Inverse Trigonometry CBSE class 12 pdf

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CBSE class 12 Inverse Trigonometry Notes:

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                          Inverse Trigonometric Function

         The inverse trigonometric functions play an important role in calculus for they serve to define many integrals. The concepts of inverse trigonometric functions is also used in science and engineering.

                                                Basic Trigonometric functions:

Trigonometric      functions	Period	Domain	Range
Sin θ	2π	all Real numbers	[–1  1]
Cos θ	2π	all Real numbers	[–1  1]
Tan θ	π	all Real – (2n + 1)	All Real numbers
Sec θ	2π	all Real – (2n + 1)	(– ∞, – 1] U [1, ∞)
Cosec θ	2π	all Real – nπ	(– ∞, – 1] U [1, ∞)
Cot θ	π	all Real – nπ	All Real numbers

                  

                                   Domain and range of inverse functions:

Inverse Trig.       functions	Domain	Range
Sin–1θ	[–1  1]	[ –     ]
Cos–1θ	[–1  1]	[ 0  π ]
Tan–1θ	all Real	( –     )
Sec-–1θ	all R – (–1, 1)	[ 0  π ] – {  }
Cosec–1θ	all R – (–1, 1)	[ –     ] – { 0 }
Cot–1θ	all Real	( 0  π )

Properties of inverse functions:- 1.      Relation between inverse functions: i.          Sin-1 x = cosec-1  ; x ɛ (-∞  -1] U [1 ∞ ) ii.       Cos-1 x = sec-1  ; x ɛ (-∞  -1] U [1 ∞ ) iii.      tan-1 x = cot-1  ; x > 0 2.      Negative values : i.          Sin-1( - x ) = – Sin-1( x ), x ɛ [-1   1] ii.       tan-1( - x ) = – tan-1( x ), x ɛ R iii.     cosec-1( - x ) = – cosec-1( x ), x ɛ [1   ∞) iv.      Cos-1( - x )   = π – Cos-1( x ) ; x ɛ [-1   1] v.        sec-1( - x ) = π – sec-1( x ) ; x ɛ [1   ∞) vi.      cot-1( - x ) = π  – cot-1( x ) ; x ɛ R. 3.      Inverse formulas : - i.          Sin-1( x ) + Cos-1( x ) = ,  x ɛ [-1   1] ii.       tan-1( x ) + cot-1( x ) = ,  x ɛ R iii.     sec-1( x ) + cosec-1( x ) = ,  x ɛ [1  ∞] iv.      tan-1( x ) + tan-1 (y ) = tan-1[    ] ;  xy < 1, if xy > 1 the π – tan-1[   ] v.        tan-1( x )  –  tan-1(y ) = tan-1[  ] ; xy > -1 vi.      2 tan-1( x ) = Sin-1[ ] ;  |x| ≤  1 vii.   2 tan-1( x ) = Cos-1[  ] ;   x > 0 x.     Sin-1( x ) + Sin-1 (y ) = Sin-1[x ], if x.y > 0 and  +  > 1 vii. 2 tan-1( x ) = tan-1[ ] ;  -1 < x < 1 ix.        Cos-1( x ) + Cos-1 (y ) = Cos-1[    – x.y] Examples of inverse functions: 1.       Find the value of tan-12 + tan-13. 2.       Find the value of tan-1  + tan-1  + tan-1 . 3.      Find the value of Sin-1  + Sin-1  . 4.      Find the value of x in Sin-1(1 – x) – 2Sin-1x = .                                                 Download Inverse trigonometry file please click below https://drive.google.com/file/d/0BwSVbdov-IG3blVBeDV2TFROZk0/view?usp=sharing

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