Question

Find each of the following in rectangular form $\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$ and check your results by computer. Remember to save time by doing as much as you can in your head.

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

Short answer

The rectangular form of the given question is, $sin\left(iIni\right)=-1$.

Step-by-step solution

Given Information.

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

Meaning of rectangular form.

Representing the complex number in rectangular form means writing the given complex number in the form of $\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$, in which $\mathit{x}$is the real part and $\mathit{y}$ 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 as,

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

Therefore, the rectangular form of $\sin \left(i\mathrm{In}i\right)$is $-1$.