Arcsin — Formula, Graph, Domain and Range

TL;DR

Arcsin (written sin⁡−1x or arcsin⁡x) is the inverse sine function. It takes a number in [-1,1] and returns the angle in [-\pi/2,\pi/2] whose sine equals that number. Its graph is a smooth, strictly-increasing S-curve passing through the origin, with endpoints (−1,−π/2) and (1,π/2).

A Function Built on a Promise

To define the inverse of sine, mathematicians had to make a promise: pick one angle per sine value, and stick to it. Sine is periodic — infinitely many angles share each sine value. The promise was made in the 1820s by Cauchy and locked in ever since: arcsin returns the angle in [-\pi/2,\pi/2], the slice of sine closest to zero where the function is one-to-one.

What Is Arcsin?

Arcsin is the inverse of the sine function. If sin⁡θ=x for some ( \theta \in [-\pi/2,\pi/2] ), then arcsin⁡x=θ. Equivalently:

[ y = \arcsin x \iff \sin y = x \text{ and } y \in [-\pi/2, \pi/2] ]

Arcsin is also written sin⁡−1x — but the "−1" is not a reciprocal. The reciprocal of sin⁡x is csc⁡x (cosecant). sin⁡−1x means "the inverse function evaluated at x."

The Arcsin Formula

The defining relationship:

[ y = \arcsin x \iff \sin y = x \text{ with } −1≤x≤1 ext{ and } −\pi/2≤y≤\pi/2 ]

Standard values you should know cold:

The pattern is symmetric — arcsin is an odd function, so arcsin⁡(−x)=−arcsin⁡(x). This is what gives the table its mirror symmetry around zero.

Domain and Range of Arcsin

The domain is the closed interval [-1,1] because sine outputs values only in [-1,1] — the inverse can't accept anything outside that range. The range [-\pi/2,\pi/2] is the principal branch — the largest interval containing zero on which sine is one-to-one and continuous.

The Arcsin Graph

The arcsin graph is the reflection of the restricted sine curve on [-\pi/2,\pi/2] across the line y=x.

Three features worth pinning down:

• Strictly increasing. Higher xxx always gives higher arcsin⁡x.
• Passes through the origin. arcsin⁡(0)=0.
• Vertical tangents at ±1. The derivative goes to infinity as x→±1 — the graph turns vertical at the endpoints. This is the geometric mirror of sine's horizontal tangents at ±π/2.

Arcsin Identities

The most useful arcsin identities:

1. arcsin⁡(−x)=−arcsin⁡(x) (odd function)
2. arcsin⁡(x)+arccos⁡(x)=π/2 for all x∈[-1,1]
3. arcsin⁡(sin⁡θ)=θ only when θ∈[-\pi/2,\pi/2]
4. sin⁡(arcsin⁡x)=x for all x∈[-1,1]

Arcsin derivative and integral

[ \frac{d}{dx}\arcsin(x) = \frac{1}{\sqrt{1-x^2}}, \quad x \in (-1, 1) ]

[ \int \arcsin(x), dx = x \arcsin(x) + \sqrt{1-x^2} + C ]

The derivative blows up at x=±1 — matching the vertical tangents of the graph.

Three Worked Examples — Quick, Standard, Stretch

Quick

Find arcsin⁡(1/2).

We want the angle θ∈[-π/2,π/2] with sin⁡θ=1/2.

From the unit circle, sin⁡(π/6)=1/2, and π/6 is within the principal range.

arcsin⁡(1/2)=π/6=30°.

A Solve You Can Trust — Standard Example

Find arcsin⁡(sin⁡(5π/6)).

The wrong path. A student writes: "Since arcsin is the inverse of sine, the two cancel: arcsin⁡(sin⁡(5π/6))=5π/6."

That's the textbook trap. The "inverse cancellation" only works one-way — the side that lives in the original function's principal range.

Sanity check. Arcsin's range is [-\pi/2,\pi/2]. The angle 5π/6 is out of that range, so arcsin⁡(sin⁡(5π/6)) cannot equal 5π/6.

The correct path. Compute sin⁡(5π/6) first. 5π/6 is in quadrant II, with reference angle π−5π/6=π/6. Sine is positive in quadrant II:

sin⁡(5π/6)=sin⁡(π/6)=1/2.

Then:

arcsin⁡(sin⁡(5π/6))=arcsin⁡(1/2)=π/6.

Stretch

Find the exact value of cos⁡(arcsin⁡(3/5)).

Let θ=arcsin⁡(3/5), so sin⁡θ=3/5 and θ∈[-π/2,π/2].

Since 3/5 is positive, θ is in the first quadrant, so cos⁡θ>0.

Imagine a right triangle with opposite side 3 and hypotenuse 5. By Pythagoras, the adjacent side is √(25−9)=4.

cos⁡θ=4/5.

So:

cos⁡(arcsin⁡(3/5))=4/5.

Where Arcsin Shows Up in the Real World

Arcsin is the function that recovers an angle from a vertical measurement — the height-over-hypotenuse ratio.

1. Snell's law and refraction.
2. Astronomy.
3. Aviation.
4. Computer vision.

The Mathematicians Who Shaped Arcsin

1. Aryabhata (476–550 CE, Indian)
2. Daniel Bernoulli (1700–1782, Swiss)
3. Augustin-Louis Cauchy (1789–1857, French)

Why it matters: Every modern calculator's arcsin button is executing Cauchy's convention.

Key Takeaways

• Arcsin is the inverse sine — input in [-1,1], output in [-\pi/2,\pi/2].
• The graph is a strictly increasing S-curve from (−1,−π/2) to (1,π/2), passing through the origin.
• The derivative is [ \frac{1}{\sqrt{1-x^2}} ] — diverges at the endpoints.
• arcsin⁡(sin⁡x)=x only when x is already in the principal range.
• sin⁡−1x≠1/sin⁡x — the former is the inverse function; the latter is cosecant.