Inverse Circular Functions
Trignometric equations of Class 11
Principal Values For Inverse Circular Functions
|
x < 0 |
x ≥ 0 |
|
-π/2 ≤ sin-1x <0 |
0 ≤ sin-1x ≤ π/2 |
|
π/2 < cos-1x ≤ π |
0 ≤ cos-1x ≤ π/2 |
|
-π/2 < tan-1x < 0 |
0 ≤ tan-1x < π/2 |
|
π/2 < cot-1x < π |
0 < cot-1x ≤ π/2 |
|
π/2 < sec-1x ≤ π |
0 ≤ sec-1x < π/2 |
|
−π/2 ≤ cosec-1x < 0 |
0 < cosec-1x ≤ π/2 |
Example
sin-1(√3/2) = π/3, not 2π/3
tan-1 (-√3) = −π/3, not 2π/3
(ii) Some Results on Inverse Trigonometric Functions
(a) sin-1(−x) = −sin-1x −1 ≤ x ≤ 1
(b) cos-1(−x) = π − cos-1x −1 ≤ x ≤ 1
(c) tan-1(−x) = −tan-1x x ∈ R
(d) cot-1(−x) = π− cot-1x x ∈ R
(e) sec-1(−x) = π − sec-1x x ≤ −1 or x ≥ 1
(f) cosec-1(−x) = −cosec-1x x ≤ −1 or x ≥ 1
(g) sin-1x + cos-1x = π/2, −1 ≤ x ≤ 1
(h) tan-1x + cot-1x = π/2, x ∈ R
(i) sec-1x + cosec-1x = π/2, x ≤ −1 or x ≥ 1
(j) sec-1x = cos-1(1/x), x ≤ −1 or x ≥ 1
(k) cosec-1x = sin-1(1/x), x ≤ −1 or x ≥ 1
(l) cot-1x = tan-11/x, x > 0 = π + tan-1 1/x, x < 0
(m) If x > 0, y > 0, xy < 1, then tan-1x + tan-1y = tan-1
(n) If x > 0, y > 0, xy > 1, then tan-1x + tan-1y = π + tan-1
(o) x < 0, y < 0, xy = 1 then tan-1x + tan-1y = π/2
(p) If x ≥ 0, y ≥ 0, x2 + y2 ≤ 1 then sin-1x + sin-1y = sin-1
(q) If x > 0, y > 0, x2 + y2 > 1 then sin-1x + sin-1y = π − sin-1 
(r) If 0 ≤ x, y ≤ 1 then sin-1x −sin-1y = sin-1
(s) If |x| ≤ 1, then 2tan-1x = sin-1 
= cos-1 
= tan-1 
(t) If |x| > 1 then π − 2 tan-1x = sin-1 
= cos-1 
= tan-1
