Inverse Sine Calculator

Inverse sine is the function that returns the angle whose sine equals a given number. Inverse sine is written as sin⁻¹(x), arcsin(x) or asin(x), and it reverses the sine function: sin(30°) = 0.5, so sin⁻¹(0.5) = 30°.

Inverse sine (arcsine) is one of the 6 inverse trigonometric functions: arcsin, arccos, arctan, arccsc, arcsec and arccot. Arcsine accepts numbers from −1 to 1 and returns a principal value from −90° to 90° (−π/2 to π/2 radians).

In a right triangle, the sine of an angle θ equals the opposite side divided by the hypotenuse. The inverse sine takes that opposite/hypotenuse ratio and returns θ. In algebra, sin⁻¹ solves equations such as sin θ = 0.8: θ = sin⁻¹(0.8) = 53.13°.

• Input: a number from −1 to 1, such as a sine value or a side ratio.
• Output: an angle in degrees or radians.
• Range: −90° ≤ θ ≤ 90°, or −π/2 ≤ θ ≤ π/2.

The −1 in sin⁻¹(x) marks the inverse function, not an exponent. The reciprocal 1/sin(x) is a different function, the cosecant csc(x).

To calculate inverse sine, divide the opposite side by the hypotenuse, then apply sin⁻¹ to the result. Follow these 4 steps:

1. Find the sine value x. For a right triangle, x = opposite ÷ hypotenuse. Example: 5 ÷ 13 = 0.3846.
2. Check that x lies from −1 to 1. A value outside that interval has no real arcsine.
3. Apply the inverse sine: θ = sin⁻¹(0.3846) = 22.62°.
4. Convert the unit if needed: 22.62° × π/180 = 0.3948 rad.

Check the answer by taking the sine of the angle: sin(22.62°) = 0.3846. The result matches x, so the angle is correct.

For special values, use the exact angles: sin⁻¹(1/2) = 30° = π/6, sin⁻¹(√2/2) = 45° = π/4, sin⁻¹(√3/2) = 60° = π/3 and sin⁻¹(1) = 90° = π/2.

Enter a value from −1 to 1, or 2 triangle sides, and the calculator returns sin⁻¹(x) as you type. The steps:

1. Pick an input method: Decimal / x for a sine value, or Opp / Hyp for the opposite side and the hypotenuse.
2. Type the value. The field accepts decimals (0.6), fractions (3/5), roots (√3/2), π and expressions.
3. Choose the result unit: degrees, radians, gradians, turns, minutes or seconds of arc, milliradians, microradians or multiples of π.
4. Choose the rounding, from whole numbers to 15 decimal places.
5. Read the result: the angle in all 9 units, the exact value, the triangle, the unit circle, the graph and the worked steps.

The calculator rejects values outside −1 to 1 and states the reason. In Opp / Hyp mode, the opposite side must not exceed the hypotenuse. Every result can be copied, shared as a link, printed or kept in the history list.

Refraction mode solves Snell’s law. Enter the refractive index you start in (n₁), the angle of incidence and the index you enter (n₂). Air to water at 45° gives θ₂ = arcsin(1 × sin 45° / 1.333) = 32.04°. Glass to air past 41.8° gives no answer: the light reflects back, and the calculator names the critical angle.

The inverse sine formula is θ = sin⁻¹(opposite / hypotenuse). θ is the angle, the opposite side faces θ, and the hypotenuse is the longest side, across from the right angle.

Related forms of the formula:

• Identity: sin(sin⁻¹ x) = x for −1 ≤ x ≤ 1, and sin⁻¹(sin θ) = θ for −90° ≤ θ ≤ 90°.
• Arctangent form: sin⁻¹ x = tan⁻¹(x / √(1 − x²)) for −1 < x < 1.
• Complement rule: sin⁻¹ x + cos⁻¹ x = 90° (π/2).
• Derivative: d/dx sin⁻¹ x = 1/√(1 − x²).
• Integral: ∫ sin⁻¹ x dx = x·sin⁻¹ x + √(1 − x²) + C.

Example: a 5 m ladder reaches 4 m up a wall. sin θ = 4/5 = 0.8, so the ladder meets the ground at θ = sin⁻¹(0.8) = 53.13°.

Inverse sine in degrees returns an angle from −90° to 90°. A calculator in DEG mode gives sin⁻¹(0.5) = 30°, sin⁻¹(0.8) = 53.13° and sin⁻¹(−1) = −90°.

To convert a radian result to degrees, multiply by 180/π (57.2958): sin⁻¹(0.8) = 0.9273 rad × 57.2958 = 53.13°.

Degrees split into degrees, minutes and seconds (DMS). 1 degree holds 60 minutes, and 1 minute holds 60 seconds: 53.1301° = 53° 7′ 48.4″.

Common values in degrees:

• sin⁻¹(0) = 0°
• sin⁻¹(0.25) = 14.48°
• sin⁻¹(0.5) = 30°
• sin⁻¹(√2/2) = 45°
• sin⁻¹(√3/2) = 60°
• sin⁻¹(1) = 90°

Inverse sine in radians returns an angle from −π/2 to π/2 (−1.5708 to 1.5708). sin⁻¹(0.5) = π/6 = 0.5236 rad and sin⁻¹(1) = π/2 = 1.5708 rad.

1 radian is the angle that cuts an arc equal in length to the radius. On the unit circle, sin⁻¹(x) is the arc length from the point (1, 0) to the point at height x on the right half of the circle.

To convert degrees to radians, multiply by π/180: 30° × π/180 = π/6. Excel, Python, JavaScript, C and Java return arcsine in radians by default.

• sin⁻¹(1/2) = π/6 = 0.5236 rad
• sin⁻¹(√2/2) = π/4 = 0.7854 rad
• sin⁻¹(√3/2) = π/3 = 1.0472 rad
• sin⁻¹(1) = π/2 = 1.5708 rad

The domain of inverse sine is −1 ≤ x ≤ 1, and the range is −90° ≤ θ ≤ 90° (−π/2 ≤ θ ≤ π/2).

The sine of a real angle always lies from −1 to 1, so no angle has a sine of 1.5 or −2, and sin⁻¹(1.5) has no real value.

Sine repeats every 360°, so many angles share one sine value: sin(30°) = sin(150°) = sin(390°) = 0.5. Inverse sine restricts sine to −90° to 90°, where sine rises from −1 to 1 and takes each value once. The restriction gives sin⁻¹ a single output, the principal value.

• Domain: [−1, 1]
• Range: [−π/2, π/2]
• Monotonic: increasing over the whole domain
• Odd function: sin⁻¹(−x) = −sin⁻¹(x)
• End points: sin⁻¹(−1) = −90° and sin⁻¹(1) = 90°

To find every angle with sin θ = x, use θ = sin⁻¹(x) + 360°·k or θ = 180° − sin⁻¹(x) + 360°·k, for any integer k.

On a scientific calculator, press SHIFT (or 2nd), then sin, type the value and press =. On a Casio fx-991 or fx-82, the keys for sin⁻¹(0.5) are SHIFT, sin, 0.5, ), = and the screen shows 30.

1. Set the angle mode: D on the screen means degrees, R means radians.
2. Press SHIFT (Casio, Sharp) or 2nd (Texas Instruments) to reach sin⁻¹ above the sin key.
3. Type the value, close the bracket and press =.

On a TI-84, press 2nd, SIN, 0.5, ENTER. To switch the TI-84 between degrees and radians, press MODE and select DEGREE or RADIAN.

A Math ERROR (Casio) or ERR:DOMAIN (TI) means the value lies outside −1 to 1. A result of 0.5236 instead of 30 means the calculator is in radian mode.

On the Android (Google) Calculator, open the scientific keys, tap INV, tap sin⁻¹, type the value and tap =. sin⁻¹(0.5) returns 30 in DEG mode.

1. Tap the expand arrow, or turn the phone to landscape, to show the scientific keys.
2. Check the mode label: DEG for degrees, RAD for radians.
3. Tap INV. The sin key changes to sin⁻¹.
4. Tap sin⁻¹, type 0.5 and tap =.

The display shows sin⁻¹(0.5) while you type and 30 after =. A value above 1 shows an error message in place of a result.

On the iPhone Calculator, switch to Scientific, type the value, tap 2nd, then tap sin⁻¹. Type 0.5, tap 2nd and tap sin⁻¹: the display shows 30.

1. Turn the iPhone sideways, or tap the calculator icon and choose Scientific (iOS 18 and later).
2. Check the mode: Rad on the display means radians, no label means degrees.
3. Type 0.5.
4. Tap 2nd. The sin key changes to sin⁻¹.
5. Tap sin⁻¹. The result appears at once.

The iPhone applies sin⁻¹ to the number on the display, so type the value first. A value above 1 shows Error.

In Windows Calculator, choose Scientific, type the value, open Trigonometry, select 2nd, then select sin⁻¹. For 0.5 in DEG mode, the result is 30.

1. Open the menu (☰) and choose Scientific.
2. Check the angle button: DEG, RAD or GRAD.
3. Type 0.5.
4. Open Trigonometry and select 2nd.
5. Select sin⁻¹. The result is 30.

Keyboard shortcut: type 0.5, then press Shift + S for sin⁻¹. Press F3, F4 or F5 to switch to degrees, radians or gradians.

Software calls inverse sine ASIN or asin, and the result comes out in radians. Convert the result to get degrees.

• Excel and Google Sheets: =ASIN(0.5) returns 0.5236; =DEGREES(ASIN(0.5)) returns 30.
• Python: math.asin(0.5) returns 0.5236; math.degrees(math.asin(0.5)) returns 30.000000000000004, the floating-point form of 30. NumPy uses np.arcsin.
• JavaScript: Math.asin(0.5) * 180 / Math.PI returns 30.000000000000004.
• MATLAB and Octave: asin(0.5) for radians, asind(0.5) for degrees.
• C, C++ and Java: asin(0.5) and Math.asin(0.5) return radians.
• Wolfram Alpha: ArcSin[1/2] returns π/6.

Values outside −1 to 1 produce errors: Excel returns #NUM!, Python raises ValueError: math domain error, and JavaScript returns NaN.

Without a calculator, use exact values for special angles, or the arcsine series for other numbers.

1. Special values: learn sin 30° = 1/2, sin 45° = √2/2 and sin 60° = √3/2. Then sin⁻¹(1/2) = 30°, sin⁻¹(√2/2) = 45° and sin⁻¹(√3/2) = 60°.
2. Taylor series: sin⁻¹ x = x + x³/6 + 3x⁵/40 + 5x⁷/112 + … for |x| ≤ 1.
3. Triangle sketch: draw a right triangle with opposite side x and hypotenuse 1, then measure θ with a protractor.

Example: sin⁻¹(0.3) ≈ 0.3 + 0.0045 + 0.000182 + 0.0000098 = 0.304692 rad = 17.458°. The true value is 0.3046927 rad, so 4 terms land within 0.000001.

The series converges slowly near x = ±1. For x above 0.7, use sin⁻¹ x = 90° − sin⁻¹(√(1 − x²)) to move the input closer to 0 first.

The graph of y = sin⁻¹(x) is an increasing curve from (−1, −π/2) to (1, π/2) that passes through the origin.

• Starts at (−1, −90°), passes (0, 0°) and ends at (1, 90°).
• Slope 1 at x = 0; the slope 1/√(1 − x²) grows without bound near x = ±1.
• Point symmetry about the origin: the function is odd.
• Mirror image of y = sin x on [−π/2, π/2] across the line y = x.
• No asymptotes: the graph stops at x = −1 and x = 1.

The origin is an inflection point. The curve is concave down for x < 0 and concave up for x > 0, since the second derivative x/(1 − x²)^(3/2) changes sign at 0.

The inverse sine table lists sin⁻¹(x) for common values of x from −1 to 1, in degrees and radians. Click a row to load it into the calculator.

Exact values to know: sin⁻¹(0) = 0, sin⁻¹(1/2) = π/6, sin⁻¹(√2/2) = π/4, sin⁻¹(√3/2) = π/3 and sin⁻¹(1) = π/2. Negative inputs give the same angles with a minus sign.

The 15° and 75° rows use (√6 − √2)/4 = 0.2588 and (√6 + √2)/4 = 0.9659, the sines of π/12 and 5π/12.

The inverse sine of a negative number is a negative angle: sin⁻¹(−x) = −sin⁻¹(x). sin⁻¹(−0.5) = −30°, sin⁻¹(−√2/2) = −45° and sin⁻¹(−1) = −90°.

Arcsine is an odd function. Its graph has point symmetry about the origin, so a negative input mirrors the positive result below the x-axis. The angle sits in quadrant IV, from −90° to 0°.

What is sin⁻¹(−0.5)?

sin⁻¹(−0.5) = −30° = −π/6 rad. sin(−30°) = −0.5, and −30° lies inside the range −90° to 90°. As a positive angle, −30° + 360° = 330°. The angle 210° has the same sine, −0.5, but sits outside the principal range.

arcsin takes opposite/hypotenuse, arccos takes adjacent/hypotenuse and arctan takes opposite/adjacent; each returns the angle for its ratio.

Identities that link them: sin⁻¹ x + cos⁻¹ x = 90°, and sin⁻¹ x = tan⁻¹(x / √(1 − x²)). For x = 0.6: sin⁻¹(0.6) = 36.87°, cos⁻¹(0.6) = 53.13°, and 36.87° + 53.13° = 90°.

The inverse cosecant links to arcsine through the reciprocal: csc⁻¹ x = sin⁻¹(1/x) for |x| ≥ 1.

Inverse sine finds an angle from a height and a slope length, and it appears in optics, construction, physics and aviation.

• Optics: Snell's law n₁ sin θ₁ = n₂ sin θ₂ gives the refraction angle θ₂ = sin⁻¹(n₁ sin θ₁ / n₂). Light entering water (n = 1.33) at 40° bends to 28.9°.
• Ramps: a 12 m ramp that rises 1 m has an angle of sin⁻¹(1/12) = 4.78°.
• Projectiles: the launch angle for range R at speed v is θ = ½ sin⁻¹(gR / v²).
• Aviation: a plane that climbs 1,000 m along a 10 km flight path has a climb angle of sin⁻¹(0.1) = 5.74°.
• AC circuits: the phase angle of a sampled voltage is sin⁻¹(v / V_peak).

The critical angle for total internal reflection is θc = sin⁻¹(n₂ / n₁). From glass (n = 1.5) to air, θc = sin⁻¹(1/1.5) = 41.81°.