Question

Find ℒ{f(t)} by first using a trigonometric identity. (Write your answer as a function of s.)

Answer 1

Answer:

$L\{f(t)\}=\frac{8(\sqrt3s+1)}{s^2+1}$

Step-by-step explanation:

Given : $f(t)=16\cos (t-\frac{\pi}{6})$

To find : ℒ{f(t)} by first using a trigonometric identity ?

Solution :

First we solve the function,

$f(t)=16\cos (t-\frac{\pi}{6})$

Applying trigonometric identity, $\cos (A-B)=\cos A\cos B+\sin A\sin B$

$f(t)=16(\cos t\cos (\frac{\pi}{6})+\sin t\sin(\frac{\pi}{6})$

$f(t)=16(\frac{\sqrt3}{2}\cos t+\frac{1}{2}\sin t)$

$f(t)=\frac{16}{2}(\sqrt3\cos t+\sin t)$

$f(t)=8(\sqrt3\cos t+\sin t)$

We know, $L(\cos at)=\frac{s}{s^2+a^2}$ and $L(\sin at)=\frac{a}{s^2+a^2}$

Applying Laplace in function,

$L\{f(t)\}=8\sqrt3L(\cos t)+8L(\sin t)$

$L\{f(t)\}=8\sqrt3(\frac{s}{s^2+1})+8(\frac{1}{s^2+1})$

$L\{f(t)\}=\frac{8\sqrt3s+8}{s^2+1}$

$L\{f(t)\}=\frac{8(\sqrt3s+1)}{s^2+1}$

Therefore, The Laplace transformation is $L\{f(t)\}=\frac{8(\sqrt3s+1)}{s^2+1}$