**Inverse Trigonometric Formulas:**Â Trigonometry is a part of geometry, where we learn about the relationships between angles and sides of a right-angled triangle. In Class 11 and 12 Maths syllabus, you will come across a list ofÂ trigonometry formulas, based on the functions and ratios such as, sin, cos and tan. Similarly, we have learned about inverse trigonometry concepts also. The inverse trigonometric functions are written as sin^{-1}x, cos^{-1}x, cot^{-1}Â x, tan^{-1}Â x, cosec^{-1}Â x, sec^{-1}Â x. Now, let us get the formulas related to these functions.

## What is Inverse Trigonometric Function?

The inverse trigonometric functions are also known as the anti trigonometric functions or sometimes called arcus functions or cyclometric functions. TheÂ inverse trigonometric functionsÂ of sine, cosine, tangent, cosecant, secant and cotangent are used to find the angle of a triangle from any of the trigonometric functions. It is widely used in many fields like geometry, engineering, physics, etc. But in most of the time, the convention symbol to represent the inverse trigonometric function using arc-prefix like arcsin(x), arccos(x), arctan(x), arccsc(x), arcsec(x), arccot(x). To determine the sides of a triangle when the remaining side lengths are known.

Consider, the function y = f(x), and x = g(y) then the inverse function is written as g = f^{-1},

This means that if y=f(x), then x = f^{-1}(y).

Such that f(g(y))=y and g(f(y))=x.

Example of Inverse trigonometric functions: x= sin^{-1}y

The list of inverse trigonometric functions with domain and range value is given below:

Functions |
Domain |
Range |

Sin^{-1}Â x |
[-1, 1] | [-Ï€/2, Ï€/2] |

Cos^{-1}x |
[-1, 1] | [0, Ï€/2] |

Tan^{-1}Â x |
R | (-Ï€/2, Ï€/2) |

Cosec^{-1}Â x |
R-(-1,1) | [-Ï€/2, Ï€/2] |

Sec^{-1}Â x |
R-(-1,1) | [0,Ï€]-{ Ï€/2} |

Cot^{-1}Â x |
R | [-Ï€/2, Ï€/2]-{0} |

## Inverse Trigonometric Formulas List

To solve the different types of inverseÂ trigonometric functions, inverse trigonometry formulas are derived from some basic properties of trigonometry. The formula list is given below for reference to solve the problems.

S.No |
Inverse Trigonometric Formulas |

1 | sin^{-1}(-x) = -sin^{-1}(x), x âˆˆ [-1, 1] |

2 | cos^{-1}(-x) = Ï€ -cos^{-1}(x), x âˆˆ [-1, 1] |

3 | tan^{-1}(-x) = -tan^{-1}(x), x âˆˆ R |

4 | cosec^{-1}(-x) = -cosec^{-1}(x), |x| â‰¥ 1 |

5 | sec^{-1}(-x) = Ï€ -sec^{-1}(x), |x| â‰¥ 1 |

6 | cot^{-1}(-x) = Ï€ â€“ cot^{-1}(x), x âˆˆ R |

7 | sin^{-1}x + cos^{-1}x = Ï€/2 , x âˆˆ [-1, 1] |

8 | tan^{-1}x + cot^{-1}x = Ï€/2 , x âˆˆ R |

9 | sec^{-1}x + cosec^{-1}x = Ï€/2 ,|x| â‰¥ 1 |

10 | sin^{-1}(1/x) = cosec^{-1}(x), if x â‰¥ 1 or x â‰¤ -1 |

11 | cos^{-1}(1/x) = sec^{-1}(x), if x â‰¥ 1 or x â‰¤ -1 |

12 | tan^{-1}(1/x) = cot^{-1}(x), x > 0 |

13 | tan^{-1}Â x + tan^{-1}Â y = tan^{-1}((x+y)/(1-xy)), if the value xy < 1 |

14 | tan^{-1}Â x â€“ tan^{-1}Â y = tan^{-1}((x-y)/(1+xy)), if the value xy > -1 |

15 | 2 tan^{-1}Â x = sin^{-1}(2x/(1+x^{2})), |x| â‰¤ 1 |

16 | 2tan^{-1}Â x = cos^{-1}((1-x^{2})/(1+x^{2})), x â‰¥ 0 |

17 | 2tan^{-1}Â x = tan^{-1}(2x/(1-x^{2})), -1<x<1 |

18 | 3sin^{-1}x = sin^{-1}(3x-4x^{3}) |

19 | 3cos^{-1}x = cos^{-1}(4x^{3}-3x) |

20 | 3tan^{-1}x = tan^{-1}((3x-x^{3})/(1-3x^{2})) |

21 | sin(sin^{-1}(x)) = x, -1â‰¤ x â‰¤1 |

22 | cos(cos^{-1}(x)) = x, -1â‰¤ x â‰¤1 |

23 | tan(tan^{-1}(x)) = x, â€“ âˆž < x < âˆž. |

24 | cosec(cosec^{-1}(x)) = x, â€“ âˆž < x â‰¤ 1 or -1 â‰¤ x < âˆž |

25 | sec(sec^{-1}(x)) = x,- âˆž < x â‰¤ 1 or 1 â‰¤ x < âˆž |

26 | cot(cot^{-1}(x)) = x, â€“ âˆž < x < âˆž. |

27 | sin^{-1}(sin Î¸) = Î¸, -Ï€/2 â‰¤ Î¸ â‰¤Ï€/2 |

28 | cos^{-1}(cos Î¸) = Î¸, 0 â‰¤ Î¸ â‰¤ Ï€ |

29 | tan^{-1}(tan Î¸) = Î¸, -Ï€/2 < Î¸ < Ï€/2 |

30 | cosec^{-1}(cosec Î¸) = Î¸, â€“ Ï€/2 â‰¤ Î¸ < 0 or 0 < Î¸ â‰¤ Ï€/2 |

31 | sec^{-1}(sec Î¸) = Î¸, 0 â‰¤ Î¸ â‰¤ Ï€/2 or Ï€/2< Î¸ â‰¤ Ï€ |

32 | cot^{-1}(cot Î¸) = Î¸, 0 < Î¸ < Ï€ |

33 | \(\sin ^{-1}x +\sin ^{-1}y=\sin ^{-1}(x\sqrt{1-y^{2}}+y\sqrt{1-x^{2}}), if x, y \geq 0 and x^{2}+y^{2} \leq 1\) |

34 | \(\sin ^{-1}x +\sin ^{-1}y=\pi -\sin ^{-1}(x\sqrt{1-y^{2}}+y\sqrt{1-x^{2}})\), if x, y â‰¥ 0 and x^{2}+y^{2}>1. |

35 | \(\sin ^{-1}x -\sin ^{-1}y=\pi -\sin ^{-1}(x\sqrt{1-y^{2}}-y\sqrt{1-x^{2}})\), if x, y â‰¥ 0 and x^{2}+y^{2}â‰¤1. |

36 | \(\sin ^{-1}x -\sin ^{-1}y=\pi -\sin ^{-1}(x\sqrt{1-y^{2}}-y\sqrt{1-x^{2}})\), if x, y â‰¥ 0 and x^{2}Â +y^{2}>1. |

37 | \(\cos ^{-1}x +\cos ^{-1}y=\cos ^{-1}(xy-\sqrt{1-x^{2}}\sqrt{1-y^{2}})\), if x, y >0 and x ^{2}+y^{2}Â â‰¤1. |

38 | \(\cos ^{-1}x +\cos ^{-1}y=\pi -\cos ^{-1}(xy-\sqrt{1-x^{2}}\sqrt{1-y^{2}})\), if x, y >0 and x^{2}+y^{2}>1. |

39 | \(\cos ^{-1}x -\cos ^{-1}y=\cos ^{-1}(xy+\sqrt{1-x^{2}}\sqrt{1-y^{2}})\), if x, y > 0 and x^{2}+y^{2}â‰¤1. |

40 | \(\cos ^{-1}x -\cos ^{-1}y=\pi -\cos ^{-1}(xy+\sqrt{1-x^{2}}\sqrt{1-y^{2}})\),if x, y > 0 and x^{2}Â +y^{2}>1. |

The inverse trigonometric formula list helps the students to solve the problems in an easy way by applying those properties to find out the solutions.

Related Links | |

Trigonometric Identities | Inverse Trigonometric Functions Properties |

Derivative Inverse Trigonometric Functions | Graphic Representation Inverse Trigonometric Function |