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Instant Calculation
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The double angle identities express sin(2θ), cos(2θ), and tan(2θ) purely in terms of the sine, cosine, or tangent of θ itself, so you never need to know 2θ directly or look it up on a unit circle. They're derived directly from the sum identities by setting both angles equal (sin(θ+θ) = sinθcosθ + cosθsinθ = 2sinθcosθ), which is why they're sometimes introduced as a special case rather than a separate rule. These identities are a staple of calculus (simplifying derivatives and integrals of trig expressions), physics (analyzing wave interference and oscillation energy, which often involves squared sine or cosine terms best rewritten with a double-angle form), and electrical engineering (deriving RMS power from a sinusoidal AC signal, since power is proportional to voltage squared). This calculator lets you start from either the base angle θ or a known trig ratio of θ.
How it works
Enter either the angle θ directly, or one known ratio (sin, cos, or tan of θ) plus which quadrant θ falls in — the quadrant is needed because a single ratio like sinθ = 0.6 could belong to two different angles with opposite-signed cosines. The calculator then applies sin(2θ) = 2sinθcosθ for the sine identity, and one of three equivalent forms for cosine — cos(2θ) = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ — all of which give the same numeric answer but are useful in different algebraic contexts. Tangent follows from tan(2θ) = 2tanθ/(1 − tan²θ), which is undefined whenever tanθ = ±1 (i.e., θ = 45° or 135°), since the denominator becomes zero there.
- Enter what do you know?.
- Enter angle θ (if known).
- Enter angle unit.
- Enter ratio value (if sin/cos/tan chosen).
- Enter quadrant of θ (for sin/cos/tan mode).
- Click Calculate to see your results.
Examples
θ = 30°
sin(2×30°) = sin(60°) = 0.866, matching the identity: 2×sin(30°)×cos(30°) = 2×0.5×0.866 = 0.866.
θ = 45°
cos(2×45°) = cos(90°) = 0, matching cos²(45°) − sin²(45°) = 0.5 − 0.5 = 0. Note tan(2×45°) = tan(90°) is undefined here, since 1 − tan²(45°) = 1 − 1 = 0.
θ = 15°
cos(2×15°) = cos(30°) ≈ 0.866, matching 2cos²(15°) − 1 ≈ 2(0.933) − 1 ≈ 0.866 — useful when 15° itself isn't a memorized reference angle but 30° is.
Common mistakes to avoid
- Forgetting that cos(2θ) has three equivalent forms and mixing terms from two different versions mid-calculation.
- Assuming sin(2θ) = 2sin(θ) — the correct identity is 2sinθcosθ, not a simple doubling.
- Picking the wrong sign for cosθ when only a ratio and no quadrant is given, which flips the sign of the final cos(2θ) or tan(2θ) result.
- Not checking for the undefined case in tan(2θ) when tanθ happens to equal ±1.