Arcsine Calculator

What is Arcsin?

Arcsin (arc sine) is the sine's inverse: feed it a value x and it returns the angle θ such that sin(θ) = x. Whatever the valid input, the answer lands in [−90°, 90°].

To anchor the intuition: arcsin(0) = 0°, arcsin(0.5) = 30°, arcsin(1) = 90°, arcsin(−1) = −90°.

Domain: −1 ≤ x ≤ 1. Outside that range, no real arcsin exists.

The arcsine function

Give the arcsine a number and it hands back the angle whose sine matches it: asen(x) = θ such that sen(θ) = x. You can only feed it values from [−1, 1], and what comes out sits in [−π/2, π/2] by convention, the same as [−90°, 90°]. You will see it written as asin, sen⁻¹ or arcsen. Since sine repeats itself and isn't one-to-one across its whole domain, we pin the inverse down to a single principal branch. Its derivative is (asen x)' = 1/√(1 − x²), and the Maclaurin series opens with asen(x) = x + x³/6 + 3x⁵/40 + …. For instance, asen(0,5) = 30°.

Applications: from ramps to GPS

Whenever you have a sine value and need the angle back, the arcsine is the tool. It turns rise over hypotenuse into the slope of a ramp, gives the launch angle in projectile problems, pulls latitude out of a GPS unit vector, recovers phase angles in AC circuits, helps fit parameters during non-linear regression, and converts normalized dot products into angles in computer graphics.

FAQ

Why is the domain limited to [−1, 1]? Sine never produces anything outside that interval, so no real angle has a sine above 1 or below −1. Ask for one and you get nothing back.

Why is the range only [−90°, 90°]? A function can return just one value per input, so we keep to the principal branch, the stretch where sine climbs steadily without repeating.

What is the difference between asen and 1/sen? They have nothing to do with each other. asen(x) is the inverse function, whereas 1/sen(x) = csc(x) is just one divided by the sine.

How does it relate to acos? The two add up to a right angle: asen(x) + acos(x) = π/2.