What is the Laplace transform of sin(t)?

The Laplace transform of sin(t) is 1/(s²+1). More generally, L{sin(ωt)} = ω/(s²+ω²). The numerator equals the frequency parameter ω, and the denominator s²+ω² has roots at s = ±jω representing the oscillation frequency.

How are the Laplace transforms of sin and cos related?

They share the same denominator s²+a² (same poles/frequency) but differ in the numerator: s for cosine, a for sine. The derivative property connects them: since (d/dt)sin(at) = a·cos(at), multiplying the sine transform by s and adjusting gives the cosine transform. This reflects the 90° phase difference between sin and cos.

What is the Laplace transform of a damped cosine?

The Laplace transform of e^(−bt)cos(at) is (s+b)/((s+b)²+a²), obtained by applying the frequency shifting property s → s+b to the standard cosine transform. This represents an oscillation at frequency a that decays exponentially with rate b, typical of underdamped physical systems.

Laplace Transform of cos(at) and sin(at)

Quick Answer

The Laplace transformation of cos(at) is L{cos(at)} = s/(s²+a²), and the Laplace transform of sin(at) is L{sin(at)} = a/(s²+a²), both valid for Re(s) > 0. These trigonometric transform pairs are essential for analyzing oscillatory systems in circuits, control theory, and vibration analysis. Compute cos and sin Laplace transforms instantly at www.lapcalc.com.

Calculate this instantly on LAPLACE Calculator →

Laplace Transformation of cos(at): Derivation and Formula

The Laplace transformation of cos(at) is derived from the definition: L{cos(at)} = ∫₀^∞ e^(−st)cos(at)dt. Using integration by parts twice, or applying Euler's formula cos(at) = (e^(jat) + e^(−jat))/2 and the exponential transform, the result is L{cos(at)} = s/(s²+a²) for Re(s) > 0. The numerator s indicates that the cosine transform has a zero at s = 0, distinguishing it from the sine transform. The poles at s = ±ja correspond to the oscillation frequency a rad/s. This pair appears in every RLC circuit analysis and mechanical vibration problem involving undamped oscillation.

Key Formulas

Laplace Transform of sin(t) and sin(ωt)

The Laplace transform of sin(t) is L{sin(t)} = 1/(s²+1), and more generally L{sin(ωt)} = ω/(s²+ω²). The derivation parallels the cosine case: using Euler's formula sin(ωt) = (e^(jωt) − e^(−jωt))/(2j), each exponential transforms to 1/(s−jω) and 1/(s+jω), and combining gives ω/(s²+ω²). Notice that the sine transform has no zero at s = 0 (numerator is just ω), while cosine has a zero there (numerator is s). This difference reflects the fact that sin(0) = 0 while cos(0) = 1, connected through the initial value theorem.

Compute laplace transformation of cosat Instantly

Get step-by-step solutions with AI-powered explanations. Free for basic computations.

Open Calculator

Cos Laplace and Damped Cosine Transforms

The basic cos Laplace pair extends naturally to damped oscillations using the frequency shifting property. For L{e^(−bt)cos(at)} = (s+b)/((s+b)²+a²), the shift s → s+b moves the poles from ±ja to −b ± ja, placing them in the left half-plane when b > 0. This damped cosine represents the natural response of underdamped systems—RLC circuits with resistance, mass-spring-damper systems with friction. The parameter b controls decay rate while a determines oscillation frequency. Similarly, L{e^(−bt)sin(at)} = a/((s+b)²+a²) gives the damped sine, often appearing alongside the damped cosine in system responses.

Laplace Transform of Cosine: Applications in Circuit Analysis

The Laplace transform of cosine is central to AC circuit analysis in the s-domain. When a cosine voltage source V₀cos(ωt) drives an RLC circuit, the source transforms to V₀s/(s²+ω²). The circuit's response Y(s) = H(s)·V₀s/(s²+ω²) includes both transient poles (from H(s)) and steady-state poles at s = ±jω (from the cosine input). Partial fraction decomposition separates these: the transient terms decay exponentially while the forced terms give the steady-state sinusoidal response at frequency ω with amplitude |H(jω)|·V₀ and phase ∠H(jω). This analysis is readily performed at www.lapcalc.com.

Laplace Transform of cos(t) vs sin(t): Comparison and Relationship

Comparing L{cos(at)} = s/(s²+a²) and L{sin(at)} = a/(s²+a²) reveals a fundamental relationship. Both share the same denominator s²+a², meaning they have identical poles at s = ±ja—the same natural frequency. The difference is the numerator: s for cosine versus a for sine. This connects to the derivative property: since d/dt[sin(at)] = a·cos(at), we have L{a·cos(at)} = s·L{sin(at)} − sin(0) = s·a/(s²+a²) = as/(s²+a²), confirming L{cos(at)} = s/(s²+a²). The sine and cosine transforms are thus related by multiplication by s/a, reflecting the 90° phase relationship between sin and cos.

Frequently Asked Questions

The Laplace transform of cos(at) is s/(s²+a²), valid for Re(s) > 0. The poles at s = ±ja correspond to the oscillation frequency, and the zero at s = 0 distinguishes the cosine transform from the sine transform a/(s²+a²).

Master Your Engineering Math

Join thousands of students and engineers using LAPLACE Calculator for instant, step-by-step solutions.

Start Calculating Free →