INVERSE TRIGONOMETRIC FUNCTIONS
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1 MIT INVERSE TRIGONOMETRIC FUNCTIONS C Domins Rnge Principl vlue brnch nd Grphs of Inverse Trigonometric/Circulr Functions : Function Domin Rnge Principl vlue brnch = [ / /] Domin Rnge Principl vlue brnch = [0 ] 0 Domin Rnge Principl vlue brnch = R ( / /) Einstein Clsses Unit No. 0 0 Vrdhmn Ring Rod Plz Viks Puri Etn. Outer Ring Rod New Delhi 0 08 Ph. :
2 Domin Rnge Principl vlue brnch MIT (iv) = ec or [ / /] {0} 0 Domin Rnge Principl vlue brnch (v) = sec or [0 ] {/} 0 ; Domin Rnge Principl vlue brnch (vi) = cot R (0 ) 0 < < Import Points : () (b) st qudrnt is common to the rnge of ll the inverse functions. rd qudrnt is not used in inverse functions. (c) th qudrnt is used in the clockwise direction i.e. 0 (d) (e) No inverse function is periodic. If no brnch of n inverse trigonometric function is mentioned then it mens the principl vlue brnch of the function. Einstein Clsses Unit No. 0 0 Vrdhmn Ring Rod Plz Viks Puri Etn. Outer Ring Rod New Delhi 0 08 Ph. :
3 MIT Prctice Problems :. Find the principl vlues of : (iv) ec ( ) [Answers : / / (iv) /] C Properties of Inverse Trigonometric Functions : Propert - I ( ) = ( ) = ( ) = R (iv) cot (cot ) = R (v) sec (sec ) = (vi) ec (ec ) = Propert - II ( ) = ( ) = ; 0 ( ) = (iv) cot (cot ) = ; 0 (v) sec (sec ) = ; 0 (vi) ec (ec ) = ; 0 Propert - III ( ) = ( ) = R ( ) = (iv) cot ( ) = cot R (v) ec ( ) = ec (vi) sec ( ) = sec The function nd ec re odd functions nd rest re neither even nor odd. Propert - IV ec ; sec ; cot ; ; 0 0 Einstein Clsses Unit No. 0 0 Vrdhmn Ring Rod Plz Viks Puri Etn. Outer Ring Rod New Delhi 0 08 Ph. :
4 MIT Propert - V cot R ec sec Propert - VI or or ( ) = ( ) = (cot ) = cot ( ) = R 0 ec(sec ) sec(ec ) Prctice Problems :. Evlute ech of the following : (iv) 7 (v) 6. Evlute the following : ( 0) ( ) ( 0) (iv) { 6}. Evlute ech of the following :. Evlute ech of the following : cot 8 ec 7 [Answers : () (iv) (v) () 0 0 (iv) 6 6 () () ] 7 Einstein Clsses Unit No. 0 0 Vrdhmn Ring Rod Plz Viks Puri Etn. Outer Ring Rod New Delhi 0 08 Ph. :
5 MIT C Identities of Addition nd Substrction : Propert I : 0 0 0nd 0nd 0 0 0nd 0nd Import Points : If + + z z z = > 0 > 0 z > 0 nd ( + z + z) < z z If... n R then... n S S S S7... S S S6... where S k denotes the sum of the products of... n tken k t time. If + + z = then + + z = z (iv) If + + z = then + z + z = (v) + + = (vi) Propert II : nd or 0nd 0 nd 0nd Einstein Clsses Unit No. 0 0 Vrdhmn Ring Rod Plz Viks Puri Etn. Outer Ring Rod New Delhi 0 08 Ph. :
6 MIT 6 Einstein Clsses Unit No. 0 0 Vrdhmn Ring Rod Plz Viks Puri Etn. Outer Ring Rod New Delhi 0 08 Ph. : nd 00 0nd 0 0nd or nd Propert III : 0 nd 0 nd nd 00 nd Propert IV : Propert V : 0 ); ( 0 ); ( ) ( ) ( ) (
7 MIT 7 Propert VI : ; ; ; ; ; ; Propert VII : ; ; ; Prctice Problems :. Prove tht : (v) ; ; (iv) (vi). (vii) (viii) 8 7. Evlute : ( 0.6) ( 0.8) 8. Solve for : ( 0) ( ) = [Answers : () / zero] Einstein Clsses Unit No. 0 0 Vrdhmn Ring Rod Plz Viks Puri Etn. Outer Ring Rod New Delhi 0 08 Ph. :
8 MIT 8 Miscellneous Problems :. Write the following functions in the simplest form : (iv) Ans. :. Simpl ech of the following : (iv) cot Ans. :. Simpl ech of the following :. Simpl ech of the following : b b b b b b Ans. : b b. If z [ ] such tht + + z = find the vlue of z 008 Ans. : zero Solve the following equtions : z 008. ( ) = ( ec ) (iv) ( ) 6 Ans. : = (iv) = 0 Einstein Clsses Unit No. 0 0 Vrdhmn Ring Rod Plz Viks Puri Etn. Outer Ring Rod New Delhi 0 08 Ph. :
9 7. The number of rel solutions of ( ) + MIT 9 is : () zero (b) one (c) two (d) infinite Ans. : c 8. If then the vlue of is () (b) (c) (d) Ans. : 9. If + + z = then + + z equls to () 0 (b) (c) (d) Ans. : 0. If + + z =. If then the vlue of z 00 () 0 (b) (c) (d) Ans. : c for 0 < < then equls to () / (b) (c) / (d) Ans. : b 0 0 z 0 Einstein Clsses Unit No. 0 0 Vrdhmn Ring Rod Plz Viks Puri Etn. Outer Ring Rod New Delhi 0 08 Ph. :
10 MIT 0 SINGLE CORRECT CHOICE TYPE. The vlue is where. () (b) / (c) / (d). Let b c be positive rel numbers nd ( b c) bc b( b c c( b c). Then equls b () zero (b) / (c) (d) /. Consider two ngles A = ( ) nd B = (/) + (/). Then () A = B (b) A > B (c) A < B (d) A B. If + + z = r then the vlue of z r z r zr () / (b) / (c) (d) / ec[ cot sec sec ] sec[cot ec ]. If where [0 ] then () = = (b) = = (c) = = (d) 6. If + + z = then the vlue of + + z + z is () 0 (b) (c) (d) is c) + 8. The vlue of is cot () 0 (b) (c) (d) 9. The number of solution of the eqution [ {cot( )}] = 0 is () (b) (c) (d) 6 sec 0. The number of solution of the eqution ; + + ( ) = 0 is () (b) (c) (d) ANSWERS (SINGLE CORRECT CHOICE TYPE) 7. n 0n.6 If cot n N then the 6 minimum vlue of n is () (b) (c) (d)... b. b. 6. b 7. c d 0. Einstein Clsses Unit No. 0 0 Vrdhmn Ring Rod Plz Viks Puri Etn. Outer Ring Rod New Delhi 0 08 Ph. :
11 MIT EXERCISE BASED ON NEW PATTERN (A) MATRIX-MATCH TYPE Mtching- Column - A 6 6 Column - B (A) Mtching- Column - A Column - B (p) / (p) (B) (q) (B) (C) 6 6 (q) 7 (D) cot + cot 7 + cot 8 (A) (B) (C) (D) Mtching- Column - A If If If If (r) (s) 8 6 Column - B (p) (q) ( ) (r) ( ) (s) (t) then then then 8 then = 6 7 (C) (D) Mtching- 6 (r) / (s) Let ( ) be such tht () + () + (b) =. Column - A Column - B (A) If = nd b = 0 then (p) lies on the ( ) circle + = (B) If = nd b = then (q) lies on ( ) ( ) ( ) = 0 (C) If = nd b = then (r) lies on = ( ) (D) If = nd b = then (s) lies on ( ) ( ) ( ) = 0 MULTIPLE CORRECT CHOICE TYPE. Let f () e () (b) ( / ) 8 f e 9 8 f e 9 /8 /8 7 / (c) f e. Then (d) 7 f e / Einstein Clsses Unit No. 0 0 Vrdhmn Ring Rod Plz Viks Puri Etn. Outer Ring Rod New Delhi 0 08 Ph. :
12 MIT. Consider the function f() = ( ) + ( ). Then () the lest vlue of function is (b) the lest vlue of function is 6 (c) (d) the gretest vlue of function is the gretest vlue of function is. If + + z = then () z z z (b) + + z + z = ( + z + z ) (c) z z z (d) + + z + z = ( + z + z ). The vlue of A () 0 A (c) A (cot A) (cot A) is (b) 0 A (d) 0 0 A. If re the roots of the eqution = 0 then which re rel () (b) cot (c) cot (d) ASSERTION-REASON TYPE Ech question contins STATEMENT- (Assertion) nd STATEMENT- (Reson). Ech question hs choices (A) (B) (C) nd (D) out of which ONLY ONE is correct. (A) (B) (C) (D) Sttement- is True Sttement- is True; Sttement- is correct eplntion for Sttement- Sttement- is True Sttement- is True; Sttement- is NOT correct eplntion for Sttement-. Let : = : Sttement- is True Sttement- is Flse Sttement- is Flse Sttement- is True STATEMENT- : eqution + = 6. STATEMENT- :. Consider f () 0 ( ) 6 STATEMENT- : f () 0 is root of the STATEMENT- : f() is const function for ll. STATEMENT- : If < 0 + = STATEMENT- : + cot = R (Answers) EXCERCISE BASED ON NEW PATTERN MATRIX-MATCH TYPE. [A-r; B-r; C-p; D-p]. [A-s; B-p; C-q t; D-r]. [A-p; B-q; C-r; D-s]. [A-p; B-q C-p D-s] MULTIPLE CORRECT CHOICE TYPE. b c. c. b. b. b c ASSERTION-REASON TYPE. A. A. D Einstein Clsses Unit No. 0 0 Vrdhmn Ring Rod Plz Viks Puri Etn. Outer Ring Rod New Delhi 0 08 Ph. :
13 MIT INITIAL STEP EXERCISE (SUBJECTIVE). Find the simplest vlue of rc + rc. Solve the following equtions / sstem of eqution : () (b) ( ) + () + ( + ) = () (c) (e) + = = nd. In tringle ABC A = 90 0 then prove tht b c c. b. Find ll the positive integrl solutions of +. 0 (d). Prove tht cot =. 6. Prove tht the eqution ( ) + ( ) = hs no roots for. FINAL STEP EXERCISE (SUBJECTIVE). I f k nd k k prove tht one vlue of ( ) is If z re in A.P. then prove tht ( + z) + ( z) = + z where ( ) ; z < & > 0 > 0.. Convert trigonometric function ( (cot( ))) into n lgebric function f(). Then from the lgebric function find ll the vlues of for which f() is zero. Epress the vlues of in the form ± b where nd b re rtionl numbers.. If = ( ) then find the generl vlue of.. Ug the principl vlues epress the following s gle ngle : b 6. Solve where c c + b = c c 0 7. Solve the eqution : 6 8. Find the sum of the series : () n n... n(n ) 6 Einstein Clsses Unit No. 0 0 Vrdhmn Ring Rod Plz Viks Puri Etn. Outer Ring Rod New Delhi 0 08 Ph. :
14 (b) (c). n n MIT 9. Find the integrl vlues of K for which the sstem of equtions ; K rc (rc ) (rc ).(rc ) 6 solutions nd find those solutions. possesses.. () = (b) = 0 (c) = (d) = (e) =. = ; = nd = ; = 7 ANSWERS SUBJECTIVE (INITIAL STEP EXERCISE) ANSWERS SUBJECTIVE (FINAL STEP EXERCISE). f() = ± ± ±. nn n ( ). where n I 6. { 0 } () (b) (c) 9. K = = Einstein Clsses Unit No. 0 0 Vrdhmn Ring Rod Plz Viks Puri Etn. Outer Ring Rod New Delhi 0 08 Ph. :
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