Inverse Sine Calculator

Instantly calculate the value of the inverse sine stepwise with the arcsin calculator. The tool removes the need for complex manual calculations and gives accurate results in seconds. With clear stepwise calculation, the inverse sine calculator helps users easily follow each step and understand the final result.

How to Use the Arcsin Calculator?

Step 1: Enter the sine value and ensure the value lies between -1 and 1.

Step 2: Now, select the angle unit (radians or degrees).

Step 3: Click the “Calculate” button to process the input.

What is the Inverse of Sin Function?

The inverse sine function is used to find the angle whose sine value is given. The inverse of sin is denoted as sin⁡−1(x) or asin(x), where x is the sine value. The input value x must lie within the range of the sine function, which is -1 to 1.

Inverse Sine Formula (+ Resolved Example)

The inverse sin formula is

\(\theta = \sin^{-1}(x)\)

Where x is the sine value, and θ is the angle (in radians or degrees) whose sine equals x.

Resolved Example: 

Given Value: \sin \theta = 0.5

Step 1: Write the inverse sine formula

\(\theta = \sin^{-1}(x)\)

Step 2: Place the given value.

\(\theta = \sin^{-1}(0.5)\)

Step 3: Find the angle whose sine is 0.5

\(\sin(30^\circ) = 0.5\)

Step 4: Convert to radians

\(30^\circ = \frac{\pi}{6}\)

Final Answer

\(\theta = 30^\circ \quad \text{or} \quad \frac{\pi}{6}\)

Inverse Sine Graph

When you graph the arcsin (sin⁻¹) function, it forms a smooth, increasing curve. 

The domain of the inverse sine curve begins at x = -1 and ends at x = 1 because the sine function's value varies between -1 and 1.

Moreover, the y value ends at the sine wave's peak (maximum) at π/2 radians and its dip (minimum) at -π/2 radians.

Inverse Sin Table

The table below shows the arcsin, or angle, for each of the common sine values.

Domain and Range of Sin Inverse in Simple Terms

The domain is the set of possible input values (x-values), while the range is the set of possible output values (y-values). The domain of the inverse sine function (y = arcsin (x) or sin⁡−1 (x)) is all real numbers from -1 to 1. Meanwhile, its range is limited to [-π/2, π/2] (in radians) or [-90°, 90°] (in degrees). This limitation makes sure that the function is one-to-one, enabling an inverse that is unique.

Domain:

•  -1 ≤ x ≤ 1 or [-1,1]

Range:

• -π/2 ≤ y ≤ π/2 or [-π/2, π/2]

Applications of the Sin Inverse Calculator

A sin inverse calculator makes it easy to determine an angle from a known sine value. It is helpful when it is difficult or time-consuming to figure it out manually. Some of these uses include:

• Construction and Architecture: When given the ratio of a side to the hypotenuse of a right triangle, a sin inverse calculator can help instantaneously. It figures out the required angle for structural layouts, roof pitches, ramps, or staircases in seconds.
• Engineering: The sin inverse calculator allows engineers to figure out the angles for mechanical components, electrical waveforms, signals, robotics, and control systems. Therefore, they don’t have to do the equations by hand. 
• Engineering and Navigation: When working with maps, GPS, or land measurements, a sin-1 calculator can be used to determine angles for direction, distance, and position.
• Physics: The inverse of sin calculator can be used to quickly and accurately determine angles of incidence, reflection, and refraction in physics problems involving waves, optics, or forces.
• Astronomy and Space Science: Calculating satellite, star, and planetary position angles, as well as Earth to pet, is another use for a sin-1 calculator.

Frequently Asked Questions

What is an arcsine (sin⁻¹) calculator?

An arcsine calculator finds the inverse sine of a given value and returns the angle whose sine equals that value.

What values can I enter in an arcsine calculator?

You can only enter values between −1 and 1, because sine values never exceed this range.

Does the arcsine calculator return results in degrees or radians?

Most arcsine calculators allow you to choose degrees or radians depending on your preference.

Can I calculate arcsine without a calculator?

Yes, but it usually requires tables, graphs, or numerical methods, making an arcsine calculator much faster and more accurate.

How accurate is this arcsin calculator?

The calculator provides high-precision results, suitable for academic, engineering, and scientific use.

Is arcsin used in real-world applications?

Yes. It’s commonly used in physics, engineering, navigation, signal processing, and trigonometry problems.

What is the difference between arcsin and sine?

• Sine: Angle → Ratio
• Arcsin: Ratio → Angle