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```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:

S. No.

Function

Domain

Range

1.

y = sin-1x

-1 ≤ x ≤ 1

-π/2 ≤ y ≤ π/2

2.

y = cos-1x

-1 ≤ x ≤ 1

0 ≤ y ≤ π

3.

y = tan-1x

x ∈ R

-π/2 < x < π/2

4.

y = cot-1x

x ∈ R

0 < y < π

5.

y = cosec-1x

x ≤ -1 or x ≥ 1

-π/2 ≤ y ≤ π/2, y ≠ 0

6.

y = sec-1x

x ≤ -1 or x ≥ 1

0 ≤ y ≤ π, y ≠ π/2

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:

sin-1x is bounded in [-π/2, π/2].

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

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

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:

cos-1x is bounded in [0, π].

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

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

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:

tan-1x is bounded in (-π/2, π/2).

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

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:

cot-1x is bounded in (0, π).

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

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:

cosec-1x is bounded in [-π/2, π/2].

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

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

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:

sec-1x is bounded in [0, π].

sec-1x is neither odd nor even function.

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

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

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