Find the Laplace transform of
𝑡
sin
2
⁡
𝑡

hint: use an appropriate trigonometric identity

Question

Find the Laplace transform of  
𝑡
sin
2
⁡
𝑡

hint: use an appropriate trigonometric identity

BlackTom AI · Accepted Answer

The problem asks for the Laplace transform of a function involving t and a sine term; however, the provided answer options are missing, so I will derive the transform step by step to obtain a correct expression you can compare to any future options.

First, identify the function to transform. Interpreting the prompt as finding L{ t sin^2 t }, we can use a standard trigonometric identity to simplify: sin^2 t = (1 - cos 2t)/2. This allows us to write t sin^2 t = (t/2) - (t/2) cos 2t.

Next, apply linearity of the Laplace transform: L{ t sin^2 t } = L{ t/2 } - L{ (t/2) cos 2t } = (1/2) L{ t } - (1/2) L{ t cos 2t }.

We know two key transforms:
- L{ t } = 1/s^2 for Re(s) > 0.
- L{ t cos at } = (s^2 - a^2) / (s^2 + a^2)^2 for Re(s) > 0.

Using a = 2, compute L{ t cos 2t } = (s^2 - 4) / (s^2 + 4)^2.

Put these into the expression: L{ t sin^2 t } = (1/2)(1/s^2) - (1/2) * [(s^2 - 4) / (s^2 + 4)^2].

Therefore, the Laplace transform is L{ t sin^2 t } = 1/(2 s^2) - (s^2 - 4) / [2 (s^2 + 4)^2].

If you want to present it with a common denominator, you could write it as L{ t sin^2 t } = [ (s^2 + 4)^2 - s^2 (s^2 - 4) ] / [ 2 s^2 (s^2 + 4)^2 ], but the separated form above is often clearer for interpretation.

In summary, the derived transform is:
L{ t sin^2 t } = 1/(2 s^2) - (s^2 - 4) / [2 (s^2 + 4)^2].

MAP2302 L19 section 7.3

Find the Laplace transform of 𝑡 sin 2 ⁡ 𝑡 hint: use an appropriate trigonometric identity

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