Chapter 14, Problem 1E

Determine the conjugate of each of the following: (a) 8 − j; (b) 8e⁻⁹t; (c) 22.5; (d) 4e^j9; (e) j2e^−j11.

To determine

The conjugate of the given expression.

Answer to Problem 1E

The conjugate of the given expression is $\underset{\_}{8+j}$.

Explanation of Solution

Given data:

The expression is,

$K_1=8-j$

Calculation:

Since, the complex conjugate of any number is obtained by replacing all occurrences of $j$ with $-j$ hence the conjugate of given expression is written as,

$K_1{}^{\ast }=8+j$

Conclusion:

Therefore, the conjugate of the given expression is $\underset{\_}{8+j}$.

To determine

The conjugate of the given expression.

Answer to Problem 1E

The conjugate of the given expression is $\underset{\_}{8e^{-9}t}$.

Explanation of Solution

Given data:

The expression is,

$K_1=8e^{-9}t$

Calculation:

Since, in the given expression there is no occurrence of $j$ hence the conjugate of given expression is written as,

$K_1{}^{\ast }=8e^{-9}t$

Conclusion:

Therefore, the conjugate of the given expression is $\underset{\_}{8e^{-9}t}$.

To determine

The conjugate of the given expression.

Answer to Problem 1E

The conjugate of the given expression is $\underset{\_}{22.5}$.

Explanation of Solution

Given data:

The expression is,

$K_1=22.5$

Calculation:

Since, in the given expression there is no occurrence of $j$ hence the conjugate of given expression is written as,

$K_1{}^{\ast }=22.5$

Conclusion:

Therefore, the conjugate of the given expression is $\underset{\_}{22.5}$.

To determine

The conjugate of the given expression.

Answer to Problem 1E

The conjugate of the given expression is $\underset{\_}{4e^{-j9}}$.

Explanation of Solution

Given data:

The expression is,

$K_1=4e^{j9}$

Calculation:

The general form for expression is written as,

$e^{j\theta }=\cos \theta +j\sin \theta$

The given expression is written as,

$K_1=4\left(\cos 9+j\sin 9\right)$

Since, the complex conjugate of any number is obtained by replacing all occurrences of $j$ with $-j$ hence the conjugate of given expression is written as,

$K_1{}^{\ast }=4\left(\cos 9-j\sin 9\right)$

The conjugate of expression is written as,

$K_1{}^{\ast }=4e^{-j9}$

Conclusion:

Therefore, the conjugate of the given expression is $\underset{\_}{4e^{-j9}}$.

To determine

The conjugate of the given expression.

Answer to Problem 1E

The conjugate of the given expression is $\underset{\_}{-j2e^{j11}}$.

Explanation of Solution

Given data:

The expression is,

$K_1=j2e^{-j11}$

Calculation:

The general form for expression is written as,

$e^{j\theta }=\cos \theta +j\sin \theta$

The given expression is written as,

$K_1=j2\left(\cos \left(-11\right)+j\sin \left(-11\right)\right)$

Since, the complex conjugate of any number is obtained by replacing all occurrences of $j$ with $-j$ hence the conjugate of given expression is written as,

$\begin{array}{c}{K}_1{}^{\ast }=-j2\left(\cos \left(-11\right)-j\sin \left(-11\right)\right)\\ =-j2\left(\cos \left(11\right)+j\sin \left(11\right)\right)\end{array}$

The conjugate of expression is written as,

$K_1{}^{\ast }=-j2e^{j11}$

Conclusion:

Therefore, the conjugate of the given expression is $\underset{\_}{-j2e^{j11}}$.