Arcsin Calculator

Sine value (x)

Angle

deg (°)

Angle in radians (multiples of pi)
Cosine of the angle (sqrt of 1 minus x squared)
Complementary angle (arccos of x): deg (°)

The sine of 30 deg (°) is 0.5, which confirms the inverse: arcsin undoes sine.

Arcsin always returns the principal angle between -90 and 90 degrees (-pi/2 to pi/2 radians), even though many angles share the same sine.

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Show the arcsin graph

Plot the inverse sine curve from x = -1 to x = 1.

Show the angle in every unit

List your angle in degrees, radians, multiples of pi, gradians, and turns.

Your angle in every unit
Unit	Your angle
Degrees	30 deg
Radians	0.523599 rad
Radians (multiples of pi)	0.1667 pi
Gradians	33.3333 grad
Turns	0.083333 turn

The arcsine calculator finds the angle whose sine is equal to a given number. The sine function takes an angle and gives a ratio between -1 and 1. The arcsine does the opposite: it takes a value between -1 and 1, and gives back the corresponding angle. You can either enter a sine value to find its corresponding angle, or you can enter an angle to check its sine. The unit of the angle can be changed at any time between degrees, radians, grads, and revolutions.

What is arcsin?

The arcsine is the inverse function of the sine, written as arcsin(x) or sin-1(x). An inverse function reverses a previous operation. If the sine of an angle x is y, then the arcsine of y is that angle x.

\sin(\theta) = x \quad\Longleftrightarrow\quad \arcsin(x) = \theta

There are two important restrictions to note. First, only values between -1 and 1 can be entered because the sine function never produces ratios outside of this range. Second, since the sine function repeats itself every 360 degrees, many angles will have the same sine value. To return a single value, the arcsine function chooses the principal value, which is an angle between -90 and 90 (equivalent to -pi/2 and pi/2 radians).

Note the notation of a negative exponent for sine. This indicates an inverse function, not the reciprocal. The reciprocal of sine is cosecant and is a different function.

Here's how to use this calculator.

If you enter a sine value between -1 and 1 in the first input field, the corresponding angle will be immediately shown. The result is displayed under the heading "Result" and also in the field for the angle where you can switch the unit to degrees, radians, gradians or revolutions. The panel also shows the multiple of Pi of the angle, the cosine of the angle and the complementary angle.

This tool allows calculations in both directions. If you leave the field for sine empty and enter an angle in the field for the angle, it will return the corresponding sine value. By opening the graph you can see the entire arcsin curve, and by opening the table you can display the angle in all units at once.

The formula for arcsine is:

This relationship can be expressed in a single line of code. The angle is the arcsine of x and if you take the sine of that angle, it will give you back x.

\theta = \arcsin(x) \qquad \sin(\theta) = x

Here's an example: The sine of 30 degrees is 0.5, so the arcsine of 0.5 is 30 degrees, which equals pi/6 radians.

\sin(30^\circ) = 0.5 \quad\Longrightarrow\quad \arcsin(0.5) = 30^\circ = \frac{\pi}{6}

And now for the angle.

Calculating an angle in a right triangle.

The arcsine is a standard method for calculating the angle in a right triangle when two sides are known. The sine of an angle is the value obtained by dividing the length of the side opposite that angle by the longest side (hypotenuse). So, the angle is equal to the arcsine of this ratio.

\alpha = \arcsin\!\left(\frac{\text{opposite}}{\text{hypotenuse}}\right)

Assume the opposite leg is 52 and the hypotenuse is 60. The ratio is found by dividing 52 by 60, which equals approximately 0.866. Therefore, the angle is the arcsine of 0.866, or 60 degrees.

\sin(\alpha) = \frac{52}{60} \approx 0.866 \quad\Longrightarrow\quad \alpha = \arcsin(0.866) = 60^\circ

If the opposite side is still unknown, it must first be calculated using the Pythagorean theorem and then divided by the hypotenuse before taking the arcsine.

Commonly used values for arcsine:

There are a few values that come up again and again. This table shows these values in both degrees and radians, and if you enter one of the sine values into the calculator it will show one of these angles. Because arcsine is an odd function, for negative inputs only the sign of the angle changes.

Sine value (x)	Angle in degrees	Angle in radians
-1	-90°	$-\frac{\pi}{2}$
-0.866	-60°	$-\frac{\pi}{3}$
-0.707	-45°	$-\frac{\pi}{4}$
-0.5	-30°	$-\frac{\pi}{6}$
0	0°	$0$
0.5	30°	$\frac{\pi}{6}$
0.707	45°	$\frac{\pi}{4}$
0.866	60°	$\frac{\pi}{3}$
1	90°	$\frac{\pi}{2}$

Practical identities for arcsine.

Some identities allow for easier manipulation of the arcsine and also help to understand its relationship with other inverse trigonometric functions.

\sin(\arcsin(x)) = x

\cos(\arcsin(x)) = \sqrt{1 - x^2}

\arcsin(x) + \arccos(x) = \frac{\pi}{2}

The third identity shows that the arcsine and arccosine are complementary angles. The sum of the angle x in sine and the angle x in cosine is always 90 degrees. There's also an addition formula which allows you to combine two values of arcsine into one.

\arcsin(\alpha) + \arcsin(\beta) = \arcsin\!\left(\alpha\sqrt{1 - \beta^2} + \beta\sqrt{1 - \alpha^2}\right)

Applications of arcsine

The arcsine is needed in many fields because it converts ratios back into angles. In geometry, it can be used to solve right triangles when the opposite leg and hypotenuse are known. In physics, it can be used to calculate the launch angle from the vertical and total velocity of a projectile or determine the amplitude of a pendulum that is released from rest. In computer graphics, it can be used to extract angles from normalized direction vectors. Lighting programs use this to calculate the orientation of a surface relative to the sun.

Tips and hints

The arcsine only accepts inputs between -1 and 1. If the calculated ratio is slightly outside of this range due to rounding (e.g. 1.0000001), you will need to bring it back within the range before calculating the arcsine.

The result is always the principal value between -90 and 90 degrees. To find another angle in the range from 0 to 360 degrees that has the same sine value, subtract the principal value from 180 degrees.

In everyday life and engineering degrees are used while radians are used in analysis. You can convert between the two using the unit conversion function in the Angle field.

There are three values worth remembering: The arcsine of 0 is 0 degrees, the arcsine of 0.5 is 30 degrees and the arcsine of 1 is 90 degrees.

Negative powers of sin represent the inverse function and are not the result of dividing 1 by sine. The reciprocal of sine is cosecant.

Frequently asked questions

What is the domain of arcsine?

The domain is between -1 and 1. Since the sine function does not produce values outside this range, its inverse cannot accept inputs outside of it either. An input like arcsin(2) has no real solution. Thus, an error will be returned if values greater than -1 or less than 1 are entered. Ratios that are slightly out of range due to rounding errors must first be adjusted within the range.

Is the negative power of sine equal to 1 over sine?

No they are not the same thing. This notation can be confusing. The negative exponent of sine represents the inverse function arcsin. The reciprocal of sine, 1/sine, is cosecant. These are different functions that happen to use similar notation.

Why is the result of arcsin always in the range of -90 to 90 degrees?

Since the sine function repeatedly takes on all values between −1 and 1 infinitely often, when using arcsin as an inverse function a particular branch must be selected. The range of this branch is between −90 and 90 degrees, or between −π/2 and π/2 radians. So although sin(150°) = 0.5, arcsin(0.5) returns 30°, not 150°.

What is the relationship between arcsine and arccosine?

These two functions are complementary to each other. The arccosine of x is equal to the value that results when you subtract the arcsine of x from 90 degrees, which is why their sum always equals 90 degrees or pi/2 radians. The arcsine gives the angle whose sine is x, while the arccosine indicates the angle whose cosine is x.

What is the arcsine of 0.5?

It is 30 degrees or pi/6 radians. Since the sine of 30 degrees is 0.5, it follows that the arcsine of 0.5 is equal to 30 degrees. For other commonly used values: The arcsine of 0 is 0 degrees and the arcsine of 1 is 90 degrees.

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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Wikipedia: Inverse trigonometric functions
Definitions, principal values, domains, and identities for arcsin and its siblings.
Math is Fun: Inverse Sine, Cosine, Tangent
Plain-language explanation of how the inverse trig functions undo sine, cosine, and tangent.