laplace transform of sin(t)cos(t) vs laplace transform sin(t)*cos(t)

TL;DR

The Laplace transform of sine T times cosine T can be simplified using the convolution theorem and trigonometric identities.

Transcript

okay I'll to the Austrian questions on the spot the first one the Laplace transform of sine T times cosine T and the second one the Laplace transform of sine T start cosine T and this right here is the convolution anyway which one do you think it's easier well these questions can be easy what they can be hard the hardware is that we don't know the ... Read More

Key Insights

• 🦖 The Laplace transform of sine T times cosine T can be simplified by using the convolution theorem and trigonometric identities.
• 🕴️ The Laplace transform of sine T is 1/(s^2 + 1), and the Laplace transform of cosine T is s/(s^2 + 1).
• 🕴️ The Laplace transform of sine 2T is 2/(s^2 + 4).
• 🦖 By multiplying the Laplace transforms of sine T and cosine T, we can find the Laplace transform of sine T times cosine T as s/(s^2 + 1)^2.
• 👨‍💼 The Laplace transform of sine T times cosine T can be further simplified by applying the double angle identity for sine.

Questions & Answers

Q: How can the Laplace transform of sine T times cosine T be simplified?

The Laplace transform of sine T times cosine T can be simplified using the convolution theorem. Instead of using the integral definition, we can use the Laplace transforms of sine T and cosine T separately. The Laplace transform of sine T is 1/(s^2 + 1), and the Laplace transform of cosine T is s/(s^2 + 1). By multiplying these transforms, we get (s/(s^2 + 1))*(1/(s^2 + 1)) = s/(s^2 + 1)^2.

Q: Can the Laplace transform of sine T times cosine T be further simplified?

Yes, the Laplace transform of sine T times cosine T can be further simplified using trigonometric identities. By applying the double angle identity for sine, we can rewrite sine 2T as (1/2)sin T * cos T. We can then substitute this expression back into the Laplace transform, resulting in (1/2)(2/(s^2 + 4)) = 1/(s^2 + 4).

Q: Why is there a 1/2 in front of the Laplace transform of sine 2T?

The 1/2 in front of the Laplace transform of sine 2T comes from the double angle identity. The double angle identity states that sine 2T can be expressed as (1/2)*sin T * cos T. Therefore, when we substitute this expression back into the Laplace transform, the 1/2 remains in front.

Q: What is the final Laplace transform of sine T times cosine T?

The final Laplace transform of sine T times cosine T is 1/(s^2 + 4). By applying the convolution theorem, simplifying using trigonometric identities, and evaluating the Laplace transform of sine 2T, we arrive at this expression.

Summary & Key Takeaways

• The Laplace transform of sine T times cosine T can be found using the convolution theorem and the Laplace transforms of sine T and cosine T.

• The Laplace transform of sine T times cosine T simplifies to 1/2 times the Laplace transform of sine 2T.

• The Laplace transform of sine 2T is 2/(s^2 + 4).