Laplace Transform of sin(ωt)
The Laplace transformation of sin(t) is L{sin(t)} = 1/(s²+1), and more generally L{sin(ωt)} = ω/(s²+ω²) for Re(s) > 0. This fundamental transform pair represents oscillatory behavior in the s-domain with poles at s = ±jω on the imaginary axis. Compute sine transforms and visualize pole-zero plots at www.lapcalc.com.
Laplace Transformation of sin(t): Derivation and Formula
The Laplace transformation of sin(t) is derived from the integral L{sin(t)} = ∫₀^∞ e^(−st)sin(t)dt. Using integration by parts twice creates a self-referencing equation: the integral I satisfies I = (1/s²)(1 − I·0 + ... ), which resolves to I = 1/(s²+1). Alternatively, using Euler's formula sin(t) = (e^(jt) − e^(−jt))/(2j) and the exponential transform: L{sin(t)} = (1/2j)[1/(s−j) − 1/(s+j)] = (1/2j)·2j/(s²+1) = 1/(s²+1). Both methods confirm the clean result, with poles at s = ±j representing the unit angular frequency of sin(t).
Key Formulas
Laplace of sin(ωt): General Frequency Parameter
The Laplace of sin(ωt) generalizes to L{sin(ωt)} = ω/(s²+ω²), where ω is the angular frequency in radians per second. The numerator ω scales linearly with frequency, while the denominator s²+ω² places conjugate poles at s = ±jω. This means higher-frequency sine waves have poles farther from the origin on the imaginary axis. The transform is valid for Re(s) > 0, meaning the region of convergence is the entire right half-plane. Physically, the pure imaginary poles reflect that sin(ωt) neither grows nor decays—it oscillates indefinitely with constant amplitude.
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Open CalculatorLaplace Transform sin(ωt) with Damping and Shifting
Combining the Laplace transform sin(ωt) with the frequency shifting property gives damped sine transforms. L{e^(−at)sin(ωt)} = ω/((s+a)²+ω²) shifts the poles from s = ±jω to s = −a ± jω, moving them into the left half-plane when a > 0. This damped sinusoid represents the natural response of underdamped systems—oscillation at frequency ω decaying with time constant 1/a. The time-shifted version L{sin(ω(t−c))u(t−c)} = e^(−cs)ω/(s²+ω²) introduces a pure delay. These variations cover virtually all sinusoidal signals encountered in engineering practice.
Comparing sin(ωt) and cos(ωt) Laplace Transforms
The sine and cosine transforms form a complementary pair: L{sin(ωt)} = ω/(s²+ω²) and L{cos(ωt)} = s/(s²+ω²). Both share the same denominator (same poles, same frequency), but the sine has numerator ω (constant) while cosine has numerator s (a zero at origin). This difference reflects initial conditions: sin(0) = 0 while cos(0) = 1, confirmed by the initial value theorem lim(s→∞) s·[ω/(s²+ω²)] = 0 and lim(s→∞) s·[s/(s²+ω²)] = 1. The derivative relationship d/dt[sin(ωt)] = ω cos(ωt) maps to s·[ω/(s²+ω²)] − 0 = ωs/(s²+ω²) = ω·L{cos(ωt)}.
Applications of the Sine Transform in AC Circuit Analysis
The sine transform is central to AC circuit analysis in the Laplace domain. When a voltage source v(t) = V₀sin(ωt) drives a circuit, its transform V(s) = V₀ω/(s²+ω²) combines with the circuit's transfer function H(s) to give the output Y(s) = H(s)V₀ω/(s²+ω²). Partial fraction decomposition separates the response into transient terms (from H(s) poles) and steady-state terms (from the ±jω poles of the input). The steady-state amplitude is V₀|H(jω)| and the phase shift is ∠H(jω), connecting Laplace analysis directly to phasor methods used in AC steady-state analysis at www.lapcalc.com.
Related Topics in advanced laplace transform topics
Understanding laplace transformation of sint connects to several related concepts: laplace of sin wt, and laplace transform sin wt. Each builds on the mathematical foundations covered in this guide.
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