Question

Using $\mathbf{a}\mathbf{=}\mathbf{1}\mathbf{\angle }{\mathbf{120}}^{\mathbf{\circ }}$, evaluate the following in rectangular form:

a. ${\mathbf{a}}^{\mathbf{10}}$

b. ${\mathbf{\left(}\mathbf{ja}\mathbf{\right)}}^{\mathbf{10}}$

c. ${\mathbf{(}\mathbf{1}\mathbf{-}\mathbf{a}\mathbf{)}}^{\mathbf{3}}$

d. ${\mathbf{e}}^{\mathbf{a}}$

Hint for (d): ${\mathbf{e}}^{\mathbf{(}\mathbf{x}\mathbf{+}\mathbf{jy}\mathbf{)}}\mathbf{=}{\mathbf{e}}^{\mathbf{x}}\mathbf{\angle }\mathbf{y}$ , where y is in radians.

Short answer

(a) After evaluating the expression: ${\mathrm{a}}^{10}$, the value obtained in rectangular form isrole="math" localid="1655382317474" $-\frac{1}{2}+\frac{\sqrt{3}}{2}\mathrm{j}$ .

(b) After evaluating the expression:${\left(\mathrm{ja}\right)}^{10}$, the value obtained in rectangular form is $\frac{1}{2}-\frac{\sqrt{3}}{2}\mathrm{j}$.

(c) After evaluating the expression:${(1-\mathrm{a})}^3$, the value obtained in rectangular form is $-5.19\mathrm{j}$.

(d) Therefore, after evaluating the expression: ${\mathrm{e}}^{\mathrm{a}}$, the value obtained in rectangular form is $0.3929+\mathrm{j}0.4620$.

Step-by-step solution

Express a10  in the rectangular form

a)

Determine the value of the expression:

As given in the data, substitute the value $1\angle {120}^{\circ }$for in the above expression.

role="math" localid="1655382746717" $$\begin{gathered} {\mathrm{a}}^{10}={\left({\mathrm{a}}^3\right)}^3\times {\mathrm{a}}^1 \\ ={\left({(1\angle {120}^{\circ })}^3\right)}^3\times {(1\angle {120}^{\circ })}^1 \\ ={\left((1\angle {360}^{\circ })\right)}^3\times {(1\angle {120}^{\circ })}^1 \\ ={(1\angle 0^{\circ })}^3\times (1\angle {120}^{\circ }) \end{gathered}$$

Further solving,

$$\begin{gathered} {\mathrm{a}}^{10}=\left(1\right)(1\angle {120}^{\circ }) \\ =(1\angle {120}^{\circ }) \end{gathered}$$

Now convert the above polar expression into the rectangular form;

${\mathrm{a}}^{10}=-\frac{1}{2}+\frac{\sqrt{3}}{2}\mathrm{j}$

Therefore, after evaluating the expression:${\mathrm{a}}^{10}$ , the value obtained in rectangular form is$-\frac{1}{2}+\frac{\sqrt{3}}{2}\mathrm{j}$ .

Express  (ja)10 in the rectangular form

(b)

Determine the value of the expression: ${\left(\mathrm{ja}\right)}^{10}$

As given in the data, substitute the value $-\frac{1}{2}+\frac{\sqrt{3}}{2}\mathrm{j}$ for ${\mathrm{a}}^{10}$ in the above expression.

$$\begin{gathered} {\left(\mathrm{ja}\right)}^{10}={\left[\mathrm{j}\left(\mathrm{a}\right)\right]}^{10} \\ ={\left(\mathrm{j}\right)}^{10}{\left(\mathrm{a}\right)}^{10}={\left(\mathrm{j}\right)}^2{\left(\mathrm{j}\right)}^4{\left(\mathrm{j}\right)}^4\left(\mathrm{a}10\right) \\ =\left(-1\right)\left(1\right)\left(1\right)\left(-\frac{1}{2}+\frac{\sqrt{3}}{2}\mathrm{j}\right) \end{gathered}$$

Further solving,

$$\begin{gathered} {\left(\mathrm{ja}\right)}^{10}=-\left(\frac{1}{2}+\frac{\sqrt{3}}{2}\mathrm{j}\right) \\ =\frac{1}{2}-\frac{\sqrt{3}}{2}\mathrm{j} \end{gathered}$$

Therefore, after evaluating the expression:${\left(\mathrm{ja}\right)}^{10}$, the value obtained in rectangular form is $\frac{1}{2}-\frac{\sqrt{3}}{2}\mathrm{j}$.

Express  (1-a)3 in the rectangular form

(c)

Determine the value of the expression:

As given in the data, substitute the value for $1\angle {120}^{\circ }$ in the above expression.

${(1-\mathrm{a})}^3={\left[1-(1\angle 120\circ )\right]}^3$

Convert the above given expression into rectangular form;

$$\begin{gathered} {(1-\mathrm{a})}^3={\left[1-\left(-\frac{1}{2}+\frac{\sqrt{3}}{2}\mathrm{j}\right)\right]}^3 \\ ={\left[1+\frac{1}{2}-\frac{\sqrt{3}}{2}\right]}^3 \\ ={\left[\frac{3}{2}-\frac{\sqrt{3}}{2}\mathrm{j}\right]}^3 \\ =3.37-5.84\mathrm{j}+3.37{\mathrm{j}}^2-0.65{\mathrm{j}}^3 \end{gathered}$$

Further solving,

$$\begin{gathered} {(1-\mathrm{a})}^3=3.37-5.84\mathrm{j}-3.37+0.65\mathrm{j} \\ =-5.19\mathrm{j} \end{gathered}$$

Therefore, after evaluating the expression: ${(1-\mathrm{a})}^3$, the value obtained in rectangular form is $-5.19\mathrm{j}$ .

Express ea  in the rectangular form

(d)

Determine the value of the expression:${\mathrm{e}}^{\mathrm{a}}$

As given in the data, substitute the value $1\angle {120}^{\circ }$for a and $1\angle \mathrm{\theta }$ for role="math" localid="1655440010088" ${\mathrm{e}}^{j\theta }$in the above expression.

${\mathrm{e}}^{\mathrm{a}}={\mathrm{e}}^{(1\angle {120}^{\circ })}$

Convert the above given expression into rectangular form;

$$\begin{gathered} {\mathrm{e}}^{\mathrm{a}}={\mathrm{e}}^{\left(-\frac{1}{2}+\frac{\sqrt{3}}{2}\mathrm{j}\right)} \\ =\left({\mathrm{e}}^{-\frac{1}{2}}\right)\left(1\angle {\left(\frac{\sqrt{3}}{2}\right)}^{\circ }\mathrm{radians}\right) \\ =(0.6065)\left(1\angle \left(\frac{0.866\times {180}^{\circ }}{\mathrm{\pi }}\right)\right) \end{gathered}$$

Further solving,

$$\begin{gathered} {\mathrm{e}}^{\mathrm{a}}=0.6065\angle 49.{62}^{\circ } \\ =0.3929+\mathrm{j}0.4620 \end{gathered}$$

Therefore, after evaluating the expression: ${\mathrm{e}}^{\mathrm{a}}$, the value obtained in rectangular form is $0.3929+\mathrm{j}0.4620$.