Inverse Trigonometric Function Formula

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\(^{-1}\) (\(\frac{3x - x^{3}}{1 - 3x^{2}}\))




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