Question

Find each of the following in rectangular form x + iyand check your results by computer. Remember to save time by doing as much as you can in your head.

$\mathit{s}\mathit{i}\mathit{n}\left(\pi -i\mathrm{In}3\right)$.

Short answer

The rectangular form of the given question is, sin $\left(\pi -i\mathrm{In}3\right)=\frac{4i}{3}$.

Step-by-step solution

Given Information.

The given expression is $\sin \left(\pi -i\mathrm{In}3\right)$.

Meaning of rectangular form.

Representing the complex number in rectangular form means writing the given complex number in the form of x = iy, in which is the real part and y is the imaginary part.

Step 3: Put the value in the formula.

Use the complex formula $\sin \theta =\frac{e^{i\theta }-e^{-i\theta }}{2i}$to rewrite the above expression.

And $\sin \left(\pi -i\mathrm{In}3\right)$can written as,

$$\begin{gathered} \sin \left(\pi -i\mathrm{In}3\right)=\sin \pi .\cos \left(i\mathrm{In}3\right)-\cos \pi \left(i\mathrm{In}3\right) \\ \sin \left(\pi -i\mathrm{In}3\right)=\sin \left(i\mathrm{In}3\right) \end{gathered}$$

Now,

$$\begin{gathered} \sin (i\mathrm{In}3)=\frac{{\mathrm{e}}^{\left(\mathrm{i}\mathrm{In}3\right)i}-{\mathrm{e}}^{-\left(\mathrm{i}\mathrm{In}3\right)i}}{2\mathrm{i}} \\ =\frac{{\mathrm{e}}^{\left(-\mathrm{In}3\right)}-{\mathrm{e}}^{\left(\mathrm{In}3\right)}}{2i} \\ =\frac{{\mathrm{e}}^{\left(\mathrm{In}{3}^{-1}\right)}-{\mathrm{e}}^{\left(\mathrm{In}3\right)}}{2i} \end{gathered}$$

$=\frac{{\displaystyle \frac{1}{3}}-3}{2i}$

$$\begin{gathered} \sin (i\mathrm{In}3)=-\frac{8}{2i\times 3} \\ \sin (i\mathrm{In}3)=-\frac{4}{3i} \\ \sin (i\mathrm{In}3)=-\frac{4i}{3} \end{gathered}$$

So,

$\sin \left(\pi -i\mathrm{In}3\right)=\sin \left(i\mathrm{In}3\right)$

Substitute the value of $\sin \left(\pi -i\mathrm{In}3\right)$in the above equation.

$\sin \left(\pi -i\mathrm{In}3\right)=\frac{4i}{3}$

Therefore, the rectangular form of$\sin \left(\pi -i\mathrm{In}3\right)$ is $\frac{4i}{3}$.