Question

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

$0.2\left(\cos {100}^{\circ }+i\sin {100}^{\circ }\right)$.

Short answer

The rectangular form of the complex number $0.2\left(\cos {100}^{\circ }+i\sin {100}^{\circ }\right)$is,$-0.035+0.197i$.

Step-by-step solution

Step 1 Given complex number is,

$0.2\left(\cos {100}^{\circ }+i\sin {100}^{\circ }\right)$.

The given complex number is in the form of $r\left(\cos \theta +i\sin \theta \right)$where $x=r\cos \theta ,y=r\sin \theta$.

The $x$co-ordinates of $0.2\left(\cos {100}^{\circ }+i\sin {100}^{\circ }\right)$is,

$$\begin{gathered} x=r\cos \theta \\ x=0.2\cos {100}^{\circ } \\ x=0.2\left(-0.175\right) \\ x=-0.035 \end{gathered}$$

Step 2 The y co-ordinate of 0.2cos 100∘+i sin 100∘ is,

$$\begin{gathered} y=r\sin \theta \\ y=0.2\sin {100}^{\circ } \\ y=0.2\left(0.985\right) \\ y=0.197 \end{gathered}$$

The rectangular form is,$$\begin{gathered} z=x+iy \\ z=-0.035+0.197 \end{gathered}$$