How to Find the Laplace Transform of sin(3t)cos(2t)

TL;DR

The Laplace transform of sin(3t)cos(2t) is computed using the identity sin(a)cos(b) = 1/2(sin(a+b) + sin(a-b)), resulting in 1/2sin(5t) + 1/2sin(t). The final Laplace transform expression is 1/2*(1/(s^2 + 25)) + 1/2*(1/(s^2 + 1)).

Transcript

hey what's up YouTube and this problem we have to find the Laplace transform of the sine of 3t times the cosine of 2t solution so when you see something like this you should recall a trig identity so if you have the sine of a time's the cosine of B this is equal to so it's one half parenthesis and then it's sine of a plus B plus sine of a minus B s... Read More

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.

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(a-b).

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(a-b).

• 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)).