Sin-1 Calculator

How to use the inverse sine calculator

What this calculator does

This calculator evaluates the real principal inverse sine relationship y = arcsin(x), also written as sin⁻¹(x). In angle mode, it finds the one principal angle whose sine equals the entered value. In sine mode, it reverses the workflow and evaluates the sine of an angle that lies inside the principal interval. The calculator reports the same angle in radians and degrees, verifies the result by calculating sin(y), plots the inverse sine curve, and prepares a current-state Excel workbook. It does not list every coterminal angle whose sine is the same; it intentionally returns the standard principal value only.

When to use it

Use it when a right-triangle ratio is known and you need the corresponding acute or signed principal angle, when checking a trigonometry exercise, when converting an inverse-sine result between radians and degrees, or when validating code or spreadsheet output against a transparent back-check. The real-valued input domain and principal range follow the standard inverse-function convention described in the OpenStax treatment of inverse trigonometric functions.

How to calculate

1. The calculator opens with a ready-to-use example: x = 0.5, which produces y = 30° or approximately 0.523599 rad. Download Excel is immediately available for this example.
2. Under Solve for, choose Angle y from sine x to evaluate arcsin(x), or choose Sine x from angle y to evaluate sin(y) within the principal range.
3. Replace the active input. Results update automatically; there is no separate Calculate button. Use a period for decimals. Commas, scientific notation, and extra symbols are rejected rather than silently reinterpreted.
4. Choose Degrees (°) or Radians (rad) under Angle unit. When the angle is editable, changing units converts its current value rather than merely changing the label.
5. Read the primary result, supporting result cards, graph marker, and calculation table. Select Download Excel to export the current validated model. Select Reset to clear the demonstration and all calculated content; export remains unavailable until a complete valid input is entered again.

Input guide

Solve for is a required two-option mode control. Choose angle mode when x is known, or sine mode when y is known. Switching modes makes the appropriate field editable and keeps the other field as the computed counterpart. A common mistake is to treat sin⁻¹(x) as 1/sin(x); here the superscript means the inverse function, not a reciprocal.

Sine value x is required in angle mode. Enter a plain real decimal from – 1 through 1, inclusive, such as 0.5. Values outside that interval have no real arcsine result. Raising x moves the principal result upward along the curve; x = – 1 gives – 90°, x = 0 gives 0°, and x = 1 gives 90°. Do not enter an angle in this field.

Angle y is required in sine mode. Enter a plain decimal angle, such as 30 degrees or 0.523599 radians. To preserve the one-to-one inverse relationship, the accepted range is – 90° to 90° in degrees or – π/2 to π/2 in radians. Increasing y generally increases x within this interval. A frequent error is entering 30 while radians are selected, which is outside the supported principal range.

Angle unit is required and determines the editable angle format, the primary angle display, the chart's vertical labels, and workbook units. Degrees use the interval [ – 90°, 90°]; radians use [ – π/2, π/2]. The mathematical relationship is unchanged because 180° equals π radians.

Output guide

The header pills labeled Mode, x, and y summarize the current direction and the canonical input-output pair. The dynamic primary result is labeled Principal angle in angle mode and Sine value in sine mode. Radians and Degrees show the same principal angle in both common units. Sine value shows x, while Back-check displays sin(y) so a correct calculation reproduces x within floating-point precision. Zero is a valid central result; boundary values ±1 correspond exactly to ±90° or ±π/2.

The chart series Inverse sine curve shows the real principal branch over – 1 ≤ x ≤ 1, and Current point marks the active pair. The Calculation details table lists Sine value x, Principal angle in radians, Principal angle in degrees, and Back-check. These are exact identities at the model level, displayed to a practical precision rather than recommendations or statistical estimates.

Worked example

With the startup values, x = 0.5. The calculator evaluates y = arcsin(0.5). Since sin(30°) = 0.5, the principal result is 30°. Converting to radians gives 30 × π/180 = π/6 ≈ 0.5235987756 rad, displayed as 0.523599 rad. The Back-check evaluates sin(0.5235987756) and returns 0.5. The graph marker appears at (0.5, 30°), and the workbook stores the canonical numeric values rather than the rounded text shown on screen.

Learn more

For a more formal account of principal branches, see the NIST Digital Library of Mathematical Functions section on inverse trigonometric functions. For notation, special values, and derivative properties, consult Wolfram MathWorld's inverse sine reference.