Free · Trigonometry
Double-Angle Sine, Cosine and Tangent
Evaluate a doubled-angle trigonometric function or solve its principal inverse, with units, exact special-angle references and optional working.
- Formula & worked example
- Private in your browser
- No signup
Calculator inputs
Loading calculator…
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
At θ=15°, the doubled sine is sin30°=1/2.
sin2θ=2sinθcosθ; cos2θ=1−2sin²θ; tan2θ=2tanθ/(1−tan²θ), where defined.
Worked example
Enter these known values and leave the other values blank.
- Angle θ
- 15 deg
- Sine of angle 2θ
- 0.5
Assumptions and limitations
- The function selector retains doubled sine, cosine and tangent. The step toggle retains the selected identity, substitution, intermediate numerical or exact reference values, and final result.
- Inverse branches are the captured principal choices: asin(s)/2, asin(sqrt((1−c)/2)), and atan(t)/2. They do not enumerate all periodic solutions. Sine and cosine inverse values must lie within [−1,1].
- Doubled tangent is undefined at cosine-zero poles. For numerical support, tangent mode also requires |cos(2θ)|>10^-10; closer values are held with an explanation rather than returning an unreliable large number. The original undefined-step note remains retained.
- Exact reference notes recognize the retained literal standard-angle table modulo360° within10^-60degrees of internal conversion roundoff. They do not replace the numerical calculation or prove exactness of rounded measurements.
- Doubled-function output controls retain five decimal places, with the underlying calculated detail beneath each result. Captured value placeholders in the working notes retain the arithmetic values rather than silently changing their precision. A tiny nonzero result may display0 in the output control. The explanation uses the actual angle rather than folding it through arcsine.
- All three angle units, captured editable inverse controls, locks, clear and reset remain available. Require |2θ|<10^30 radians as an absolute phase-support budget; reduce larger angles first. This is not a physical domain limit or a relative-accuracy guarantee near zeros.
- The captured result group opens after an angle is known. To use an inverse in the workspace, enter an initial angle, then edit the displayed result; edit priority replaces the older angle with the principal inverse. The API also accepts a result directly.
Common questions
What is the double-angle sine identity?
sin(2θ)=2sinθcosθ.
Does the inverse give every angle?
No. The retained inverse branch gives one principal angle. Periodic alternatives must be selected separately.
Which expressions are supported?
Enter decimal or scientific numbers or provably exact finite-decimal arithmetic such as1/8. Lossy, repeating, function and symbolic expressions are held: enter a decimal value instead.0^0 is rejected as ambiguous. The same bounded input proof used by the first three models applies.
References
Translate the captured fixed special-angle tables into declarative numeric predicates; recognize within1e-60degree internal conversion resolution. Use the actual angle in radians in the explanation, rather than asin(sinθ), which folds120°to60°. Retain all three direct doubled-function formulas and their captured principal inverse branches. Use the equivalent principal acos(c)/2 cosine inverse to avoid losing endpoint differences in 2−2c or 1−c near ±1. Require exact closed inverse sine/cosine bounds. Preserve all controls and step chunks through the whole-source/block-pinned literal-note translation; route the undefined doubled-tangent explanation by cos2 alone, make exact-surd steps nonshadowing, and keep actual finite zero/nonzero routing exact. Require |2θ|<10^30 radians for the numerical phase budget, and |cos(2θ)|>10^-10 in tangent mode so the retained 10^-30 absolute phase budget is not amplified beyond the five-place output support. Near-pole tangent values are safely held, including unresolved exact poles. Inputs use the same source-bound exact finite-decimal arithmetic contract; hidden and forced inputs are rejected. Where tan(θ) itself is undefined or unresolved within |cosθ|≤10^-10, retain the original quotient steps conditionally and show a direct doubled-angle sine/cosine explanation instead; do not present a spurious enormous intermediate tangent. Apply zero-preserving angle conversion rejection and require nonzero canonical radian inputs or inverse supplied values to have magnitude at least10^-10000, matching the bounded trig working budget; zero remains allowed.