Question

In Problems $23-32$, write each complex number in rectangular form.

$0.4\left(\cos {200}^{\circ }+i\sin {200}^{\circ }\right)$.

Short answer

The rectangular form of the complex number$0.4\left(\cos {200}^{\circ }+i\sin {200}^{\circ }\right)$is,$-0.3758-0.1368i$.

Step-by-step solution

Step 1 Given complex number is,

$0.4\left(\cos {200}^{\circ }+i\sin {200}^{\circ }\right)$.

We can simplify the terms inside the brackets and write it in the rectangular form.

$$\begin{gathered} z=0.4\left(\cos {200}^{\circ }+i\sin {200}^{\circ }\right) \\ z=0.4\left(\cos \left({180}^{\circ }+{20}^{\circ }\right)+i\sin \left({180}^{\circ }+{20}^{\circ }\right)\right) \\ z=0.4\left(-\cos {20}^{\circ }-i\sin {20}^{\circ }\right) \\ [\cos \left({180}^{\circ }+\theta \right)=-\cos \theta \\ \sin \left({180}^{\circ }+\theta \right)=-\sin \theta ] \end{gathered}$$

Step 2 Now substitute the values for cos 20∘,sin 20∘.

$$\begin{gathered} z=0.4\left(-0.9396-0.342i\right) \\ z=-0.3758-0.1368i \end{gathered}$$

Thus the required rectangular form is,$-0.3758-0.1368i$.