What are Inverse Trigonometric Functions, and what are their applications?

Inverse Trigonometric Functions are also referred to as:

  • Anti trigonometric functions

  • Cyclometric functions

  • Arcus functions

They are obtained by inverting the essential trigonometric functions. The functions that are reversed are sine, cosine, tangent, cotangent, secant, cosecant. The range and domain of Inverse trigonometric functions are obtained by the conversion of part and content of trigonometric functions, respectively.

Inverse trigonometric functions are widely used in:

  • Engineering

  • Navigation

  • Physics

  • Geometry 

 The inverse trigonometric functions are denoted as 

  • Sin-1x 

  • Cos-1x 

  • cot-1 x

  • tan-1 x

  • cosec-1 x

  • sec-1 x

We will further study them in detail.

Applications of inverse trigonometric functions :

Inverse trigonometric functions are widely used in Engineering, Navigation, Physics and Geometry. Basic concepts where Inverse trigonometric functions are used are given beneath:

  • A right-angled triangle is used for measuring unknown angles.

  • It measures the angle of inclination.

  • Angles of bridges are also measured using it.

  • Carpenters use it for measuring required cut angles.

Notations of inverse trigonometric functions :

  1. y = sin-1(x)

  2. y=cos-1(x)

  3. y=tan-1(x)

  4. y=cot-1(x)

  5. y = sec-1(x)

  6. y=csc-1(x)

Types of Inverse Trigonometric Functions:

  1. Sin-1x or Arcsine- It represents an inverse of the sine function. Sin-1x is used to denote it. Its domain is -1 ≤ x ≤ 1. Its range is -π/2 ≤ y ≤ π/2.
  2. Cos-1x or Arccosine- It represents an inverse of the cosine function. Cos-1x is used to denote it. Its domain is -1≤x≤1. Its range is 0 ≤ y ≤ π.
  3. tan-1 x or Arctangent – It represents an inverse of tangent function. tan-1 x is used to denote it. Its domain is -∞ < x < ∞. Its range is -π/2 < y < π/2.
  4. cot-1 x or Arccotangent – It represents an inverse of cotangent function. cot-1 x is used to denote it. Its domain is -∞ < x < ∞. Its range is 0 < y < π.
  5. sec-1 x or Arcsecant – It represents an inverse of the secant function. sec-1 x is used to denote it. Its domain is -∞ ≤ x ≤ -1 or 1 ≤ x ≤ ∞. Its range is 0 ≤ y ≤ π, y ≠ π/2.
  6. cosec-1 x or Arccosecant – It represents an inverse of cosecant function. cosec-1 x is used to denote it. Its domain is -∞ ≤ x ≤ -1 or 1 ≤ x ≤ ∞. Its range is -π/2 ≤ y ≤ π/2, y ≠ 0.

Derivatives (dy/dx) of inverse trigonometric functions :

  • y = sin-1(x) : 1/√(1-x2)

  • y=cos-1(x) : -1/√(1-x2)

  • y=tan-1(x) : 1/(1+x2)

  • y=cot-1(x): -1/(1+x2)

  • y = sec-1(x): 1/[|x|√(x2-1)]

  • y=csc-1(x) : -1/[|x|√(x2-1)]

Properties of Inverse trigonometric functions: 

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 (or) cot−1(1/x) −π, if x < 0

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

  • 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)

  • 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)

  • 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}

  • Sin−1(x) = cos−1[√(1−x2)], 0≤x≤1 = −cos−1[√(1−x2)], −1≤x<0

  • Sin−1x + Cos−1x = π/2

  • Tan−1x + Cot−1(x) = π/2

  • Sec−1x + Cosec−1x = π/2

CONCLUSION:

In this article, we learned some topics related to inverse trigonometric functions. We looked after the types of ITF and used them. We studied the notations and derivatives of Inverse trigonometric functions. Also, we learnt about the properties of ITF. It was all about Inverse Trigonometric Functions. I hope you will understand all these topics after reading this article.