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 (sin\(^{-1}\) x) = x and sin\(^{-1}\) (sin θ) = θ, provided that - \(\frac{π}{2}\) ≤ θ ≤ \(\frac{π}{2}\) and - 1 ≤ x ≤ 1.
(ii) cos (cos\(^{-1}\) x) = x and cos\(^{-1}\) (cos θ) = θ, provided that 0 ≤ θ ≤ π and - 1 ≤ x ≤ 1.
(iii) tan (tan\(^{-1}\) x) = x and tan\(^{-1}\) (tan θ) = θ, provided that - \(\frac{π}{2}\) < θ < \(\frac{π}{2}\) and - ∞ < x < ∞.
(iv) csc (csc\(^{-1}\) x) = x and sec\(^{-1}\) (sec θ) = θ, provided that - \(\frac{π}{2}\) ≤ θ < 0 or 0 < θ ≤ \(\frac{π}{2}\) and - ∞ < x ≤ 1 or -1 ≤ x < ∞.
(v)
sec (sec\(^{-1}\) x) = x and sec\(^{-1}\) (sec θ) = θ, provided that 0 ≤ θ ≤
\(\frac{π}{2}\) or \(\frac{π}{2}\) <
θ ≤ π and - ∞ < x ≤ 1 or 1 ≤ x < ∞.
(vi) cot (cot\(^{-1}\) x) = x and cot\(^{-1}\) (cot θ) = θ, provided that 0 < θ < π and - ∞ < x < ∞.
(vii) The function sin\(^{-1}\) x is defined if – 1 ≤ x ≤ 1; if θ be the principal value of sin\(^{-1}\) x then - \(\frac{π}{2}\) ≤ θ ≤ \(\frac{π}{2}\).
(viii) The function cos\(^{-1}\) x is defined if – 1 ≤ x ≤ 1; if θ be the principal value of cos\(^{-1}\) x then 0 ≤ θ ≤ π.
(ix) The function tan\(^{-1}\) x is defined for any real value of x i.e., - ∞ < x < ∞; if θ be the principal value of tan\(^{-1}\) x then - \(\frac{π}{2}\) < θ < \(\frac{π}{2}\).
(x) The function cot\(^{-1}\) x is defined when - ∞ < x < ∞; if θ be the principal value of cot\(^{-1}\) x then - \(\frac{π}{2}\) < θ < \(\frac{π}{2}\) and θ ≠ 0.
(xi) The function sec\(^{-1}\) x is defined when, I x I ≥ 1 ; if θ be the principal value of sec\(^{-1}\) x then 0 ≤ θ ≤ π and θ ≠ \(\frac{π}{2}\).
(xii) The function csc\(^{-1}\) x is defined if I x I ≥ 1; if θ be the principal value of csc\(^{-1}\) x then - \(\frac{π}{2}\) < θ < \(\frac{π}{2}\) and θ ≠ 0.
(xiii) sin\(^{-1}\) (-x) = - sin\(^{-1}\) x
(xiv) cos\(^{-1}\) (-x) = π - cos\(^{-1}\) x
(xv) tan\(^{-1}\) (-x) = - tan\(^{-1}\) x
(xvi) csc\(^{-1}\) (-x) = - csc\(^{-1}\) x
(xvii) sec\(^{-1}\) (-x) = π - sec\(^{-1}\) x
(xviii) cot\(^{-1}\) (-x) = cot\(^{-1}\) x
(xix) In numerical problems principal values of inverse circular functions are generally taken.
(xx) sin\(^{-1}\) x + cos\(^{-1}\) x = \(\frac{π}{2}\)
(xxi) sec\(^{-1}\) x + csc\(^{-1}\) x = \(\frac{π}{2}\).
(xxii) tan\(^{-1}\) x + cot\(^{-1}\) x = \(\frac{π}{2}\)
(xxiii) sin\(^{-1}\) x + sin\(^{-1}\) y = sin\(^{-1}\) (x \(\sqrt{1 - y^{2}}\) + y\(\sqrt{1 - x^{2}}\)), if x, y ≥ 0 and x\(^{2}\) + y\(^{2}\) ≤ 1.
(xxiv) sin\(^{-1}\) x + sin\(^{-1}\) y = π - sin\(^{-1}\) (x \(\sqrt{1 - y^{2}}\) + y\(\sqrt{1 - x^{2}}\)), if x, y ≥ 0 and x\(^{2}\) + y\(^{2}\) > 1.
(xxv) sin\(^{-1}\) x - sin\(^{-1}\) y = sin\(^{-1}\) (x \(\sqrt{1 - y^{2}}\) - y\(\sqrt{1 - x^{2}}\)), if x, y ≥ 0 and x\(^{2}\) + y\(^{2}\) ≤ 1.
(xxvi) sin\(^{-1}\) x - sin\(^{-1}\) y = π - sin\(^{-1}\) (x \(\sqrt{1 - y^{2}}\) - y\(\sqrt{1 - x^{2}}\)), if x, y ≥ 0 and x\(^{2}\) + y\(^{2}\) > 1.
(xxvii) cos\(^{-1}\) x + cos\(^{-1}\) y = cos\(^{-1}\)(xy - \(\sqrt{1 - x^{2}}\)\(\sqrt{1 - y^{2}}\)), if x, y > 0 and x\(^{2}\) + y\(^{2}\) ≤ 1.
(xxviii) cos\(^{-1}\) x + cos\(^{-1}\) y = π - cos\(^{-1}\)(xy - \(\sqrt{1 - x^{2}}\)\(\sqrt{1 - y^{2}}\)), if x, y > 0 and x\(^{2}\) + y\(^{2}\) > 1.
(xxix) cos\(^{-1}\) x - cos\(^{-1}\) y = cos\(^{-1}\)(xy + \(\sqrt{1 - x^{2}}\)\(\sqrt{1 - y^{2}}\)), if x, y > 0 and x\(^{2}\) + y\(^{2}\) ≤ 1.
(xxx) cos\(^{-1}\) x - cos\(^{-1}\) y = π - cos\(^{-1}\)(xy + \(\sqrt{1 - x^{2}}\)\(\sqrt{1 - y^{2}}\)), if x, y > 0 and x\(^{2}\) + y\(^{2}\) > 1.
(xxxi) tan\(^{-1}\) x + tan\(^{-1}\) y = tan\(^{-1}\) (\(\frac{x + y}{1 - xy}\)), if x > 0, y > 0 and xy < 1.
(xxxii) tan\(^{-1}\) x + tan\(^{-1}\) y = π + tan\(^{-1}\) (\(\frac{x + y}{1 - xy}\)), if x > 0, y > 0 and xy > 1.
(xxxiii) tan\(^{-1}\) x + tan\(^{-1}\) y = tan\(^{-1}\) (\(\frac{x + y}{1 - xy}\)) - π, if x < 0, y > 0 and xy > 1.
(xxxiv) tan\(^{-1}\) x + tan\(^{-1}\) y + tan\(^{-1}\) z = tan\(^{-1}\) \(\frac{x + y + z - xyz}{1 - xy - yz - zx}\)
(xxxv) tan\(^{-1}\) x - tan\(^{-1}\) y = tan\(^{-1}\) (\(\frac{x - y}{1 + xy}\))
(xxxvi) 2 sin\(^{-1}\) x = sin\(^{-1}\) (2x\(\sqrt{1 - x^{2}}\))
(xxxvii) 2 cos\(^{-1}\) x = cos\(^{-1}\) (2x\(^{2}\) - 1)
(xxxviii) 2 tan\(^{-1}\) x = tan\(^{-1}\) (\(\frac{2x}{1 - x^{2}}\)) = sin\(^{-1}\) (\(\frac{2x}{1 + x^{2}}\)) = cos\(^{-1}\) (\(\frac{1 - x^{2}}{1 + x^{2}}\))
(xxxix) 3 sin\(^{-1}\) x = sin\(^{-1}\) (3x - 4x\(^{3}\))
(xxxx) 3 cos\(^{-1}\) x = cos\(^{-1}\) (4x\(^{3}\) - 3x)
(xxxxi) 3 tan\(^{-1}\) x = tan\(^{-1}\) (\(\frac{3x - x^{3}}{1 - 3x^{2}}\))
● Inverse Trigonometric Functions
11 and 12 Grade Math
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