Double Angle Formula Calculator — Step-by-Step
Enter any angle θ to see sin(2θ), cos(2θ) and tan(2θ) calculated step by step, with each formula labelled and all three equivalent forms of cos(2θ) compared.
Angle unit
Result from applying sin(2θ) = 2 sin θ cos θ
Step 1 — compute sin θ and cos θ
Step 2 — sin(2θ) = 2 · sin θ · cos θ
Step 3 — cos(2θ) = cos²θ − sin²θ
Step 4 — tan(2θ) = 2 tan θ / (1 − tan²θ)
- 1
Convert to radians
45 × π ÷ 180 = 0.785398 - 2
sin θ
sin(0.7854) = 0.707107 - 3
cos θ
cos(0.7854) = 0.707107 - 4
sin(2θ) = 2 × sin θ × cos θ
2 × 0.707107 × 0.707107 = 1
How does this calculator work?
The double-angle formulas — sin(2θ) = 2 sin θ cos θ, cos(2θ) = cos²θ − sin²θ, and tan(2θ) = 2 tan θ / (1 − tan²θ) — follow from the angle-addition identities with A = B = θ. This calculator shows the substitution step by step and confirms all three equivalent forms of cos(2θ) give the same result.
Formula
How this is calculated
The double-angle formulas are derived from the angle-addition identities by setting both angles equal to θ. For sine: sin(θ + θ) = sin θ cos θ + cos θ sin θ = 2 sin θ cos θ. For cosine: cos(θ + θ) = cos θ cos θ − sin θ sin θ = cos²θ − sin²θ. Substituting the Pythagorean identity sin²θ + cos²θ = 1 yields two more equivalent forms: cos(2θ) = 2cos²θ − 1 and cos(2θ) = 1 − 2sin²θ.
All three cos(2θ) forms give identical numerical results — the table confirms this. The step panel shows the actual substitution in order, so you can follow the arithmetic from the original values of sin θ and cos θ through to the final answers. Each form is useful in different contexts: cos²θ − sin²θ is the most direct, 2cos²θ − 1 is handy when only cosine is known, and 1 − 2sin²θ when only sine is known.
Degree inputs are converted to radians before evaluation (θ_rad = θ_deg × π / 180). tan(2θ) is undefined when tan²θ = 1, i.e., at 45° + 90°k, because the denominator 1 − tan²θ equals zero there.
Frequently asked questions
All three follow from the basic form cos²θ − sin²θ by applying the identity sin²θ + cos²θ = 1 to eliminate either sin²θ (giving 2cos²θ − 1) or cos²θ (giving 1 − 2sin²θ). They are algebraically identical; you choose the form that is most convenient given what you already know.
They come from the angle-addition formulas sin(A + B) and cos(A + B) by setting A = B = θ. That substitution immediately gives sin(2θ) = 2 sin θ cos θ and cos(2θ) = cos²θ − sin²θ.
Yes — the stat grid shows all three forms and they all produce the same decimal to six places. Any small differences beyond about the 15th decimal place are floating-point rounding noise, not mathematical disagreement.
TG we-Calculate Editorial Team. (2026). Double Angle Formula Calculator — Step-by-Step [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/double-angle-formula-calculator
TG we-Calculate Editorial Team. "Double Angle Formula Calculator — Step-by-Step." TG we-Calculate. 2026. https://we-calculate.com/calculator/double-angle-formula-calculator.
TG we-Calculate Editorial Team, "Double Angle Formula Calculator — Step-by-Step," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/double-angle-formula-calculator
@misc{wecalculate_double_angle_formula_calculator, title = {Double Angle Formula Calculator — Step-by-Step}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/double-angle-formula-calculator}}, year = {2026}, note = {TG we-Calculate} }
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