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= s in−1 −1,1 − , 22
−1 yx= c o s −1,1 0,
−1 yx= tan R − , 22
yx= cot−1 R (0, )
−1 yx= sec R −−( 1,1) 0, − 2
yecx= cos −1 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 ta n ( ta n ) xx= : x −, 22 c o t (− c1 o t ) xx= : x(0, )
−1 s e c ( s e c ) xx= : x −0, 2 cos(cos)ececxx−1 = : x −− ,0 22
7. s in ( s in )−1 xx= : x− 1,1 c o s ( c o s )−1 xx= : x− 1,1 t a n ( t a n )−1 xx= : xR c o t ( c o t )−1 xx= : xR s e c ( s e c )−1 xx= : xR − − ( 1,1) cos(cos)ececx −1 = : xR − − ( 1,1) 1 8. sincos−−11= ecx : x 1 c os sec−−11= x : x 1 tancot−−11= x : x 0 x 1 tancot−−11= −+ x : x 0 x 9. sin()sin()−−11−=xx − : cos−−11 (−xx ) = − cos : tan()tan−−11−=xx − : cot()cot−−11−=−xx : sec()sec−−11−=−xx : cos()cosecxecx−−11−= − : 10. sincos−−11xx+= : 2 tan−−11xx+= cot : 2 ssececxcox−−11+= : 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:
−−−11111 1) Evaluate: tancottansin−++− (Ans: − ) 332 12
2 2) Evaluate: a) coscos680−1 ( (−)) (Ans: ) 9
b) sinsin600−1 ( (−)) ( Ans: ) 3
−1 34 3 −1 4 3) Simplify: coscossinxx+ where − x . ( Ans: x − tan ) 55 44 3
4) Evaluate: tan2121( sec−− 2cot) += cos311( ec )
−1 9 5) Evaluate : tan tan ( Ans: ) 8 8
1 6) Solve: sin6sin6−−11xx+= 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 sintan3cos(−+− ) 2
10) If coscoscos−−−111xyz++= , prove that xyzxyz222+++= 21
2 −−11x +1 11) Prove the following: sincotcostan ( x) = 2 x + 2
212abx − 2 ab− 12) If sincostan−−−111 −=, then prove that x = 111++−abx222 1+ ab
An useful Web links: https://youtu.be/LNe8rst2G4g