Answer to Question #303054 in Complex Analysis for RAIN

Question

z₁=3.45∠980°

z₂₌-5+9i

1. z1+z2 final answer in polar form
2. z1-z2 final answer in trigonometric form
3. z2*z1 final answer in exponential form
4. z1/z2 final answer in rectangular form

Answer 1

$$z_1=3.45\angle980\degree=3.45e^{i980\degree}=3.45e^{-i100\degree}$$

$$=3.45(\cos(-100\degree)+i\sin(-100\degree))$$

$$=3.45(\cos(5\pi/9)-i\sin(5\pi/9))$$

$$=3.45\cos(5\pi/9)-3.45\sin(5\pi/9)i$$

$$z_2=-5+9i=\sqrt{106}e^{i\tan^{-1}(9/-5)}$$

$$=\sqrt{106}e^{-i\tan^{-1}(1.8)}$$

$$=\sqrt{106}(\cos(\tan^{-1}(1.8))-i\sin(\tan^{-1}(1.8)))$$

$$=\sqrt{106}\cos(\tan^{-1}(1.8)-\sqrt{106}\sin(\tan^{-1}(1.8))i$$

1.

$$z_1+z_2=3.45\cos(5\pi/9)-3.45\sin(5\pi/9)i$$

$$+(-5+9i)=(3.45\cos(5\pi/9)-5)+$$

$$+(-3.45\sin(5\pi/9)+9)i$$

$$\approx-5.60+5.60i=5.60\sqrt{2}(\cos(135\degree)+i\sin(135\degree))$$

$$z_1+z_2=5.60\sqrt{2}(\cos(135\degree)+i\sin(135\degree))$$

$$=5.60\sqrt{2}\angle135\degree=5.60\sqrt{2}\ cis(135\degree)$$

2.

$$z_1-z_2=3.45\cos(5\pi/9)-3.45\sin(5\pi/9)i$$

$$-(-5+9i)=(3.45\cos(5\pi/9)+5)+$$

$$+(-3.45\sin(5\pi/9)-9)i$$

$$\approx4.40-12.40i=13.16(\cos(-70.5\degree)+i\sin(-70.5\degree))$$

$$z_1-z_2=13.16(\cos(-70.5\degree)+i\sin(-70.5\degree))$$

$$=13.16\sqrt{2}\angle289.5\degree=13.16\sqrt{2}\ cis(289.5\degree)$$

3.

$$z_1\cdot z_2=3.45e^{-i100\degree}\cdot\sqrt{106}e^{-i\tan^{-1}(1.8)}$$

$$=3.45\sqrt{106}e^{-i(100+\tan^{-1}(1.8))}$$

$$=35.52e^{-i161\degree}$$

4.

$$z_1/z_2=3.45e^{-i100\degree}/(\sqrt{106}e^{-i\tan^{-1}(1.8)})$$

$$=3.45\sqrt{106}e^{i(-100+\tan^{-1}(1.8))}$$

$$=0.3351e^{-i39.0546\degree}$$

$$=0.26-0.21i$$