Laplace Transformation
Question
Find \(\mathcal{L}\{f(t)\}\) by first using a trigonometric identity. Write your answer as a function of \(s\), where \(f(t) = \cos^2(t)\). The function \(f(t)\) is given by: \[ f(t) = \cos^2(t) \]
Answer: To find the Laplace transform of \(f(t) = \cos^2(t)\), you can use the following trigonometric identity: \[\cos^2(t) = \frac{1 + \cos(2t)}{2}.\] Now, you can express \(f(t)\) in terms of this identity: \[f(t) = \frac{1 + \cos(2t)}{2}.\] Now, you can take the Laplace transform of each term separately. The Laplace transform of a constant is simply the constant multiplied by the Laplace transform of 1, which is \(1/s\). The Laplace transform of \(\cos(2t)\) is \(\frac{s}{s^2 + 4}\) (you can find this in Laplace transform tables or derive it using trigonometric identities). So, applying the Laplace transform to each term: \[\mathcal{L}\{f(t)\} = \frac{1}{2} \cdot \frac{1}{s} + \frac{1}{2} \cdot \frac{s}{s^2 + 4}.\] Now, you can simplify this expression further if needed.
Find \(\mathcal{L}\{f(t)\}\) by first using a trigonometric identity. Write your answer as a function of \(s\), where \(f(t) = \cos^2(t)\). The function \(f(t)\) is given by: \[ f(t) = \cos^2(t) \]
Answer: To find the Laplace transform of \(f(t) = \cos^2(t)\), you can use the following trigonometric identity: \[\cos^2(t) = \frac{1 + \cos(2t)}{2}.\] Now, you can express \(f(t)\) in terms of this identity: \[f(t) = \frac{1 + \cos(2t)}{2}.\] Now, you can take the Laplace transform of each term separately. The Laplace transform of a constant is simply the constant multiplied by the Laplace transform of 1, which is \(1/s\). The Laplace transform of \(\cos(2t)\) is \(\frac{s}{s^2 + 4}\) (you can find this in Laplace transform tables or derive it using trigonometric identities). So, applying the Laplace transform to each term: \[\mathcal{L}\{f(t)\} = \frac{1}{2} \cdot \frac{1}{s} + \frac{1}{2} \cdot \frac{s}{s^2 + 4}.\] Now, you can simplify this expression further if needed.
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