Determine the Laplace transform of: (a) cos(ωt+θ) (b) sin(ωt+θ)

Name: Problem 2, Chapter 15: Fundamentals of Electric Circuits
Uploaded: 2022-02-07T10:31:01-08:00
Duration: 2 min 59 s
Channel: Arpit Gupta
Description: Determine the Laplace transform of: (a) cos(ωt+θ) (b) sin(ωt+θ)

step 1 In this problem of Laplace transform we have to determine the laplace transform of say first thi

Determine the Laplace transform of: (a) cos(ωt+θ) (b)...

Key Concepts

Laplace Transform

The Laplace transform is an integral transform that converts a function of time into a function of a complex variable, usually denoted by s. It is defined by the integral L{f(t)} = ??? e^(–st) f(t) dt and is widely used to solve differential equations and analyze systems, particularly in engineering and physics.

Standard Laplace Transforms of Sine and Cosine

A fundamental part of Laplace transform theory is the existence of standard formulas for the sine and cosine functions. Specifically, the Laplace transform of cos(?t) is s/(s² + ?²) and that of sin(?t) is ?/(s² + ?²). These formulas provide a basis for handling more complex forms of these functions.

Phase Shift in Trigonometric Functions

When dealing with trigonometric functions that include a phase shift, such as cos(?t + ?) or sin(?t + ?), it is important to use trigonometric identities to express them in terms of standard sine and cosine functions. The identities cos(?t+?) = cos(?t)cos? – sin(?t)sin? and sin(?t+?) = sin(?t)cos? + cos(?t)sin? allow one to break down the phase-shifted functions into components with known Laplace transforms.

Linearity Property of the Laplace Transform

The Laplace transform is a linear operator, meaning that the transform of a linear combination of functions is equal to the same linear combination of their transforms. This property is crucial when applying the transform to expressions obtained from trigonometric identities, as it permits the Laplace transform to be computed term-by-term.

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