Inverse Trigonometric Functions in Maths

Inverse Trigonometric Functions in Maths is simply defined as the inverse of some basic trigonometric functions such as sine, cosine, tan, sec, cosec and cot. The other names of Inverse trigonometric functions are arcus function, anti-trigonometric function or cyclomatic function. We can understand the concept of inverse trigonometric function in a better way with the help of the triangle given below.

In the above triangle ABC, the basic trigonometric function will be defined as:

• Sin θ = a/b

• Cos θ = c/b

• Tan θ = a/c

In the above basic trigonometric function, the angles are termed as input and the ratio of the sides of a triangle is considered as their outcomes. However, the inverse trigonometric function considers the ratio of the side of a triangle as input and obtains the value of angles.

• Sin-1 (a/b) = θ

• Cos-1 (c/b)  = θ

• Tan-1 (a/c) = θ

It implies inverse trigonometric functions are beneficial whenever the sides of a triangle are known and we want to determine the angles of a triangle.

Note: The representation Sin-1 might create confusion because we usually use a negative exponent to represent reciprocals. Although in this situation, Sin-1θ =  1/Sin θ. We generally use Cosecant or csc to determine the reciprocal of sin. To avoid this confusion, some people choose to represent the inverse function by using arc as a prefix. For example.,  

• Arcsin θ = Sin-1 θ

Domain and Principal Value of Inverse Trigonometric Functions

We know that the principal value of the trigonometric function at a point is the value of the inverse function at a point , which falls in the range of the principal values unit. The principal value of cos √(3/2) is π/6 as π/6 ∈ \[[0,\pi ]\].

Whenever any positive value and the negative values are given in a way that these two values are equal, then the principal value of the inverse trigonometric function will always be the positive value. Let us now discuss the domain and range of all the six inverse trigonometric functions.

Inverse Trigonometric Function Formulas

Some basic inverse trigonometric functions formulas are discussed below:

• Sin-1 (-α) = - Sin-1 (α)

• Cos-1 (-α) = π - Cos-1 (α)

• Sin-1 (α) + Cos-1 (α) = π/2

• Tan-1 (α) + Tan-1 (β) = π + tan-1 (\[\frac{\alpha +\beta }{1-\alpha \beta }\])

• 2 Sin-1 (α) = Sin-1 (2α\[\sqrt{1-\alpha ^2}\])

• 3 Sin-1 (α) = Sin-1 (3α - 4α3)

• Sin-1 (α) + Sin-1 (β) = Sin-1 (α\[\sqrt{1-\beta ^2}\] + β\[\sqrt{1-\alpha ^2}\]), if α, β ≥0 and α² + β² ≤ 1]

•  Cos-1 (α) + Cos-1 (β) = Si (αβ -\[\sqrt{1-\alpha ^2}\]\[\sqrt{1-\beta ^2}\]), if α, β ≥ 0 and α² + β² ≤ 1]

Inverse Trigonometric Function Properties

Let us discuss now some inverse trigonometric function properties

• Sin(arcsin α ) = α and arcsin (sin β)= β, where -1 ≤  α ≤ 1 and -π/2 ≤ β ≤ π/2

• Tan(arctan α) = α and arctan (tan β)= β, where -π/2 ≤ β ≤ π/2

• Sec(arcsec α ) = α and arcsec (sec β)= β, where α ≥ 1 and 0 ≤ β ≤ π/2

• Sin-1  (1/θ) = Csc-1 θ

• Cos-1  (1/θ) = Sec-1 θ

• Tan-1  (1/θ) = Cot-1 θ

• Sin-1 θ+  Cos-1 θ= π/2, θ ∈ \[[-1.1]\]

• Tan-1θ  +  Cot1 θ = π/2, θ ∈ R

• Sec-1 θ+  Cosec1 θ = π/2, |x| ≤ 1

Inverse Circular Functions Graph

The six inverse circular function graphs are discussed below:

1. ArcSin x

Let y = sin-1 x,|x| ≤ 1 , y ∈ \[[(-\frac{\pi }{2},\frac{\pi }{2})]\]

Important Points To be Considered

Sin-1 x is an odd function (symmetric about its origin) and it falls in the range of (-π/2,π/2)

The maximum value of sin-1 x is π/2, and it lies at x = 1

The minimum value of sin-1 x is -π/2, and it lies at x = -1

Sin-1 x is a function that is decreasing in its domains

Sin-1 x is also a periodic function.

2. ArcCos x

Let y = cos-1 x,|x| ≤ 1 , y ∈ \[[0,\pi ]\]

Important Points To be Considered

• Cos-1 x is neither an odd nor an even function and it falls in the range of (0,π)

• The maximum value of cos-1 x is π, and it lies at x = -1

• The minimum value of cos-1 x is 0, and it lies at x = 1

• Cos-1 x is a function that is decreasing in its domains

• Cos-1 x is also a periodic function.

3. ArcTan x

Let y = tan-1 x, x ∈ R, y ∈ \[[(-\frac{\pi }{2},\frac{\pi }{2})]\]

Important Points To be Considered

• Tan-1 x is an odd function (symmetric about its axis) and it falls in the range of (-π/2,π/2)

• The maximum and minimum value of tan-1 x cannot be defined.

• Tan-1 x is a function that is decreasing in its domains

• Tan-1 x is also a periodic function.

4. Arc Cot x

Let y = cot-1 x, x ∈ R, y ∈ \[[0,\pi ]\] 

Important Points To be Considered

• Cot-1 x is neither odd nor even function and it falls in the range of 0,π

• The maximum and minimum value of cot-1 x cannot be defined.

• Cot-1 x is a function that is decreasing in its domains

• Cot-1 x is also a periodic function.

5. ArcSec x

Let y = sec-1 x, |x| ≥ 1, y ∈ (0, π/2) ∪ (π/2,π )

Important Points To be Considered

• Sec-1 x is neither an odd nor an even function.

• Sec-1 x falls in the range of \[[0,π]\]

• The maximum value of sec-1 x is π, and it lies at x = 1

• The minimum value of sec-1 x is -π, and it lies at x = -1

• Sec-1 x is a function that is decreasing in two different intervals

• Sec-1 x is also a periodic function.

6. Arc Cosec x

Let y = cosec-1 x, |x| ≥ 1, y ∈ (–π/2,0) ∪ ( 0,π/2)

Important Points To be Considered

• cosec-1 x is an odd function that falls in the range of –π/2, π/2.

• The maximum value of cosec-1 x is π/2, and it lies at x = 1

• The minimum value of cosec-1 x is -π/2, and it lies at x = -1

• cosec-1 x is a function that is decreasing in two different intervals cosec-1 x is also a periodic function.

Some Significant Properties Of Inverse Trigonometric Functions

The domain and range of the functions decide the properties of the inverse trigonometric functions. These properties are very important for solving the questions and for a clear understanding of the concepts. Following are the inverse trigonometric functions properties:

• Property I

Sin−1 (x) = cosec−1 (1/x), x ∈ [−1, 1] − {0}

Cos−1 (x) = sec−1 (1/x), x ∈ [−1, 1] − {0}

Tan−1 (x) = cot−1 (1/x), if x > 0 and cot−1 (1/x) −π, if x < 0

Cot−1(x) = tan−1 (1/x), if x > 0 and tan−1 (1/x) + π, if x < 0

• Property II

Sin−1 (−x) = −Sin−1(x)

Tan−1 (−x) = −Tan−1(x)

Cos−1 (−x) = π − Cos−1(x)

Cosec−1 (−x) = − Cosec−1(x)

Sec−1 (−x) = π − Sec−1(x)

Cot−1 (−x) = π − Cot−1(x)

• Property III

Sin−1 (1/x) = cosec−1x, x ≥ 1 or x ≤ −1

Cos−1 (1/x) = sec−1x, x ≥ 1 or x ≤ −1

Tan−1 (1/x) = −π + cot−1(x)

• Property IV

Sin−1 (cos θ) = π/2 − θ, if θ ∈ [0,π]

Cos−1 (sin θ) = π/2 − θ, if θ ∈ [−π/2, π/2]

Tan−1 (cot θ) = π/2 − θ, θ ∈ [0,π]

Cot−1 (tan θ) = π/2 − θ, θ ∈ [−π/2, π/2]

Sec−1 (cosec θ) = π/2 − θ, θ ∈ [−π/2, 0]∪[0, π/2]

Cosec−1 (sec θ) = π/2 − θ, θ ∈ [0,π]−{π/2}

• Property V

Sin−1x + Cos−1x = π/2

Tan−1x + Cot−1(x) = π/2

Sec−1x + Cosec−1x = π/2

• Property VI

tan−1(x) – tan−1(y) = tan−1 [(x−y)/ (1+xy)], xy > −1

tan−1(x) + tan−1(y) = tan− 1[(x + y)/ (1-xy)], xy < 1