How this calculator works
Replacing both angles in a sum identity by θ gives a double-angle identity. The sine formula uses both sine and cosine, while cosine has three equivalent forms. A zero cosine of 2θ makes tangent undefined. Reverse mode selects one principal value of 2θ before dividing by two; periodic and supplementary solutions are not uniquely determined by a single ratio.
Formula
sin(2θ)=2 sin θ cos θ; cos(2θ)=cos²θ-sin²θ=1-2sin²θ; tan(2θ)=2tan θ/(1-tan²θ), where defined.
Worked example
For θ = 30°, sin 60° = √3/2, cos 60° = 1/2 and tan 60° = √3. For θ = 45°, sin 90° = 1 and tangent is undefined.
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Frequently asked questions
Is sin(2θ) twice sin θ?
Generally no. It equals 2 sin θ cos θ; the cosine factor matters.
Why does reverse mode return only one θ?
A function value does not specify the revolution or quadrant. The tool explicitly chooses its principal branch.
Can tangent be undefined although tan θ exists?
Yes. At θ = 45°, tan θ = 1 but tan 90° is undefined because the double-angle denominator is zero.