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 the From Angle tab whenever you have θ itself, in degrees or radians. Use the From sin/cos tab when a problem hands you sinθ and cosθ directly — common in vector and triangle problems where θ was never stated as a number. Both tabs produce the same sin2θ/cos2θ/tan2θ values when they describe the same angle; the calculator flags sin/cos pairs that don't satisfy sin²θ + cos²θ = 1, since that means they can't both come from a real angle.
Half-angle identities and when tangent is undefined
The Half Angle tab runs the relationship in reverse: given θ, it returns sin(θ/2), cos(θ/2), and tan(θ/2). Watch for undefined results — tanθ is undefined at θ = 90° and 270° (where cosθ = 0), and tan(2θ) is separately undefined at θ = 45° and 135° (where 1 − tan²θ = 0), because 2θ then lands exactly on 90° or 270°. The calculator displays 'undefined' rather than an error or infinity symbol in these cases.