Formula and Method for the Double Angle Calculator
The double-angle identities express sin(2θ), cos(2θ), and tan(2θ) in terms of sinθ, cosθ, and tanθ alone. They follow directly from the angle-sum formulas sin(a+b), cos(a+b), and tan(a+b) by setting a = b = θ: sin(2θ) = sin(θ+θ) = 2 sinθ cosθ, and cos(2θ) = cos(θ+θ) = cos²θ − sin²θ, which — using the Pythagorean identity sin²θ + cos²θ = 1 — can also be written as 2cos²θ − 1 or 1 − 2sin²θ. Dividing sin(2θ) by cos(2θ) and simplifying gives tan(2θ) = 2tanθ / (1 − tan²θ).
How the calculation works
Enter the angle θ and choose whether it is measured in degrees or radians. The calculator converts θ to radians (radians = degrees × π/180) if needed, doubles it to get 2θ, then evaluates sin(2θ), cos(2θ), and tan(2θ) directly. These match the values you would get from the identities above, since both approaches describe the same quantity two different ways. When cos(2θ) is zero — at 2θ = 90°, 270°, and so on — tan(2θ) is undefined, and the calculator reports that instead of a number.
Common mistakes
- Mixing degrees and radians: make sure the unit you select matches how you're thinking about the angle — 30° and 30 rad give very different double-angle results.
- Assuming cos(2θ) = 2cosθ: it does not; cosine does not distribute over doubling. You must use one of the three equivalent double-angle forms.
- Forgetting where tan(2θ) is undefined: the denominator 1 − tan²θ hits zero whenever θ = 45° + 90°n (tanθ = ±1), so tan(2θ) has no defined value at those angles.
Where double-angle identities are used
- Projectile motion: the range of a projectile launched at angle θ is R = v²sin(2θ)/g, which is maximized at θ = 45° because sin(2θ) peaks there.
- Calculus: the power-reduction identities cos²θ = (1+cos2θ)/2 and sin²θ = (1−cos2θ)/2 — rearrangements of the double-angle formulas — are the standard technique for integrating sin² and cos².
- Signal processing and AC circuits: doubling a phase angle shows up in harmonic and frequency-doubling analysis of periodic waveforms.
- Deriving other identities: half-angle and triple-angle formulas are obtained algebraically from the double-angle identities.