Question

Inverse trigonometric functions formulas list 2

Answer 1

sin (sin−1−1 x) = x and sin−1−1 (sin θ) = θ, provided that - π2π2 ≤ θ ≤ π2π2 and - 1 ≤ x ≤ 1.

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

tan (tan−1−1 x) = x and tan−1−1 (tan θ) = θ, provided that - π2π2 < θ < π2π2 and - ∞ < x < ∞.

csc (csc−1−1 x) = x and sec−1−1 (sec θ) = θ, provided that - π2π2 ≤ θ < 0 or 0 < θ ≤ π2π2 and - ∞ < x ≤ 1 or -1 ≤ x < ∞.

sec (sec−1−1 x) = x and sec−1−1 (sec θ) = θ, provided that 0 ≤ θ ≤ π2π2 or π2π2 < θ ≤ π and - ∞ < x ≤ 1 or 1 ≤ x < ∞.

cot (cot−1−1 x) = x and cot−1−1 (cot θ) = θ, provided that 0 < θ < π and - ∞ < x < ∞.

The function sin−1−1 x is defined if – 1 ≤ x ≤ 1; if θ be the principal value of sin−1−1 x then - π2π2 ≤ θ ≤ π2π2.

The function cos−1−1 x is defined if – 1 ≤ x ≤ 1; if θ be the principal value of cos−1−1 x then 0 ≤ θ ≤ π.

The function tan−1−1 x is defined for any real value of x i.e., - ∞ < x < ∞; if θ be the principal value of tan−1−1 x then - π2π2 < θ < π2π2.

The function cot−1−1 x is defined when - ∞ < x < ∞; if θ be the principal value of cot−1−1 x then - π2π2 < θ < π2π2 and θ ≠ 0.

The function sec−1−1 x is defined when, I x I ≥ 1 ; if θ be the principal value of sec−1−1 x then 0 ≤ θ ≤ π and θ ≠ π2π2.

The function csc−1−1 x is defined if I x I ≥ 1; if θ be the principal value of csc−1−1 x then - π2π2 < θ < π2π2 and θ ≠ 0.

sin−1−1 (-x) = - sin−1−1 x

cos−1−1 (-x) = π - cos−1−1 x

tan−1−1 (-x) = - tan−1−1 x

csc−1−1 (-x) = - csc−1−1 x

sec−1−1 (-x) = π - sec−1−1 x

cot−1−1 (-x) = cot−1−1 x

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

sin−1−1 x + cos−1−1 x = π2π2

sec−1−1 x + csc−1−1 x = π2π2.

tan−1−1 x + cot−1−1 x = π2π2

sin−1−1 x + sin−1−1 y = sin−1−1 (x 1−y2−−−−−√1−y2 + y1−x2−−−−−√1−x2), if x, y ≥ 0 and x22 + y22 ≤ 1.

sin−1−1 x + sin−1−1 y = π - sin−1−1 (x 1−y2−−−−−√1−y2 + y1−x2−−−−−√1−x2), if x, y ≥ 0 and x22 + y22 > 1.

sin−1−1 x - sin−1−1 y = sin−1−1 (x 1−y2−−−−−√1−y2 - y1−x2−−−−−√1−x2), if x, y ≥ 0 and x22 + y22 ≤ 1.

sin−1−1 x - sin−1−1 y = π - sin−1−1 (x 1−y2−−−−−√1−y2 - y1−x2−−−−−√1−x2), if x, y ≥ 0 and x22 + y22 > 1.

cos−1−1 x + cos−1−1 y = cos−1−1(xy - 1−x2−−−−−√1−x21−y2−−−−−√1−y2), if x, y > 0 and x22 + y22 ≤ 1.

cos−1−1 x + cos−1−1 y = π - cos−1−1(xy - 1−x2−−−−−√1−x21−y2−−−−−√1−y2), if x, y > 0 and x22 + y22 > 1.

cos−1−1 x - cos−1−1 y = cos−1−1(xy + 1−x2−−−−−√1−x21−y2−−−−−√1−y2), if x, y > 0 and x22 + y22 ≤ 1.

cos−1−1 x - cos−1−1 y = π - cos−1−1(xy + 1−x2−−−−−√1−x21−y2−−−−−√1−y2), if x, y > 0 and x22 + y22 > 1.

tan−1−1 x + tan−1−1 y = tan−1−1 (x+y1−xyx+y1−xy), if x > 0, y > 0 and xy < 1.

tan−1−1 x + tan−1−1 y = π + tan−1−1 (x+y1−xyx+y1−xy), if x > 0, y > 0 and xy > 1.

tan−1−1 x + tan−1−1 y = tan−1−1 (x+y1−xyx+y1−xy) - π, if x < 0, y > 0 and xy > 1.

tan−1−1 x + tan−1−1 y + tan−1−1 z = tan−1−1 x+y+z−xyz1−xy−yz−zxx+y+z−xyz1−xy−yz−zx

tan−1−1 x - tan−1−1 y = tan−1−1 (x−y1+xyx−y1+xy)

2 sin−1−1 x = sin−1−1 (2x1−x2−−−−−√1−x2)

2 cos−1−1 x = cos−1−1 (2x22 - 1)

2 tan−1−1 x = tan−1−1 (2x1−x22x1−x2) = sin−1−1 (2x1+x22x1+x2) = cos−1−1 (1−x21+x21−x21+x2)

3 sin−1−1 x = sin−1−1 (3x - 4x33)

3 cos−1−1 x = cos−1−1 (4x33 - 3x)

3 tan−1−1 x = tan−1−1 (3x−x31−3x23x−x31−3x2)
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