Asked by Kris
sin(2x)
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Common useful facts about sin(2x):
- Double-angle identity: sin(2x) = 2 sin x cos x.
- In terms of tangent: sin(2x) = 2 tan x / (1 + tan^2 x) (where cos x ≠ 0).
- Complex exponential: sin(2x) = (e^{i2x} − e^{-i2x})/(2i).
Calculus:
- Derivative: d/dx[sin(2x)] = 2 cos(2x).
- Indefinite integral: ∫ sin(2x) dx = −(1/2) cos(2x) + C.
Properties:
- Domain: all real x. Range: [−1, 1].
- Period: π (since sin(2(x+π)) = sin(2x+2π) = sin(2x)).
- Zeros: sin(2x) = 0 ⇔ x = kπ/2, k ∈ Z.
- Max/min: maxima 1 at x = π/4 + kπ, minima −1 at x = 3π/4 + kπ.
Series expansion:
- sin(2x) = Σ_{n=0}^∞ (−1)^n (2x)^{2n+1}/(2n+1)!.
If you meant something else (plot, solve an equation, simplify in a particular context), tell me and I’ll do that.
- Double-angle identity: sin(2x) = 2 sin x cos x.
- In terms of tangent: sin(2x) = 2 tan x / (1 + tan^2 x) (where cos x ≠ 0).
- Complex exponential: sin(2x) = (e^{i2x} − e^{-i2x})/(2i).
Calculus:
- Derivative: d/dx[sin(2x)] = 2 cos(2x).
- Indefinite integral: ∫ sin(2x) dx = −(1/2) cos(2x) + C.
Properties:
- Domain: all real x. Range: [−1, 1].
- Period: π (since sin(2(x+π)) = sin(2x+2π) = sin(2x)).
- Zeros: sin(2x) = 0 ⇔ x = kπ/2, k ∈ Z.
- Max/min: maxima 1 at x = π/4 + kπ, minima −1 at x = 3π/4 + kπ.
Series expansion:
- sin(2x) = Σ_{n=0}^∞ (−1)^n (2x)^{2n+1}/(2n+1)!.
If you meant something else (plot, solve an equation, simplify in a particular context), tell me and I’ll do that.
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