Question

Using the operator$\mathit{a}\mathbf{=}\mathbf{1}\mathbf{\angle }\mathbf{120}\mathbf{°}$, evaluate the following in polar form: (a)$\left(a-1\right)\mathbf{/}\left(1+a-a^2\right)$, (b)$\left(a^2+a+j\right)\mathbf{/}\left(ja+a^2\right)$, (c) $\left(1+a\right)\left(1+a^2\right)$ and (d) $\left(a-a^2\right)\left(a^2-1\right)$.

Short answer

(a)After evaluating the expression:$\frac{\mathrm{a}-1}{1+\mathrm{a}-{\mathrm{a}}^2}$, the value obtained in the polar form is $0.866\angle 90°$.

(b)After evaluating the expression:$\frac{{\mathrm{a}}^2+\mathrm{a}+\mathrm{j}}{\mathrm{ja}+{\mathrm{a}}^2}$, the value obtained in the polar form is $0.732\angle -90°$.

(c)After evaluating the expression:$(1+\mathrm{a})(1+{\mathrm{a}}^2)$, the value obtained in the polar form is $1$.

(d)After evaluating the expression:$(\mathrm{a}-{\mathrm{a}}^2)({\mathrm{a}}^2-1)$, the value obtained in the polar form is $3\angle 300°$.

Step-by-step solution

Express (a-1)/(1+a-a2) in the polar form

(a)

Determine the value of the expression: $\frac{\mathrm{a}-1}{1+\mathrm{a}-{\mathrm{a}}^2}$

As given in the data, substitute the value $1\angle 120°$for a in the above expression.

$\frac{\mathrm{a}-1}{1+\mathrm{a}-{\mathrm{a}}^2}=\frac{1\angle 120°-1}{1+1\angle 120°-(1\angle 120°)}$

Convert the polar form expressions in the above equation into the rectangular form;

$$\begin{gathered} \frac{\mathrm{a}-1}{1+\mathrm{a}-{\mathrm{a}}^2}=\frac{-0.5+\mathrm{j}0.866-1}{1-0.5+\mathrm{j}0.866-(1\angle 240°)} \\ =\frac{-1.5+\mathrm{j}0.866}{0.5+\mathrm{j}0.866-(-0.5-\mathrm{j}0.866)} \\ =\frac{-1.5+\mathrm{j}0.866}{1+\mathrm{j}1.732} \end{gathered}$$

Now again convert rectangular to polar;

$$\begin{gathered} \frac{\mathrm{a}-1}{1+\mathrm{a}-{\mathrm{a}}^2}=\frac{1.732\angle 150°}{2\angle 60°} \\ =0.866\angle 90° \end{gathered}$$

Thus, after evaluating the expression: $\frac{\mathrm{a}-1}{1+\mathrm{a}-{\mathrm{a}}^2}$, the value obtained in the polar form is $0.866\angle 90°$.

Express (a2+a+j)/(ja+a2) in the polar form

(b)

Determine the value of the expression: $\frac{{\mathrm{a}}^2+\mathrm{a}+\mathrm{j}}{\mathrm{ja}+{\mathrm{a}}^2}$

As given in the data, substitute the value $1\angle 120°$for a in the above expression.

$$\begin{gathered} \frac{{\mathrm{a}}^2+\mathrm{a}+\mathrm{j}}{\mathrm{ja}+{\mathrm{a}}^2}=\frac{{(1\angle 120°)}^2+1\angle 120°+\mathrm{j}}{\mathrm{j}(1\angle 120°)+{(1\angle 120°)}^2} \\ =\frac{1\angle 240°+1\angle 120°+\mathrm{j}}{1\angle 120°(\mathrm{j}+1\angle 120°)} \end{gathered}$$

Convert the polar expressions in the above equation into the rectangular form;

$$\begin{gathered} \frac{{\mathrm{a}}^2+\mathrm{a}+\mathrm{j}}{\mathrm{ja}+{\mathrm{a}}^2}=\frac{{(1\angle 120°)}^2+1\angle 120°+\mathrm{j}}{\mathrm{j}(1\angle 120°)+{(1\angle 120°)}^2} \\ =\frac{1\angle 240°+1\angle 120°+\mathrm{j}}{1\angle 120°(\mathrm{j}+1\angle 120°)} \end{gathered}$$

Now, again convert the rectangular form expressions into the polar form;

$$\begin{gathered} \frac{{\mathrm{a}}^2+\mathrm{a}+\mathrm{j}}{\mathrm{ja}+{\mathrm{a}}^2}=\frac{-0.5-0.866\mathrm{j}-0.5+0.866\mathrm{j}+\mathrm{j}}{1\angle 120°(\mathrm{j}-0.5+0.866\mathrm{j})} \\ =\frac{-1+\mathrm{j}}{1\angle 120°(-0.5+1.866\mathrm{j})} \end{gathered}$$

Thus, after evaluating the expression: $\frac{{\mathrm{a}}^2+\mathrm{a}+\mathrm{j}}{\mathrm{ja}+{\mathrm{a}}^2}$, the value obtained in the polar form is $0.732\angle -90°$.

Expressin the polar form

(c)

Determine the value of the expression: $(1+\mathrm{a})(1+{\mathrm{a}}^2)$

As given in the data, substitute the value $1\angle 120°$for a in the above expression.

$(1+\mathrm{a})(1+{\mathrm{a}}^2)=(1+1\angle 120°)[1+\left(1\angle 120°\right)]$

Convert the polar expressions in the above equation into the rectangular form;

$$\begin{gathered} (1+\mathrm{a})(1+{\mathrm{a}}^2)=(1-0.5+\mathrm{j}0.866)(1+1\angle 240°) \\ =(0.5+\mathrm{j}0.866)(1-0.5-\mathrm{j}0.866) \\ =(0.5+\mathrm{j}0.866)(0.5-\mathrm{j}0.866) \\ =1 \end{gathered}$$

Thus, after evaluating the expression:$(1+\mathrm{a})(1+{\mathrm{a}}^2)$, the value obtained in the polar form is$1$.

Expressin the polar form

(d)

Determine the value of the expression:$(\mathrm{a}-{\mathrm{a}}^2)({\mathrm{a}}^2-1)$

$$\begin{gathered} (\mathrm{a}-{\mathrm{a}}^2)({\mathrm{a}}^2-1)=-\mathrm{a}(1-\mathrm{a})(1-{\mathrm{a}}^2) \\ =1\angle 120°(1-1\angle 120°)(1-(1\angle 120°)) \\ =1\angle 120°(\sqrt{3}\angle -30°)(1-1\angle 240°) \\ =\sqrt{3}\angle 90°(\sqrt{3}\angle -30°) \end{gathered}$$

Further solving,

$$\begin{gathered} (\mathrm{a}-{\mathrm{a}}^2)({\mathrm{a}}^2-1)=-3\angle 120° \\ =3\angle 300° \end{gathered}$$

Thus, after evaluating the expression:$(\mathrm{a}-{\mathrm{a}}^2)({\mathrm{a}}^2-1)$, the value obtained in the polar form is$3\angle 300°$.