Inverse Trigonometric Functions

Calculate arcsin, arccos, arctan, and more with high precision.

Select a function, enter a value, and get the angle in both radians and degrees.

Practical Examples

Explore common calculations to understand how the calculator works.

Arcsine of 0.5

arcsin

Find the angle whose sine is 0.5.

value: 0.5

Arccosine of -1

arccos

Find the angle whose cosine is -1.

value: -1

Arctangent of 1

arctan

Find the angle whose tangent is 1.

value: 1

Arcsecant of 2

arcsec

Find the angle whose secant is 2.

value: 2

Other Titles
Understanding Inverse Trigonometric Functions: A Comprehensive Guide
A deep dive into the world of inverse trigonometric functions, from basic concepts to practical applications.

What Are Inverse Trigonometric Functions?

  • The Core Concept
  • Principal Values
  • The Six Functions
Inverse trigonometric functions, also known as arcus functions or anti-trigonometric functions, are the inverse functions of the basic trigonometric functions (sine, cosine, tangent, cotangent, secant, and cosecant). They are used to find an angle from a trigonometric ratio.
Principal Values
Since trigonometric functions are periodic and not one-to-one, their domains must be restricted to define an inverse function. The output of an inverse trigonometric function is called a 'principal value', which falls within a specific range.

Function Ranges (Principal Values):

  • arcsin(x): [-π/2, π/2] or [-90°, 90°]
  • arccos(x): [0, π] or [0°, 180°]
  • arctan(x): (-π/2, π/2) or (-90°, 90°)

Step-by-Step Guide to Using the Calculator

  • Selecting a Function
  • Entering a Value
  • Interpreting the Results
Our calculator simplifies the process of finding inverse trigonometric values. Follow these steps for accurate results.
How to Use
1. Select the Function: Choose the desired inverse function (e.g., arcsin) from the dropdown menu. 2. Enter the Value: Input the trigonometric ratio into the value field. Ensure it's within the function's domain. 3. Calculate: Click the 'Calculate' button to see the angle in both degrees and radians.

Practical Usage Examples

  • arcsin(0.5) returns 30° or π/6 radians
  • arccos(0) returns 90° or π/2 radians
  • arctan(1) returns 45° or π/4 radians
  • Used in physics to find angles in vector problems

Real-World Applications

  • Physics and Engineering
  • Computer Science
  • Navigation
Inverse trigonometric functions are essential in many fields.
Practical Uses
In physics, they are used to analyze oscillations and waves. In engineering, they help in designing structures and mechanical systems. In computer graphics, they are crucial for rotating objects in 3D space.

Common Misconceptions and Correct Methods

  • sin⁻¹(x) vs 1/sin(x)
  • Domain and Range Errors
  • Calculator Mode
A common point of confusion is the notation. sin⁻¹(x) represents arcsin(x), not 1/sin(x) (which is csc(x)). Also, always be mindful of the domain of each function to avoid errors. For example, the domain of arcsin(x) is [-1, 1].

Mathematical Derivations and Formulas

  • Derivative Formulas
  • Integral Formulas
  • Relationship between Functions
Key Formulas
The derivatives of inverse trigonometric functions are algebraic, which makes them very useful in integration.

Example Derivatives:

  • d/dx arcsin(x) = 1 / √(1 - x²)
  • d/dx arctan(x) = 1 / (1 + x²)