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We will discuss the list of inverse trigonometric function formula which will help us to solve different types of inverse circular or inverse trigonometric function.
(i) sin (sinβ1 x) = x and sinβ1 (sin ΞΈ) = ΞΈ, provided that - Ο2 β€ ΞΈ β€ Ο2 and - 1 β€ x β€ 1.
(ii) cos (cosβ1 x) = x and cosβ1 (cos ΞΈ) = ΞΈ, provided that 0 β€ ΞΈ β€ Ο and - 1 β€ x β€ 1.
(iii) tan (tanβ1 x) = x and tanβ1 (tan ΞΈ) = ΞΈ, provided that - Ο2 < ΞΈ < Ο2 and - β < x < β.
(iv) csc (cscβ1 x) = x and secβ1 (sec ΞΈ) = ΞΈ, provided that - Ο2 β€ ΞΈ < 0 or 0 < ΞΈ β€ Ο2 and - β < x β€ 1 or -1 β€ x < β.
(v)
sec (secβ1 x) = x and secβ1 (sec ΞΈ) = ΞΈ, provided that 0 β€ ΞΈ β€
Ο2 or Ο2 <
ΞΈ β€ Ο and - β < x β€ 1 or 1 β€ x < β.
(vi) cot (cotβ1 x) = x and cotβ1 (cot ΞΈ) = ΞΈ, provided that 0 < ΞΈ < Ο and - β < x < β.
(vii) The function sinβ1 x is defined if β 1 β€ x β€ 1; if ΞΈ be the principal value of sinβ1 x then - Ο2 β€ ΞΈ β€ Ο2.
(viii) The function cosβ1 x is defined if β 1 β€ x β€ 1; if ΞΈ be the principal value of cosβ1 x then 0 β€ ΞΈ β€ Ο.
(ix) The function tanβ1 x is defined for any real value of x i.e., - β < x < β; if ΞΈ be the principal value of tanβ1 x then - Ο2 < ΞΈ < Ο2.
(x) The function cotβ1 x is defined when - β < x < β; if ΞΈ be the principal value of cotβ1 x then - Ο2 < ΞΈ < Ο2 and ΞΈ β 0.
(xi) The function secβ1 x is defined when, I x I β₯ 1 ; if ΞΈ be the principal value of secβ1 x then 0 β€ ΞΈ β€ Ο and ΞΈ β Ο2.
(xii) The function cscβ1 x is defined if I x I β₯ 1; if ΞΈ be the principal value of cscβ1 x then - Ο2 < ΞΈ < Ο2 and ΞΈ β 0.
(xiii) sinβ1 (-x) = - sinβ1 x
(xiv) cosβ1 (-x) = Ο - cosβ1 x
(xv) tanβ1 (-x) = - tanβ1 x
(xvi) cscβ1 (-x) = - cscβ1 x
(xvii) secβ1 (-x) = Ο - secβ1 x
(xviii) cotβ1 (-x) = cotβ1 x
(xix) In numerical problems principal values of inverse circular functions are generally taken.
(xx) sinβ1 x + cosβ1 x = Ο2
(xxi) secβ1 x + cscβ1 x = Ο2.
(xxii) tanβ1 x + cotβ1 x = Ο2
(xxiii) sinβ1 x + sinβ1 y = sinβ1 (x β1βy2 + yβ1βx2), if x, y β₯ 0 and x2 + y2 β€ 1.
(xxiv) sinβ1 x + sinβ1 y = Ο - sinβ1 (x β1βy2 + yβ1βx2), if x, y β₯ 0 and x2 + y2 > 1.
(xxv) sinβ1 x - sinβ1 y = sinβ1 (x β1βy2 - yβ1βx2), if x, y β₯ 0 and x2 + y2 β€ 1.
(xxvi) sinβ1 x - sinβ1 y = Ο - sinβ1 (x β1βy2 - yβ1βx2), if x, y β₯ 0 and x2 + y2 > 1.
(xxvii) cosβ1 x + cosβ1 y = cosβ1(xy - β1βx2β1βy2), if x, y > 0 and x2 + y2 β€ 1.
(xxviii) cosβ1 x + cosβ1 y = Ο - cosβ1(xy - β1βx2β1βy2), if x, y > 0 and x2 + y2 > 1.
(xxix) cosβ1 x - cosβ1 y = cosβ1(xy + β1βx2β1βy2), if x, y > 0 and x2 + y2 β€ 1.
(xxx) cosβ1 x - cosβ1 y = Ο - cosβ1(xy + β1βx2β1βy2), if x, y > 0 and x2 + y2 > 1.
(xxxi) tanβ1 x + tanβ1 y = tanβ1 (x+y1βxy), if x > 0, y > 0 and xy < 1.
(xxxii) tanβ1 x + tanβ1 y = Ο + tanβ1 (x+y1βxy), if x > 0, y > 0 and xy > 1.
(xxxiii) tanβ1 x + tanβ1 y = tanβ1 (x+y1βxy) - Ο, if x < 0, y > 0 and xy > 1.
(xxxiv) tanβ1 x + tanβ1 y + tanβ1 z = tanβ1 x+y+zβxyz1βxyβyzβzx
(xxxv) tanβ1 x - tanβ1 y = tanβ1 (xβy1+xy)
(xxxvi) 2 sinβ1 x = sinβ1 (2xβ1βx2)
(xxxvii) 2 cosβ1 x = cosβ1 (2x2 - 1)
(xxxviii) 2 tanβ1 x = tanβ1 (2x1βx2) = sinβ1 (2x1+x2) = cosβ1 (1βx21+x2)
(xxxix) 3 sinβ1 x = sinβ1 (3x - 4x3)
(xxxx) 3 cosβ1 x = cosβ1 (4x3 - 3x)
(xxxxi) 3 tanβ1 x = tanβ1 (3xβx31β3x2)
β Inverse Trigonometric Functions
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