Laplace Transform of sin(3t)cos(2t) using Trigonometric Identities  Summary and Q&A
TL;DR
The Laplace transform of the sine of 3t times the cosine of 2t is equal to 1/2 times sine of 5t plus 1/2 times sine of t.
Key Insights
 😑 Trigonometric identities are useful in simplifying expressions involving sine and cosine functions.
 🍹 The trigonometric identity used in this problem is the product to sum identity.
 👨💼 The Laplace transform can be taken using specific formulas for the sine and cosine functions.
 💁 The Laplace transform of sine has the form 1/(s^2+k^2), where k is the coefficient of t in the sine function.
 💁 The Laplace transform of cosine has the form 1/(s^2+k^2), where k is the coefficient of t in the cosine function.
 🍹 The Laplace transform of a sum of functions is equal to the sum of their individual Laplace transforms.
 🔨 The Laplace transform is a mathematical tool used in solving differential equations and analyzing systems in engineering and physics.
Transcript
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Questions & Answers
Q: What is the trigonometric identity used to solve this problem?
The trigonometric identity used is: sine(a) times cosine(b) is equal to 1/2 times sine(a+b) plus 1/2 times sine(ab).
Q: How do you apply the trig identity to find the Laplace transform in this problem?
Apply the identity by substituting the values of a and b from the given problem, which are 3t and 2t respectively. This gives us 1/2 times sine(5t) plus 1/2 times sine(t).
Q: What is the Laplace transform formula for sine?
The Laplace transform formula for sine is 1/(s^2+k^2), where k is the coefficient of t in the sine function. In this problem, the Laplace transform of sine(5t) is 1/(s^2+5^2) and the Laplace transform of sine(t) is 1/(s^2+1^2).
Q: What is the final Laplace transform of the given expression?
The final Laplace transform is 1/2 times (1/(s^2+5^2)) plus 1/2 times (1/(s^2+1^2)).
Summary & Key Takeaways

To find the Laplace transform of sine 3t times cosine 2t, use the trigonometric identity: sine(a) times cosine(b) is equal to 1/2 times sine(a+b) plus 1/2 times sine(ab).

Apply the identity to the given problem: sine(3t) times cosine(2t) becomes 1/2 times sine(5t) plus 1/2 times sine(t).

Take the Laplace transforms of each term: 1/2 times Laplace of sine(5t) plus 1/2 times Laplace of sine(t).

Use the Laplace transform formula for sine: 1/2 times (1/(s^2+5^2)) plus 1/2 times (1/(s^2+1^2)).