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Arcsin 1 — Value in Degrees and Radians

#Trigonometry

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?