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Arcsin Calculator

The arcsin calculator will determine the inverse sine values for angles and display results in radians and degrees along with the calculations involved.

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An online arcsin calculator helps you compute \( \arcsin(x) \), the inverse of sine, and displays results in both degrees and radians. It also provides step-by-step calculations for accurate results.

What is Arcsin?

The arcsin function is the inverse of the sine function. It returns the angle whose sine equals a given number:

\( \arcsin(y) = x \quad \text{if and only if} \quad \sin(x) = y \)

Many devices have an arcsin button (sometimes labeled sin⁻¹), but an online arcsin calculator ensures accuracy and ease of use.

Arcsin Formula

Since \( y = \sin(x) \), the inverse is:

\( \arcsin(y) = \sin^{-1}(y) = x + 2k\pi, \quad k \in \mathbb{Z} \)

Example:

If \( \sin 30^\circ = 0.5 \), then:

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

How to Calculate Arcsin

To calculate arcsin manually:

Given \( \sin 30^\circ = 0.5 \)
Apply \( \arcsin(y) = \sin^{-1}(y) \)
Result: \( \arcsin(0.5) = 30^\circ \)

For complex calculations, an arccos calculator works similarly for inverse cosine.

Using Arcsin to Find an Angle

Example: Right-angled triangle with opposite side \( a = 52 \) and hypotenuse \( c = 60 \):

\( \sin(\alpha) = a / c = 52 / 60 = 0.866 \)
\( \alpha = \arcsin(0.866) = 60^\circ \)

Arcsin Angle

Common Arcsin Values Table

x	arcsin(x) (°)	arcsin(x) (rad)
-1	-90°	-π/2
-√3/2	-60°	-π/3
-√2/2	-45°	-π/4
-1/2	-30°	-π/6
1/2	30°	π/6
√2/2	45°	π/4
√3/2	60°	π/3
1	90°	π/2

Graph of Arcsin

Arcsin(x) is the inverse of \( f(x) = \sin(x) \) for \( -\pi/2 \le x \le \pi/2 \).
Domain: \( x \in [-1, 1] \)
Range: \( y \in [-\pi/2, \pi/2] \)

Arcsin Graph

The unit circle calculator can also help find sine, cosine, and tangent values.

How the Arcsin Calculator Works

Input:

Enter a value between -1 and 1
Set decimal precision (up to 16 digits)
Click Calculate

Output:

Angle in radians
Angle in degrees
Optional: other units
Step-by-step calculation

FAQs

Is Cosecant the inverse of sine?

No. Cosecant is the reciprocal of sine: hypotenuse / opposite side. Learn more

Is Arcsin the same as sin⁻¹?

Yes, \( \sin^{-1}(x) = \arcsin(x) \).

What is sin(arcsin)?

\( \sin(\arcsin x) = x \)
\( \arcsin(\sin x) = x + 2k\pi, k \in \mathbb{Z} \)
\( \arcsin(-x) = -\arcsin(x) \)
Complementary: \( \arcsin x = \pi/2 - \arccos x = 90° - \arccos x \)

How do you calculate arcsin on a unit circle?

Select the angle on the right half of the circle closest to zero. Example: \( \sin^{-1}(-1/2) = -30° = -π/6 \). Range: [-90°, 90°]

Endnote

The arcsin calculator provides accurate inverse sine results. It’s free and ideal for students and educators learning \( \arcsin(x) \).

References

Wikipedia: Inverse Trigonometric Functions
Math Open Reference: Range and domain of arcsin
Mile Foot: Inverse reciprocal identities
Shmoop: Graph of inverse sine function
Brilliant: Inverse trigonometric graphs

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