SUMMARY
The Laplace transform of the function f(t) = sin(4t + π/3)u(t) can be derived using the time-shift property and trigonometric identities. The transformation simplifies to f(t) = [sin(4t)cos(π/3) + cos(4t)sin(π/3)]u(t). The key to solving this problem lies in applying the Laplace transform properties correctly, particularly for sinusoidal functions and their phase shifts.
PREREQUISITES
- Understanding of Laplace transforms and their definitions.
- Familiarity with trigonometric identities, specifically sine and cosine addition formulas.
- Knowledge of the time-shift property in Laplace transforms.
- Basic calculus skills, particularly in handling piecewise functions and unit step functions.
NEXT STEPS
- Study the time-shift property of Laplace transforms in detail.
- Learn about trigonometric identities and their applications in signal processing.
- Explore the properties of the Laplace transform for sinusoidal functions.
- Practice solving Laplace transforms of phase-shifted functions with various frequencies.
USEFUL FOR
Students in engineering or mathematics, particularly those studying control systems or signal processing, will benefit from this discussion. It is also useful for educators teaching Laplace transforms and their applications in real-world scenarios.