Identities of Inverse Trigonometric Function
The following are the identities of inverse trigonometric functions:
- sin-1 (sin x) = x provided -π/2 ≤ x ≤ π/2
- cos-1 (cos x) = x provided 0 ≤ x ≤ π
- tan-1 (tan x) = x provided -π/2 < x < π/2
- sin(sin-1 x) = x provided -1 ≤ x ≤ 1
- cos(cos-1 x) = x provided -1 ≤ x ≤ 1
- tan(tan-1 x) = x provided x ∈ R
- cosec(cosec-1 x) = x provided -1 ≤ x ≤ ∞ or -∞ < x ≤ 1
- sec(sec-1 x) = x provided 1 ≤ x ≤ ∞ or -∞ < x ≤ 1
- cot(cot-1 x) = x provided -∞ < x < ∞
- [Tex]sin^{-1}(\frac{2x}{1 + x^2}) = 2 tan^{-1}x[/Tex]
- [Tex]cos^{-1}(\frac{1 – x^2}{1 + x^2}) = 2 tan^{-1}x[/Tex]
- [Tex]tan^{-1}(\frac{2x}{1 – x^2}) = 2 tan^{-1}x[/Tex]
- 2cos-1 x = cos-1 (2x2 – 1)
- 2sin-1x = sin-1 2x√(1 – x2)
- 3sin-1x = sin-1(3x – 4x3)
- 3cos-1 x = cos-1 (4x3 – 3x)
- 3tan-1x = tan-1((3x – x3/1 – 3x2))
- sin-1x + sin-1y = sin-1{ x√(1 – y2) + y√(1 – x2)}
- sin-1x – sin-1y = sin-1{ x√(1 – y2) – y√(1 – x2)}
- cos-1 x + cos-1 y = cos-1 [xy – √{(1 – x2)(1 – y2)}]
- cos-1 x – cos-1 y = cos-1 [xy + √{(1 – x2)(1 – y2)}
- tan-1 x + tan-1 y = tan-1(x + y/1 – xy)
- tan-1 x – tan-1 y = tan-1(x – y/1 + xy)
- tan-1 x + tan-1 y +tan-1 z = tan-1 (x + y + z – xyz)/(1 – xy – yz – zx)
People Also View:
Inverse Trigonometric Identities
Inverse Trigonometric Identities: In mathematics, inverse trigonometric functions are also known as arcus functions or anti-trigonometric functions. The inverse trigonometric functions are the inverse functions of basic trigonometric functions, i.e., sine, cosine, tangent, cosecant, secant, and cotangent. It is used to find the angles with any trigonometric ratio. Inverse trigonometric functions are generally used in fields like geometry, engineering, etc. The representation of inverse trigonometric functions are:
If a = f(b), then the inverse function is
b = f-1(a)
Examples of inverse inverse trigonometric functions are sin-1x, cos-1x, tan-1x, etc.
Table of Content
- Domain and Range of Inverse Trigonometric Identities
- Properties of Inverse Trigonometric Functions
- Identities of Inverse Trigonometric Function
- Sample Problems on Inverse Trigonometric Identities
- Practice Problems on Inverse Trigonometric Identities