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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 (sin1 x) = x and sin1 (sin θ) = θ, provided that - π2 ≤ θ ≤ π2 and - 1 ≤ x ≤ 1.

(ii) cos (cos1 x) = x and cos1 (cos θ) = θ, provided that 0 ≤ θ ≤ π and - 1 ≤ x ≤ 1.

(iii) tan (tan1 x) = x and tan1 (tan θ) = θ, provided that - π2 < θ < π2 and - ∞ < x < ∞.

(iv) csc (csc1 x) = x and sec1 (sec θ) = θ, provided that - π2 ≤ θ < 0 or  0 < θ ≤ π2  and - ∞ < x ≤ 1 or -1 ≤ x < ∞.

(v) sec (sec1 x) = x and sec1 (sec θ) = θ, provided that 0 ≤ θ ≤ π2 or π2 <  θ ≤ π and - ∞ < x ≤ 1 or 1 ≤ x < ∞.

(vi)  cot (cot1 x) = x and cot1 (cot θ) = θ, provided that 0 < θ < π and - ∞ < x < ∞.

(vii) The function sin1 x is defined if – 1 ≤ x ≤ 1; if θ be the principal value of sin1 x then - π2 ≤ θ ≤ π2.

(viii) The function cos1  x is defined if – 1 ≤ x ≤ 1; if θ be the principal value of cos1 x then 0 ≤ θ ≤ π.

(ix) The function tan1 x is defined for any real value of x i.e., - ∞ < x < ∞; if θ be the principal value of tan1 x then - π2 < θ < π2.

(x)  The function cot1 x is defined when - ∞ < x < ∞; if θ be the principal value of cot1 x then - π2 < θ < π2 and θ ≠ 0.

(xi) The function sec1 x is defined when, I x I ≥ 1 ; if θ be the principal value of sec1 x then 0 ≤ θ ≤ π and θ ≠ π2.

(xii) The function csc1 x is defined if I x I ≥ 1; if θ be the principal value of csc1 x then - π2 < θ < π2 and θ ≠ 0.

(xiii) sin1 (-x) = - sin1 x

(xiv) cos1 (-x) = π - cos1 x

(xv) tan1 (-x) = - tan1 x

(xvi) csc1 (-x) = - csc1 x

(xvii) sec1 (-x) = π - sec1 x

(xviii) cot1 (-x) = cot1 x

(xix) In numerical problems principal values of inverse circular functions are generally taken.  

(xx) sin1 x + cos1 x = π2

(xxi) sec1 x + csc1 x = π2.

(xxii) tan1 x + cot1 x = π2

(xxiii) sin1 x + sin1 y = sin1 (x 1y2 + y1x2), if x, y ≥ 0 and x2  + y2 ≤ 1.

(xxiv) sin1 x + sin1 y = π - sin1 (x 1y2 + y1x2), if x, y ≥ 0 and x2  + y2 > 1.

(xxv) sin1 x - sin1 y = sin1 (x 1y2 - y1x2), if x, y ≥ 0 and x2  + y2 ≤ 1.

(xxvi) sin1 x - sin1 y = π - sin1 (x 1y2 - y1x2), if x, y ≥ 0 and x2  + y2 > 1.

(xxvii) cos1 x + cos1 y = cos1(xy - 1x21y2), if x, y > 0 and x2  + y2 ≤  1.

(xxviii) cos1 x + cos1 y = π - cos1(xy - 1x21y2), if x, y > 0 and x2  + y2 >  1.

(xxix) cos1 x - cos1 y = cos1(xy + 1x21y2), if x, y > 0 and x2  + y2 ≤  1.

(xxx) cos1 x - cos1 y = π - cos1(xy + 1x21y2), if x, y > 0 and x2  + y2 >  1.

(xxxi) tan1 x + tan1 y = tan1 (x+y1xy), if x > 0, y > 0 and xy < 1.

 (xxxii) tan1 x + tan1 y = π + tan1 (x+y1xy), if x > 0, y > 0 and xy > 1.

(xxxiii) tan1 x + tan1 y = tan1 (x+y1xy) - π, if x < 0, y > 0 and xy > 1.

(xxxiv) tan1 x + tan1 y + tan1 z = tan1 x+y+zxyz1xyyzzx

(xxxv) tan1 x - tan1 y = tan1 (xy1+xy)

(xxxvi) 2 sin1 x = sin1 (2x1x2)

(xxxvii) 2 cos1 x = cos1 (2x2 - 1)

(xxxviii) 2 tan1 x = tan1 (2x1x2) = sin1 (2x1+x2) = cos1 (1x21+x2)

(xxxix) 3 sin1 x = sin1 (3x - 4x3)

(xxxx) 3 cos1 x = cos1 (4x3 - 3x)

(xxxxi) 3 tan1 x = tan1 (3xx313x2)

 Inverse Trigonometric Functions




11 and 12 Grade Math

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