The double-angle formulas let you find sin(2θ), cos(2θ), and tan(2θ) from sinθ, cosθ, and tanθ alone — no need to compute 2θ or look up a new angle in a trig table. They come up constantly in calculus (simplifying integrals), physics (projectile range formulas), and any geometry problem where an angle gets doubled.
Where the identities come from
Every double-angle identity is a special case of the angle-addition formulas. Starting from sin(A + B) = sinA cosB + cosA sinB, setting A = B = θ gives sin(2θ) = 2 sinθ cosθ. The same substitution into cos(A + B) = cosA cosB − sinA sinB gives cos(2θ) = cos²θ − sin²θ. Because sin²θ + cos²θ = 1, that cosine form can be rewritten two more ways — 2cos²θ − 1 or 1 − 2sin²θ — which is why the calculator lists all three: pick whichever matches the value you already have.
From Angle vs. From sin/cos
Use tan(2θ) = sin(2θ)/cos(2θ). It is undefined only when cos(2θ)=0, at θ=45°+90°n. At θ=90°, tan θ is undefined but tan(2θ)=0. Supplied sine/cosine pairs must satisfy the unit-circle identity, or the explicit Normalize workflow must be selected.
Half-angle identities and when tangent is undefined
Use tan(2θ) = sin(2θ)/cos(2θ). It is undefined only when cos(2θ)=0, at θ=45°+90°n. At θ=90°, tan θ is undefined but tan(2θ)=0. Supplied sine/cosine pairs must satisfy the unit-circle identity, or the explicit Normalize workflow must be selected.
Use finite values with magnitude at most 1e50. Expressions support pi, sqrt(), fractions and arithmetic. Angles are limited to 1e12 degrees. Units are mathematical; geometry alone does not establish physical safety. Read the linked values, named formulas, diagram coordinate table and any branch or parameter comparison.
Save scenario stores version 2 inputs, workflow, units, precision and selected result on this device. Restore saved scenario replays that snapshot. Share uses the same state; Copy and CSV include the selected answer and all current derived values. Invalid inputs clear the result and disable result exports.