10.37 Represent the complex number $z=1-\cos 200^{\circ}+i \sin 200^{\circ}$ in trigonometric form.
10.37 First, let's write the given number in the form

$z=2 \sin ^{2} 100^{\circ}+2 i \sin 100^{\circ} \cos 100^{\circ}=$

$=2 \sin ^{2} 80^{\circ}-2 i \sin 80^{\circ} \cos 80^{\circ}$ and find its modulus:

$z=\sqrt{4 \sin ^{4} 80^{\circ}+4 \sin ^{2} 80^{\circ} \cos ^{2} 80^{\circ}}=\sqrt{4 \sin ^{2} 80^{\circ}}=2 \sin 80^{\circ}$,

since $\sin 80^{\circ}>0$. Next, we have

$\cos \varphi=\frac{2 \sin ^{2} 80^{\circ}}{2 \sin 80^{\circ}}=\sin 80^{\circ}$,

$\sin \varphi=-\frac{2 \sin 80^{\circ} \cos 80^{\circ}}{2 \sin 80^{\circ}}=-\cos 80^{\circ}$.

Thus, $z=2 \sin 80^{\circ}\left(\sin 80^{\circ}-i \cos 80^{\circ}\right)$, but this representation is not the trigonometric form of the given number.

To find the desired representation, we use the fact that $\sin 80^{\circ}=\cos \left(80^{\circ}-90^{\circ}\right)=\cos \left(-10^{\circ}\right),-\cos 80^{\circ}=$ $=\sin \left(80^{\circ}-90^{\circ}\right)=\sin \left(-10^{\circ}\right)$. Then we get $z=2 \sin 80^{\circ}\left(\cos \left(-10^{\circ}\right)+i \sin \left(-10^{\circ}\right)\right)$.