Inverse Trigonometric Functions 

• sin^-1x, cos^-1x, tan^-1x etc. denote angles or real numbers whose sine is x, cosine is x and tangent is x, provided that the answers given are numerically smallest available. These are also termed as arc sin x, arc cosine x etc.

• If there are two angles one positive and the other negative having same numerical value, then positive angle should be taken.

• Principal values and domains of inverse circular functions:

• Some important points:

• The first quadrant is common to all inverse functions.

• Third quadrant is not used in inverse functions.

• Fourth quadrant is used in clockwise direction i.e. -π/2 ≤ y ≤ 0.

• Graphs of all inverse circular functions:

1. y = sin^-1x, |x| ≤ 1, y ∈ [-π/2, π/2]

Some of the key properties of sin^-1 x are listed below:

1. sin^-1x is bounded in [-π/2, π/2].

2. sin^-1x is an odd function which is symmetric about the origin.

3. In its domain, sin^-1x is an increasing function.

4. sin^-1x attains its maximum value π/2 at x = 1 while its minimum value is -π/2 which occurs at x = -1.

2. y = cos^-1x, |x| ≤ 1, y ∈ [0, π]

Some of the key properties of cos^-1 x are listed below:

1. cos^-1x is bounded in [0, π].

2. cos^-1x is a neither odd nor even function.

3. In its domain, cos^-1x is a decreasing function.

4. cos^-1x attains its maximum value π at x = -1 while its minimum value is 0 which occurs at x = 1.

3. y = tan^-1x, x ∈ R, y ∈ (-π/2, π/2)

Some of the key properties of tan^-1 x are listed below:

1. tan^-1x is bounded in (-π/2, π/2).

2. tan^-1x is an odd function which is symmetric about origin.

3. In its domain, tan^-1x is an increasing function.

4. y = cot^-1x, x ∈ R, y ∈ (0, π)

Some of the key properties of cot^-1 x are listed below:

1. cot^-1x is bounded in (0, π).

2. cot^-1x is a neither odd nor even function.

3. In its domain, cot^-1x is a decreasing function.

5. y = cosec^-1x, |x| ≥ 1, y ∈ [-π/2, 0) ∪ (0, π/2].

Some of the key properties of cosec^-1 x are listed below:

1. cosec^-1x is bounded in [-π/2, π/2].

2. cosec^-1x is an odd function which is symmetric about the origin.

3. In its domain, cosec^-1x is a decreasing function.

4. cosec^-1x attains its maximum value π/2 at x = 1 while its minimum value is -π/2 which occurs at x = -1.

6. y = sec^-1x, |x| ≥ 1, y ∈ [0, π/2) ∪ (π/2, π].

Some of the key properties of sec^-1 x are listed below:

1. sec^-1x is bounded in [0, π].

2. sec^-1x is neither odd nor even function.

3. In its domain, sec^-1x is an increasing function.

4. sec^-1x attains its maximum value π at x = -1 while its minimum value is 0 which occurs at x = 1.

• tan^-1x and cot^-1x are continuous and monotonic on R need not imply that their range is R.

• If f(x) is continuous and has a range R then it need not imply that it is monotonic.

• Properties of inverse trigonometric functions:

sin (sin^-1x) = x, -1 ≤ x ≤ 1

cos (cos^-1x) = x, -1 ≤ x ≤ 1

tan (tan^-1x) = x, x ∈ R

cot (cot^-1x) = x, x ∈ R

sec (sec^-1x) = x, |x| ≥ 1

cosec (cosec^-1x) = x, |x| ≥ 1

y = sin (sin^-1x) = x, x ∈ [-1, 1], y ∈ [-1, 1], y is aperiodic

y = cos (cos^-1x) = x, x ∈ [-1, 1], y ∈ [-1, 1], y is aperiodic

y = tan (tan^-1x) = x, x ∈ R, y ∈ R, y is aperiodic

y = cot (cot^-1x) = x, x ∈ R, y ∈ R, y is aperiodic

y = cosec (cosec^-1x) = x, |x| ≥ 1, |y| ≥ 1, y is aperiodic

y = sec (sec^-1x) = x, |x| ≥ 1, |y| ≥ 1, y is aperiodic

sin^-1 (sin x) = x, -π/2 ≤ x ≤ π/2

cos^-1 (cos x) = x, 0 ≤ x ≤ π

tan^-1 (tan x) = x, -π/2 < x < π/2

cot^-1 (cot x) = x, 0 < x < π

sec^-1 (sec x) = x, 0 ≤ x ≤ π, x ≠ π/2

cosec^-1 (cosec x) = x, -π/2 ≤ x ≤ π/2, x ≠ 0

y = sin^-1 (sin x) = x, x ∈ R, y ∈ [-π/2, π/2], periodic with period 2π

y = cos^-1(cos x) = x, x ∈ R, y ∈ [0, π], periodic with period 2π

y = tan^-1(tan x) = x, x ∈ R – {(2n-1)π/2, n ∈ I}, y ∈ (-π/2, π/2).

y = cot^-1(cot x) = x, x ∈ R – {nπ}, y ∈ (0, π), periodic with π

y = cot^-1(cot x) = x, x ∈ R – {nπ}, y ∈ (0, π), periodic with π

y = cosec^-1(cosec x) = x, x ∈ R – {nπ, n ∈ I}, y ∈ [-π/2, 0) ∪ (0, π/2], y is periodic with period 2π

y = sec^-1(sec x) = x, x ∈ R – {(2n-1)π/2, n ∈ I}, y ∈ [0, π/2) ∪ (π/2, π], y is periodic with period 2π

• cosec^-1x = sin^-11/x , |x| ≥ 1

• sin^-1x = cosec^-11/x, |x| ≤ 1, x ≠ 0

• sec^-1x = cos^-11/x , |x| ≥ 1

• cos^-1x = sec^-11/x, |x| ≤ 1, x ≠ 0

• cot^-1x = tan^-11/x, x > 0

                    = π + tan^-11/x, x < 0

Points to Remember:

• cosec^-1x and sin^-11/x are identical functions.

• sin^-1x and cosec^-11/x are not identical because domains of sin^-1x and cosec^-11/x are not equal.

• sec^-1x and cos^-11/x are identical functions.

• cos^-1x and sec^-11/x are not identical because their domains are not equal.

• sin^-1(-x) = - sin^-1(x), -1 ≤ x ≤ 1

  cos^-1(-x) = π – cos^-1(x), -1 ≤ x ≤ 1

  tan^-1(-x) = - tan^-1(x), x ∈ R

  cot^-1(-x) = π – cot^-1(x), x ∈ R

  sec^-1(-x) = π – sec^-1(x) , |x| ≥ 1

  cosec^-1(-x) = – cosec^-1(x), |x| ≥ 1

• sin^-1 x + cos^-1x = π/2 , -1 ≤ x ≤ 1

  tan^-1x + cot^-1x = π/2, x ∈ R

  cosec^-1x + sec^-1x = π/2, |x| ≥ 1

• tan^-1x + tan^-1y = tan^-1 [(x + y)/(1 – xy)], xy < 1

                                = π + tan^-1 [(x + y)/(1 – xy)], if x > 0, y > 0 and xy > 1

                                = -π + tan^-1 [(x + y)/(1 – xy)], if x < 0, y < 0 and xy > 1

• x > 0, y > 0, tan^-1x – tan^-1y = tan^-1[(x – y)/(1 + xy)]

• sin^-1x + sin^-1 y = sin^-1(x√1 - y² + y√1 - x²) if x ≥ 0, y ≥ 0 and x² + y² ≤ 1.

                                = π – sin^-1(x√1 - y² + y√1 - x²) if x ≥ 0, y ≥ 0 and x² + y² > 1.

• sin^-1x - sin^-1 y = sin^-1(x√1 - y² - y√1 - x²) if x > 0, y > 0

• cos^-1x ± cos^-1 y = cos^-1(xy ± √1 - x² √1 - y²) if x > 0, y > 0 and x < y.

• tan^-1x + tan^-1y + tan^-1z = tan^-1 [(x + y + z - xyz)/ (1 – xy – yz – zx)], where x > 0, y > 0, z > 0 and xy + yz + zx < 1 and xy < 1, yz < 1, zx < 1.

• sin^-1(2x/1 + x²) = 2 tan^-1x, -1 ≤ x ≤ 1

                                 = π – 2 tan^-1x, if x ≥ 1

                                 = - π – 2 tan^-1x, if x ≤ -1

• cos^-1[(1 – x²)/(1 + x²)] = 2 tan^-1x, x ≥ 0

                                           = - 2 tan^-1x, x < 0

• tan^-1(2x/1 - x²) = π + 2 tan^-1x, x < -1

                                = 2 tan^-1x, -1 < x < 1

                                = 2 tan^-1x - π, if x > 1