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# Arcsin 1 — Value in Degrees and Radians

[#Trigonometry](/content/tag/trigonometry/index.html)

TL;DR  
Arcsin 1 equals 90°, or \( \frac{\pi}{2} \) radians — the angle in \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \) whose sine is exactly 1. This article gives the exact value in both units, the unit-circle reason it has to be 90°, a sin-inverse reference table, two worked methods, the mistakes around the restricted range, and FAQs.

BT  
Last updated on June 13, 2026 6 min read

## Arcsin 1 in Degrees and Radians

The value of **arcsin 1** is 90°, which in radians is \( \frac{\pi}{2} \) (about 1.5708 rad). It is the unique angle inside the inverse-sine range \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \) whose sine equals 1.

### Quick Answer:

- Result: arcsin 1 = 90° = \( \frac{\pi}{2} \) radians
- Notation: arcsin 1 = sin⁻¹ 1 (the inverse function, NOT 1/sin)
- Method shown: read the angle whose sine is 1 from the unit circle / special-angle table
- Exact form: \( \frac{\pi}{2} \) radians (= 90°)
- Approximate value: 1.5708 radians

Because sine peaks at 1 exactly once in the inverse-sine range — at the top of the unit circle — there is only one angle that qualifies. That angle is \( \frac{\pi}{2} \), the quarter turn.

## Sin-Inverse Reference Table

These are the sin-inverse values readers look up alongside \( \arcsin 1 \). Each is shown in radians and degrees, with the unit-circle point that produces it.

| Input     | \( \arcsin x \) (rad) | \( \arcsin x \) (deg) | Unit-circle point      |
| ---     | ---     | ---     | ---      |
| 1     | \( \frac{\pi}{2} \) | 90° | (0,1)      |
| \( \frac{\sqrt{3}}{2} \)     | \( \frac{\pi}{3} \) | 60° | (1/2, \( \frac{\sqrt{3}}{2} \))      |
| \( \frac{\sqrt{2}}{2} \)     | \( \frac{\pi}{4} \) | 45° | (\( \frac{\sqrt{2}}{2} \), \( \frac{\sqrt{2}}{2} \))      |
| \( \frac{1}{2} \)     | \( \frac{\pi}{6} \) | 30° | (\( \frac{\sqrt{3}}{2} \), 1/2)      |
| 0     | 0 | 0° | (1,0)      |
| \( -\frac{1}{2} \)     | \( -\frac{\pi}{6} \) | -30° | (\( \frac{\sqrt{3}}{2} \), -1/2)      |
| -1     | \( -\frac{\pi}{2} \) | -90° | (0,-1)      |

## Where Arcsin 1 Appears

The value 90° surfaces wherever a quantity hits its maximum. **Is arcsin 1 the same as 90 degrees?** Yes — and that matters in physics: a projectile launched at the angle whose normalized vertical component is 1 travels straight up, the 90° extreme.

In signal processing, a sine wave reaches peak amplitude exactly when its phase angle is \( \arcsin 1 = \frac{\pi}{2} \), which is why the quarter-period mark is the crest of every oscillation. Inverse sine also turns up in robotics, where a joint angle is recovered from a known sine ratio — and the \( \frac{\pi}{2} \) output flags the arm at full extension.

## What Arcsin Means

Arcsine answers the question "which angle has this sine?" Written as \( \arcsin x \) or \( \sin^{-1} x \), it is the inverse of the sine function. Sine on its own is many-to-one — \( \sin 90° = 1 \), but so would \( \sin 450° \) if we let the angle roam — so to define a single-valued inverse, mathematicians restrict sine to \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \), where it climbs steadily from -1 to 1. The inverse of that restricted sine is the arcsine, and its outputs live in that same interval.

Feed it 1 and it returns the top of that range: \( \frac{\pi}{2} \).

## How to Find Arcsin 1

### **Method 1: Read it from the special-angle table**

The sine table gives \( \sin 90° = 1 \). Arcsine reverses the lookup:

\( \arcsin 1 = 90° = \frac{\pi}{2} \).

No other angle in \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \) has sine 1, so the answer is unique.

**Final answer:** \( \arcsin 1 = 90° = \frac{\pi}{2} \).

### **Method 2: Locate it on the unit circle**

On the unit circle, the sine of an angle is the y-coordinate of its terminal point. The only point with y=1 is (0,1), sitting at the top:

\( \arcsin 1 = \text{angle to } (0,1) = \frac{\pi}{2} = 90°.\)

**Final answer:** the same \( \frac{\pi}{2} = 90° \), confirmed by the unit circle.

## Examples of Arcsin 1

### Example 1

**Convert \( \arcsin 1 \) to radians.**

From the special-angle table, \( \arcsin 1 = 90° \). Multiply by \( \frac{\pi}{180} \): \( 90×\frac{\pi}{180} = \frac{\pi}{2} \).

**Final answer:** \( \frac{\pi}{2} \) radians.

### Example 2

**A student writes \( \arcsin 1 = \frac{1}{\sin 1} \). Check the claim.**

_Wrong attempt._ Reading \( \sin^{-1} 1 \) as a power, the student computes \( (\sin 1)^{-1} \approx \frac{1}{0.0175} \approx 57.3 \) (calculator in degrees) and reports \( \arcsin 1 \approx 57.3 \).

_The break._ Arcsine returns an angle whose sine is the input; the input here is 1, and sine never exceeds 1, so the angle must be the one where sine maxes out. A value near 57.3 is not even close — and \( \sin(57.3°) ≈ 0.84 \), not 1.

_Correct._ The -1 in \( \sin^{-1} \) marks the inverse function, not a reciprocal. The angle whose sine is 1 is 90°:

\( \arcsin 1 = \frac{\pi}{2} = 90°.\)

**Final answer:** \( \arcsin 1 = \frac{\pi}{2} = 90° \), not \( \frac{1}{\sin 1} \).

### Example 3

**Find \( \arcsin(-1) \) in both units.**

The point with y=−1 is (0,−1), at the bottom of the unit circle, angle \( -\frac{\pi}{2} \).

**Final answer:** \( \arcsin(-1) = -\frac{\pi}{2} = -90° \).

### Example 4

**Evaluate \( \arcsin 1 + \arccos 1 \).**

\( \arcsin 1 = \frac{\pi}{2} \) and \( \arccos 1 = 0 \), so the sum is \( \frac{\pi}{2} \). This matches the identity \( \arcsin x + \arccos x = \frac{\pi}{2} \) at \( x=1 \).

**Final answer:** \( \frac{\pi}{2} = 90° \).

### Example 5

**A right triangle has its opposite side equal to its hypotenuse. What is the angle?**

The angle's sine is \( \frac{\text{opposite}}{\text{hypotenuse}} = 1 \), so the angle is \( \arcsin 1 = 90° \).

**Final answer:** 90° = \( \frac{\pi}{2} \).

## Where Sin-Inverse Trips Students Up

### Mistake 1: Reading \( \sin^{-1} 1 \) as a reciprocal

**Where it slips in:** A reader sees \( \sin^{-1} 1 \) and computes \( \frac{1}{\sin 1} \) instead of the inverse angle.

**Don't do this:** Treat the -1 as an exponent, the way it works on ordinary numbers.

**The correct way:** \( \sin^{-1} \) is the inverse function, \( \arcsin \). For the reciprocal of sine, write \( \csc x \) or \( \frac{1}{\sin x} \).

### Mistake 2: Forgetting the input ceiling of 1

**Where it slips in:** A reader tries \( \arcsin 2 \) expecting an answer.

**Don't do this:** Feed arcsine a number outside \( [-1,1] \).

**The correct way:** Sine never exceeds 1 in magnitude, so \( \arcsin x \) is undefined as a real number for \( |x| > 1 \). The endpoint \( \arcsin 1 = 90° \) is the largest legal output.

### Mistake 3: Picking the wrong angle for sine 1

**Where it slips in:** A reader knows \( \sin 90° = 1 \) but also recalls \( \sin 450° = 1 \) and isn't sure which to report.

**Don't do this:** Return an angle outside the inverse-sine range.

**The correct way:** Arcsine outputs only angles in \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \), so \( \frac{\pi}{2} \) is the answer and \( \frac{5\pi}{2} \) is not — even though both have sine 1.

## What to Remember About Arcsin 1

- **Arcsin 1** equals 90°, or \( \frac{\pi}{2} \) radians — the angle whose sine is exactly 1.
- On the unit circle it is the point (0,1) at the top, where the y-coordinate hits its maximum.
- \( \sin^{-1} \) means the inverse function, not the reciprocal \( \frac{1}{\sin} \).
- Arcsine is undefined for inputs outside \( [-1,1] \), so \( \arcsin 1 \) is the largest possible output.

## Practice These Three

1. Find \( \arcsin\left(\frac{\sqrt{3}}{2}\right) \) in degrees and radians.
2. Evaluate \( \arcsin 1 - \arcsin(-1) \).
3. A ramp rises so that the sine of its incline is 1. What is the incline angle, and what does that mean physically?