NCRT CLASS 12 INVERS FORMULA

INVERS FUNCTION FOR DOMAIN AND RANGE

Functions	Domain	Range
Sin-1 x	[-1, 1]	[-π/2, π/2]
Cos-1x	[-1, 1]	[0, π/2]
Tan-1 x	R	(-π/2, π/2)
Cosec-1 x	R-(-1,1)	[-π/2, π/2]
Sec-1 x	R-(-1,1)	[0,π]-{ π/2}
Cot-1 x	R	[-π/2, π/2]-{0}

# INVERS FORMUL:

S.No	Inverse Trigonometric Formulas
1	sin-1(-x) = -sin-1(x), x ∈ [-1, 1]
2	cos-1(-x) = π -cos-1(x), x ∈ [-1, 1]
3	tan-1(-x) = -tan-1(x), x ∈ R
4	cosec-1(-x) = -cosec-1(x), |x| ≥ 1
5	sec-1(-x) = π -sec-1(x), |x| ≥ 1
6	cot-1(-x) = π – cot-1(x), x ∈ R
7	sin-1x + cos-1x = π/2 , x ∈ [-1, 1]
8	tan-1x + cot-1x = π/2 , x ∈ R
9	sec-1x + cosec-1x = π/2 ,|x| ≥ 1
10	sin-1(1/x) = cosec-1(x), if x ≥ 1 or x ≤ -1
11	cos-1(1/x) = sec-1(x), if x ≥ 1 or x ≤ -1
12	tan-1(1/x) = cot1(x), x > 0
13	tan-1 x + tan-1 y = tan-1((x+y)/(1-xy)), if the value xy < 1
14	tan-1 x – tan-1 y = tan-1((x-y)/(1+xy)), if the value xy > -1
15	2 tan-1 x = sin-1(2x/(1+x2)), |x| ≤ 1
16	2tan-1 x = cos-1((1-x2)/(1+x2)), x ≥ 0
17	2tan-1 x = tan-1(2x/(1-x2)), -1<x<1
18	3sin-1x = sin-1(3x-4x3)
19	3cos-1x = cos-1(4x3-3x)
20	3tan-1x = tan-1((3x-x3)/(1-3x2))
21	sin(sin-1(x)) = x, -1≤ x ≤1
22	cos(cos-1(x)) = x, -1≤ x ≤1
23	tan(tan-1(x)) = x, – ∞ < x < ∞.
24	cosec(cosec-1(x)) = x, – ∞ < x ≤ 1 or -1 ≤ x < ∞
25	sec(sec-1(x)) = x,- ∞ < x ≤ 1 or 1 ≤ x < ∞
26	cot(cot-1(x)) = x, – ∞ < x < ∞.
27	sin-1(sin θ) = θ, -π/2 ≤ θ ≤π/2
28	cos-1(cos θ) = θ, 0 ≤ θ ≤ π
29	tan-1(tan θ) = θ, -π/2 < θ < π/2
30	cosec-1(cosec θ) = θ, – π/2 ≤ θ < 0 or 0 < θ ≤ π/2
31	sec-1(sec θ) = θ, 0 ≤ θ ≤ π/2 or π/2< θ ≤ π
32	cot-1(cot θ) = θ, 0 < θ < π
33	sin−1x+sin−1y=sin−1(x1−y2−−−−−√+y1−x2−−−−−√),ifx,y≥0andx2+y2≤1
34	sin−1x+sin−1y=π−sin−1(x1−y2−−−−−√+y1−x2−−−−−√), if x, y ≥ 0 and x2+y2>1.
35	sin−1x+sin−1y=π−sin−1(x1−y2−−−−−√−y1−x2−−−−−√), if x, y ≥ 0 and x2+y2≤1.
36	sin−1x+sin−1y=π−sin−1(x1−y2−−−−−√−y1−x2−−−−−√), if x, y ≥ 0 and x2 +y2>1.
37	cos−1x+cos−1y=cos−1(xy−1−x2−−−−−√1−y2−−−−−√), if x, y >0 and x2+y2 ≤1.
38	cos−1x+cos−1y=π−cos−1(xy−1−x2−−−−−√1−y2−−−−−√), if x, y >0 and x2+y2>1.
39	cos−1x+cos−1y=cos−1(xy+1−x2−−−−−√1−y2−−−−−√), if x, y > 0 and x2+y2≤1.
40	cos−1x+cos−1y=π−cos−1(xy+1−x2−−−−−√1−y2−−−−−√),if