Free · Trigonometry
Sine Angle and Principal Inverse
Evaluate sine in any retained angle unit, or solve the principal inverse from a sine value, with special-angle identities and a reference wave.
- Formula & worked example
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Calculator inputs
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How to use this calculator
- Enter the known values in the units shown. Results update as you type.
- Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
- Use the worked example to check the method. Reset restores the starting fields.
Private by default
Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.
Formula and method
sin 30° = 1/2. The principal inverse of 1/2 is 30°.
s = sin θ; θ = arcsin(s) for the principal inverse.
Worked example
Enter these known values and leave the other values blank.
- Angle α
- 30 deg
- Sine of α
- 0.5
Assumptions and limitations
- Sine accepts a real angle and returns a value between −1 and 1. Inverse input must lie within that closed interval.
- The inverse is the principal angle between −90° and 90°; other angles with the same sine require periodicity and symmetry. The calculator retains this source branch rather than enumerating all solutions.
- All captured angle units and editable inverse controls are preserved. The exact-value reference notes recognize the retained standard-angle table modulo 360° within 10^-60 degrees for internal conversion roundoff; this does not certify rounded measured angles as mathematically exact.
- The supported phase range is |θ| below 10^30 radians, with equivalent limits in the selected angle unit. This numerical policy reserves 30 decimal places of absolute phase resolution and four guard digits from 64 significant-digit arithmetic. It does not guarantee relative accuracy near zeros or define a physical limit. Larger angles must first be reduced by full turns.
- The sine field retains the captured rounding to seven decimal places. A tiny nonzero sine may therefore display 0; the calculated value shown beneath that field preserves the nonzero result. Angle and other fields using relative rounding switch to scientific notation when fixed decimal formatting cannot meet their precision.
Common questions
What is the sine of 30 degrees?
It is 1/2.
Does inverse sine give every matching angle?
No. It returns the principal inverse from −90° to 90°. Periodic and supplementary solutions need separate selection.
Why does a tiny sine display zero?
The sine field rounds to seven decimal places, as in the captured model. Its calculated-value detail preserves the underlying nonzero result.
Why must I reduce an extremely large angle?
At huge magnitudes, finite precision can lose the remainder of a full turn and produce the wrong sine. The calculator rejects angles outside its phase-support range; enter an equivalent angle after reducing by 360 degrees or 2π radians.
Which number expressions can I enter?
Enter a finite decimal or scientific-notation number. Exact finite-decimal arithmetic such as 1/8 or 1−1 is supported. Repeating fractions such as 1/3, expressions that lose digits, functions, symbols and other bases are held; enter a decimal value instead. Computational proof limits are stated in the reference note. The ambiguous expression 0^0 is rejected.
How small can a nonzero input be?
The canonical radian angle and a supplied sine value must have magnitude at least10^-10000 before trigonometric work. Smaller nonzero values are held with an explanation to keep computation bounded. Zero and all retained units remain supported;1e-9000 is within the working range in every angle unit. Invalid unit conversions never silently become zero.
References
Translate the captured literal sine special-angle table to declarative numeric membership and conditioned exact-value notes. Recognition allows1e-60degree for internal unit-conversion roundoff; it is not a proof that rounded measurements exactly equal a special angle. Use strict inverse input limits −1≤s≤1. Require |angle|<10^30 radians as a numerical phase-support budget derived from 64 significant digits, 30 absolute phase decimal places and four guard digits; larger finite angles must be reduced before calculation. This is an absolute phase policy, not relative accuracy near zeros or a physical domain limit. Preserve captured seven-place sine display and relative-rounding fallback for eligible fields. Replace undeclared string exports only through whole-source/block-pinned special-angle notes. Inputs accept finite decimal/scientific literals and provably exact finite-decimal arithmetic (+, −, ×, ÷ and bounded integer powers). Lossy, repeating, function, symbolic and other-base expressions are rejected: enter a decimal value instead. The ambiguous expression 0^0 is rejected. Proof work is bounded to 20000 characters/coefficient digits, 512 nodes, depth64, 10000 aligned decimal places or absolute integer exponent, and one million Euclidean digit-work units; these are computational limits, not physical domains. Only captured editable controls are accepted as manual/API inputs. Zero-preserving angle unit conversions reject overflow, underflow or unavailable conversions. Before trigonometric work, a nonzero canonical radian angle or supplied sine value must have absolute magnitude at least10^-10000; zero remains supported. This is a computational work/representation budget to avoid extreme-exponent kernel stalls, not a physical domain boundary. Unit switches and inverse display conversions never silently replace a nonzero value with zero.