Inverse Sine Calculator

Category: Algebra II
- July 18, 2025
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Calculate the angle whose sine equals a given value. The inverse sine (arcsin) function returns the angle in degrees or radians for sine values from -1 to 1.

Input Value

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About the Inverse Sine Function

The inverse sine function, also known as arcsin or sin-1, returns the angle whose sine is a given value. It is the inverse of the sine function.

Definition

If sin(θ) = x, then arcsin(x) = θ. In other words, arcsin(x) is the angle whose sine equals x.

arcsin(x) = θ such that sin(θ) = x
Properties of Inverse Sine
  • Domain: [-1, 1]
  • Range: [-π/2, π/2] radians or [-90°, 90°]
  • Odd function: arcsin(-x) = -arcsin(x)
  • Special values: arcsin(0) = 0, arcsin(1) = 90°, arcsin(-1) = -90°
Multiple Solutions

Note that for any given sine value, there are infinite angles with that sine. The arcsin function returns the angle in the principal range [-90°, 90°].

For the second solution in the [0°, 180°] range, use: π - arcsin(x) or 180° - arcsin(x).

Applications

The inverse sine function is used in:

  • Calculating angles in triangles when sides are known
  • Physics - determining angles in projectile motion and optics
  • Engineering - determining component vectors
  • Navigation and GPS systems
  • Computer graphics - calculating viewing angles

What is the Inverse Sine Calculator?

The Inverse Sine Calculator helps you find the angle when you know the sine value. It’s a simple tool that can give you results in both degrees and radians. The sine function, which is fundamental in trigonometry, has a range from -1 to 1. This means that the calculator works for any sine value within this range.

How to Use the Inverse Sine Calculator

Using the Inverse Sine Calculator is straightforward. You start by entering a sine value between -1 and 1. Next, choose whether you want the output in degrees or radians. You can also customise the number of decimal places you want in your result. Plus, you have the option to see the calculation steps if you’re interested in understanding how the answer was reached.

Results You Can Expect

When you run the calculation, you’ll receive several key results. These include:

  • The angle value in degrees.
  • The angle value in radians.
  • Any additional solutions for the angle, particularly relevant in a range of [0°, 180°].

The results are presented clearly, allowing you to see the main angle as well as any secondary solutions.

Why It Matters: Understanding the Inverse Sine Function

The inverse sine function, also known as arcsin, is crucial in trigonometry. This function finds the angle whose sine equals a specific number. For example, if you know that the sine of an angle is 0.5, the inverse sine function will tell you that the angle is 30 degrees. This understanding is essential for solving many mathematical problems, particularly in Geometry and Physics.

Key Properties of the Inverse Sine Function

Knowing the properties of the inverse sine function can help you understand its behaviour better. Here are some important points:

  • Domain: The input values must be between -1 and 1.
  • Range: The output angles will always fall between -90° and 90° or between -π/2 and π/2 radians.
  • Odd function: A property of the arcsin function is that arcsin(-x) equals -arcsin(x).

These properties define how the function behaves and help you predict its results.

Applications of the Inverse Sine Calculator

The Inverse Sine Calculator has many practical applications. It is commonly used in various fields such as:

  • Calculating angles in triangles.
  • Physics, especially in projectile motion.
  • Engineering for component vectors.
  • Navigation and GPS systems.
  • Computer graphics for determining viewing angles.

Each of these applications shows how the calculator can be a useful tool in both academic and professional settings.

Visualising the Results

To help you understand the results better, the calculator includes a visual representation of the arcsin function. This graphical display helps illustrate how the angles relate to their sine values. Visual tools like this can make learning and understanding trigonometric concepts more accessible and engaging.

Getting Started with the Inverse Sine Calculator Today

Now that you know how the Inverse Sine Calculator works and its many benefits, you can easily apply it to your own projects or studies. Whether you need to find angles for Mathematics homework or work-related tasks, this calculator is a reliable tool that simplifies the process. Just enter your value, and you’ll get the answers you need quickly and clearly!