 Trigonometric Identities What is a trigonometric identity? Trigonometric identities are equalities that involve trigonometric functions that are true for every single value