DE-BOOK Sine and Cosine, \sin(bt),\ \cos(bt)

Subsection 9.2.4 Sine and Cosine, $\sin (b t), \cos (b t)$

Now, let’s turn to the Laplace transforms of trigonometric functions, which frequently arise in systems involving oscillations or wave equations.

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Example 6.

L {\cos (3 t)} .

Solution. Solution

We start by applying the definition of the Laplace transform:

L {\cos (3 t)} = \int_{0}^{\infty} e^{- s t} \cdot \cos (3 t) d t .

Rather than directly integrating, we will use a modified Euler’s Formula to express cosine in terms of

e

\cos (3 t) = \frac{1}{2} (e^{3 i t} + e^{- 3 i t}) .

Substituting this into the integral gives:

\begin{aligned} L {\cos (3 t)} = & \frac{1}{2} \int_{0}^{\infty} e^{- s t} \frac{1}{2} (e^{3 i t} + e^{- 3 i t}) d t \\ = & \frac{1}{2} [\int_{0}^{\infty} e^{- s t} \cdot e^{3 i t} d t + \int_{0}^{\infty} e^{- s t} \cdot e^{- 3 i t} d t] \\ = & \frac{1}{2} [L {e^{3 i t}} + L {e^{- 3 i t}}] \\ = & \frac{1}{2} [\frac{1}{s - 3 i} + \frac{1}{s + 3 i}] (by L_{2}) \\ = & \frac{1}{2} [\frac{s + 3 i + s - 3 i}{(s - 3 i) (s + 3 i)}] \\ = & \frac{s}{s^{2} + 9} . \end{aligned}

Therefore, the Laplace transform of

\cos (3 t)

is:

L {\cos (3 t)} = \frac{s}{s^{2} + 9} .

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The sine function is handled in a similar way, as the next example shows.

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Example 7.

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Compute

L {\sin (- 4 t)} .

Solution. Solution

As with cosine, we begin with the definition of the Laplace transform,

L {\sin (- 4 t)} = \int_{0}^{\infty} e^{- s t} \cdot \sin (- 4 t) d t

and rewrite sine using Euler’s formula,

\sin (- 4 t) = \frac{e^{- 4 i t} - e^{4 i t}}{2 i} .

Substituting this into the integral, we get:

\begin{aligned} L {\sin (- 4 t)} = & \frac{1}{2 i} \int_{0}^{\infty} e^{- s t} (e^{- 4 i t} - e^{4 i t}) d t \\ = & \frac{1}{2 i} [\int_{0}^{\infty} e^{- (s + 4 i) t} d t - \int_{0}^{\infty} e^{- (s - 4 i) t} d t] \\ = & \frac{1}{2 i} [L {e^{- 4 i t}} - L {e^{4 i t}}] . \\ = & \frac{1}{2 i} [\frac{1}{s + 4 i} - \frac{1}{s - 4 i}] (by L_{2}) \\ = & \frac{1}{2 i} [\frac{s - 4 i - (s + 4 i)}{(s + 4 i) (s - 4 i)}] \\ = & \frac{4}{s^{2} + 16} . \end{aligned}

Thus, the Laplace transform of

\sin (- 4 t)

is:

L {\sin (- 4 t)} = \frac{4}{s^{2} + 16} .

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Both of these approaches can be generalized to show that the formula for the Laplace transforms of sine and cosine are given as follows:

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Common Laplace Transform (Sine, Cosine).

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$L_{4}$: $L {\sin (b t)} = \frac{b}{s^{2} + b^{2}}, s > 0$
$L_{5}$: $L {\cos (b t)} = \frac{s}{s^{2} + b^{2}}, s > 0$

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Reading Questions Check-Point Questions

1. $L {\sin (t)} =$ $\frac{}{s^{2} + 1}$ .

$L {\sin (t)} =$ $\frac{}{s^{2} + 1}$

$1$
Correct! The Laplace transform of $\sin (t)$ is $\frac{1}{s^{2} + 1} .$
$s$
No, the correct numerator should be $1,$ not $s .$
$b$
No, the correct numerator should be $1,$ not $b .$
$s^{2}$
No, the correct numerator should be $1,$ not $s^{2} .$

2. $L {\cos (- 2 t)} =$ $\frac{}{s^{2} + 4}$ .

$L {\cos (- 2 t)} =$ $\frac{}{s^{2} + 4}$

$2$
No, try again.
$4$
No, try again.
$s$
Correct! The Laplace transform of $\cos (- 2 t)$ is $\frac{s}{s^{2} + 4} .$
$2 s$
No, try again.

3. $L {} =$ $\frac{s}{s^{2} + \frac{1}{4}}$ .

$L {} =$ $\frac{s}{s^{2} + \frac{1}{4}}$

$\sin (\frac{1}{2} t)$
No, try again.
$\cos (\frac{1}{2} t)$
No, try again.
$\cos (2 t)$
Correct! The Laplace transform of $\cos (2 t)$ is $\frac{s}{s^{2} + \frac{1}{4}} .$
$\sin (2 t)$
No, try again.

4. $L {\sin (5 t)} =$ ?

$L {\sin (5 t)} =$ ?

$\frac{5}{s^{2} + 25}$
Correct! The Laplace transform of $\sin (5 t)$ is $\frac{5}{s^{2} + 25} .$
$\frac{s}{s^{2} + 25}$
Incorrect. This is the Laplace transform of $\cos (5 t),$ not $\sin (5 t) .$
$\frac{1}{s^{2} + 25}$
Incorrect. The correct numerator is $5,$ not $1 .$
$\frac{5}{s^{2} + 5^{2}}$
Incorrect. While $25$ is $5^{2},$ the answer should simplify to $\frac{5}{s^{2} + 25} .$

5. $L {\cos (- 3 t)} =$ ?

$L {\cos (- 3 t)} =$ ?

$\frac{3}{s^{2} + 9}$
Incorrect. The correct numerator should be $\dss,$ not $3 .$
$\frac{s}{s^{2} - 9}$
Incorrect. The denominator should be $9,$ not $- 9 .$
$\frac{s}{s^{2} + 9}$
Correct! The Laplace transform of $\cos (- 3 t)$ is $\frac{s}{s^{2} + 9} .$
$\frac{3}{s^{2} - 9}$
Incorrect. The correct numerator should be $\dss,$ not $3 .$

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Subsection 9.2.4 Sine and Cosine, sin⁡(bt), cos⁡(bt)

Example 6.

Example 7.

Common Laplace Transform (Sine, Cosine).

Reading Questions Check-Point Questions

1. L{sin⁡(t)}=Xs2+1.

2. L{cos⁡(−2t)}=Xs2+4.

3. L{X}=ss2+14.