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Inverse Trigonometric Functions _____________________________________________________________________ 1. Find the principal values of the following (i) 1 Sin -1 6 (ii) Tan -1 ( -3 ) 3 2 4 Tan -1 ( - ) 1 Sec -1 ( - ) 2 3 (i) 3 If Tan -1 = find Sin 3 5 (ii) 1 If Cot -1 = find Cos 1 5 2 (iii) 4 5 (ii) 17 6 (iv) 4 7 1 3 2 Evaluate (i) 3 5 Sin ( Cos -1 ) Cos ( Tan -1 (iii) 4 5 3 ) 4 4 2 Tan Cos 1 Tan 1 3 5 3 5 1 3 2 Tan -1 = Tan-1x, x = (v) 4. Cosec - (v) 1 If Sin -1 = Tan -1x, find x 3. (iv) 6 ( - ) 2. (iii) 1 10 1 4 Sin Cos 1 2 5 Prove that 1 a) Sin -1 b) 2 Tan -1 5 + Sin -1 1 = Sin x 2 5 -1 = 2 2x 2 x 1 5. 6. c) x Cosx Tan -1 = d) Sin -1 e) ab bc ca Tan -1 + Tan -1 + Tan -1 =0 1 Sinx 4 2 2 -1 -1 x 1 x x 1 x = Sin x - Sin 1 ab 1 ab x 1 ca Solve for x 2x Tan -1 b) Tan -1 (x + 1) + Tan -1 (x-1) = Tan -1 1 x2 Prove that if Sin -1 then x= + Cot -1 1 x2 = , (x > 0) 3 2x a) 2a 1 a ab . 1 ab 2 + Sin -1 2b 1 b2 8 13 , (x > 0) 2 3 1 4 = 2 Tan -1 x 7. If two angles of a triangle are Tan -1 2 and Tan -1 3, prove that the third angle is 8. Prove that Sec2 (Tan -1 2) + Cosec 2 (Cot -1 3) = 15. 9. Solve : Sin -1 x + Sin -1 ( 1-x ) = Cos -1 x ( 0, ½ ). 10. Simplify : a) Sec -1 2 x 1 1 (2 Cos -1 x) 2 x 1 x b) Sin -1 ( Tan -1 x ) c) Cot -1 1 x 2 x 1 Tan 2 Tan 1 5 4 11. Evaluate: 12. If Sin -1x - Cos -1x = 13. 1 1 2 Tan -1 + Tan -1 is _________ 14. Evaluate : Sin 2 Cos 1 3 6 , find x 1 1 Tan x 4 2 7 17 3 2 7 4 3 5 24 25 . 4 15. Prove that 2 1 2 Tan -1 + Tan -1 = Cos -1 16. Prove that Tan -1 17. Prove that Tan -1 18. Simplify Sin -1 4 9 5 1 Cosx 1 Cosx x = + , 0 < x < 2 1 Cosx 1 Cosx 4 2 1 x2 1 x2 = + 1 Cos-1 x2, -1 < x < 1. 2 2 4 2 1 x 1 x Sinx Cosx 2 , <x< 4 4 x 4 Sinx Cosx = x . 4 2 Cos -1 19. Prove that 20. Solve the equation : 2 Tan -1 (Cos x) = Tan -1 (2 Cosec x) 4 21. Write in simplest form a) Tan -1 x 1 x 2 , x R 1 1 Cot x 2 2 b) Tan -1 1 x 2 x , x R 1 1 Cot x 2 1 x 2 1 , x 0 c) Tan -1 1 1 Tan x 2 x 1 x 1 , x 0 d) Tan -1 x 2 ax ax , -a < x < a e) Tan -1 , 2 2 a a x f) Tan -1 x x 1 x2 g) Sin -1 2 -a < x < a , < x < Hint: (let x = sin α) 4 4 1 x 1 x , 0 < x < 1 Hint: (let x = cos 2α) h) Sin -1 2 1 x i) Sin 2 Tan 1 , 1 x 1 1 Tan x 2 2 1 1 x Cos a 2 1 1 x Sin a 2 1 Sin x 4 1 1 Cos x 4 2 1 x2 j) Cot -1 a , 2 2 x a k) Tan -1 x x , 2 2 a x 1 x Sec a x > a 1 x Sin a -a < x < a 2 2 x a l) Sin -1 22. 1 x Tan a Solve for x a) Cos -1 x + Sin -1 x = 6 2 (1) 1 1 x 1 b) Tan -1 Tan -1 x = 0 , x > 0 Hint: (let x = cos α) 1 x 2 23. Prove that Tan -1 24. 3 x xy - Tan -1 xy 4 y The value of Cos -1 Cos 13 6 is ---------- 6 1 Evaluate: sin sin 1 (Ans:√3/2 ) (CBSE 2008) 2 3 26. Solve for x: tan 1 (2 x) tan 1 3 x [ Ans : 1/6 , -1] (CBSE 2008, 2009) 4 1 1 1 1 27. Prove tan 1 tan 1 tan 1 tan 1 ( CBSE 2008) 3 5 7 8 4 x 1 1 x 1 28. Solve for x: tan 1 tan [ Ans : +/-(1/√2)] x 2 x 2 4 2008) 3 2 ) [ Ans : 29. Using principal value, evaluate sin 1 (sin ] (CBSE 2009) 5 5 25. 30. What is the principal value of 1 1 sin 1 cos 1 (CBSE 2010) 2 2 [ Ans : ] 2 (CBSE 31. 32. 33. 34. 35. 3 ? [ Ans : - ] What is the principal value of sin 1 (CBSE 2010) 3 2 1 1 x Prove tan 1 x cos 1 ,.....x 0,1 (CBSE 2010) 2 1 x Prove that tan 1 1 tan 1 2 tan 1 (3) (CBSE 2010) x 1 1 x 1 If tan 1 tan , find the value of x. (CBSE 2010) x 2 x 2 4 (Same as Q28) 12 3 56 Prove cos 1 sin 1 sin 1 (CBSE 2010) 13 5 65 36. Write the value of sin sin1 3 1 2 (CBSE 2011) 37.Prove the following: 1 sin x cot-1 1 sin x 1 sin x x , x 0, (CBSE 2011) 2 1 sin x 4 x xy )(CBSE 2011) (Sol: 4 xy 38.Find the value of tan-1 - tan-1 y