Class: XII Subject: Mathematics Chapter (2): Inverse Trigonometric Functions Summary of the Chapter 1. Inverse of a function f exists if the function is one-one and onto (bijective) function. Since trigonometric functions are many-one over their domains, we restrict their domains and co-domains to make them one-one and onto and then find their inverse. The domains and ranges(principal value branches) of inverse trigonometric functions are as follows: Functions Domain Range(Principal Value Branches) yx= sin−1 −1,1 − , 22 −1 yx= cos −1,1 0, −1 R yx= tan − , 22 yx= cot−1 R (0, ) −1 yx= sec R −−( 1,1) 0, − 2 y= cos ec−1 x R −−1,1 − ( ) ,0− 22 2. The smallest numerical value, either positive or negative, of θ is called the principal value of the function and it lies in the principal value branch. −1 −1 1 3. sin−1 x should not be confused with (sin x) . In fact, (sin x) = sin x 4. The graph of an inverse function can be obtained from the corresponding graph of original function as a mirror image (reflection) along the line yx= 5. For suitable values of domain, we have y=sin−1 x x = sin y 6. Properties of inverse trigonometric function sin−1 (sinxx ) = : x − , 22 cos−1 (cosxx ) = : x 0, −1 tan (tanxx ) = : x −, 22 cot−1 (cotxx ) = : x(0, ) −1 sec (secxx ) = : x −0, 2 cosec−1 (cos ec x ) = x : x −,0 − 22 7. sin(sin−1 xx ) = : x− 1,1 cos(cos−1 xx ) = : x− 1,1 tan(tan−1 xx ) = : xR cot(cot−1 xx ) = : xR sec(sec−1 xx ) = : xR −( −1,1) cosec (cos ec−1 ) = x : xR −( −1,1) 1 8. sin−−11= cosec x : x 1 cos−−11= sec x : x 1 tan−−11= cot x : x 0 x 1 tan−−11= − + cot x : x 0 x 9. sin−−11 (−xx ) = − sin ( ) : cos−−11 (−xx ) = − cos : tan−−11 (−xx ) = − tan : cot−−11 (−xx ) = − cot : sec−−11 (−xx ) = − sec : cosec−−11 (− x ) = − cos ec x : 10. sin−−11xx+= cos : 2 tan−−11xx+= cot : 2 sec−−11 x+= co sec x : 2 −1 − 1 − 1 xy+ 11. tanxy+= tan tan : xy 1 1− xy −1 − 1 − 1 xy− tanxy−= tan tan : xy −1 1+ xy 2x 1− x2 2 x 12. 2 tan−1x = sin − 1 = cos − 1 = tan − 1 1+x2 1 + x 2 1 − x 2 Day wise planning for the chapter: Day 1 Read the concepts in Textbook regarding inverse trigonometry and principle values. Day 2 Do the questions based on principal values from examples and ex: 2.1 Day 3 Memorise the formulae on inverse trigonometric functions Day 4 Exercise: 2.2 Day 5 Exercise: Miscellaneous Day 6 Miscellaneous Exercise and extra questions. Extra Questions: −111 − 1 − 1 1) Evaluate: tan− + cot + tan sin − (Ans: − ) 33 2 12 2 2) Evaluate: a) cos−1 ( cos(− 680 )) (Ans: ) 9 b) sin−1 ( sin(− 600 )) ( Ans: ) 3 −1 34 3 −1 4 3) Simplify: cos cosxx+ sin where − x . ( Ans: x − tan ) 55 44 3 4) Evaluate: tan2( sec−− 1 2) += cot 2( cosec 1 3) 11 −1 9 5) Evaluate : tan tan ( Ans: ) 8 8 1 6) Solve: sin−−11 6xx+ sin 6 3 = − ( Ans: − ) 2 12 1−−11a 1 a 2 b 7) Prove that tan+ cos + tan − cos = 4 2b 4 2 b a 1−1 3 4− 7 1−1 3 4+ 7 8) Prove that tan sin = and justify why the other value tan sin = is ignored. 2 4 3 2 4 3 −−113 9) Evaluate sin tan(− 3) + cos − 2 10) If cos−1x+ cos − 1 y + cos − 1 z = , prove that x2+ y 2 + z 2 +21 xyz = 2 −−11x +1 11) Prove the following: sin cot cos( tan x) = 2 x + 2 2a 1− b2 2 x ab− 12) If sin−1−= cos − 1 tan − 1 , then prove that x = 1+a2 1 + b 2 1 − x 2 1+ ab An useful Web links: https://youtu.be/LNe8rst2G4g .