What is the Laplace transform of sin(2t)?
Understand the Problem
The question is asking for the Laplace transform of the function sin(2t). The Laplace transform is a technique used to convert a function of time into a function of a complex variable, typically used in engineering and mathematics to solve differential equations.
Answer
The Laplace transform of \( \sin(2t) \) is \( \frac{2}{s^2 + 4} \).
Answer for screen readers
The Laplace transform of ( \sin(2t) ) is given by $$ \frac{2}{s^2 + 4} $$.
Steps to Solve
- Recall the Laplace Transform Formula for Sine Function
The Laplace transform of the sine function is given by the formula: $$ \mathcal{L}{ \sin(at) } = \frac{a}{s^2 + a^2} $$
Here, (a) is the coefficient of (t) in the sine function, and (s) is the complex variable.
- Identify the Parameters
For the function ( \sin(2t) ), we see that ( a = 2 ).
- Substitute into the Laplace Transform Formula
Plugging (a = 2) into the formula, we get: $$ \mathcal{L}{ \sin(2t) } = \frac{2}{s^2 + 2^2} $$
- Simplify the Expression
Now simplify the expression: $$ \mathcal{L}{ \sin(2t) } = \frac{2}{s^2 + 4} $$
This gives us the Laplace transform of the function ( \sin(2t) ).
The Laplace transform of ( \sin(2t) ) is given by $$ \frac{2}{s^2 + 4} $$.
More Information
The Laplace transform is widely used in engineering for analyzing linear time-invariant systems. It transforms differential equations into algebraic equations, making them easier to solve. The sine function represents oscillatory motion which is important in physics and control systems.
Tips
- Confusing the sine function with the cosine function. Make sure to use the correct formula for sine when applying the Laplace transform.
- Forgetting to square the coefficient (a) in the denominator. Always check your algebraic manipulation.
