Prove Quotient of Complex Numbers in Polar Form

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Prove: . z<sub>1</sub>/z<sub>2</sub> . = . (r<sub>1</sub>/r<sub>2</sub>)[cos(θ<sub>1</sub> - θ<sub>2</sub>) + i sin(θ<sub>1</sub> - θ<sub>2</sub>)]
.
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We have: . z<sub>1</sub> .= .r<sub>1</sub>e<sup>(θ<sub>1</sub>)</sup> . and: . z<sub>2</sub> .= .r<sub>2</sub>e<sup>(iθ<sub>2</sub>)</sup>


. . . . . . .z<sub>1</sub> . . . . r<sub>1</sub>e<sup>(iθ1)</sup> . . . . .r<sub>1</sub> . . . . . . . . . . . . . r<sub>1</sub>
Then: . ---- . = . ---------- . = . ---- e<sup>(iθ<sub>1</sub> - iθ<sub>2</sub>)</sup> . = . ---- e<sup>i(θ<sub>1</sub> - θ<sub>2</sub>)</sup>
. . . . . . .z<sub>2</sub> . . . . r<sub>2</sub>e<sup>(iθ<sub>2</sub>)</sup> . . . . .r<sub>2</sub> . . . . . . . . . . . . . r<sub>2</sub>


. . . . . . . . . . z<sub>1</sub> . . . . r<sub>1</sub>
Therefore: . --- . = . ---- [cos(θ<sub>1</sub> - θ<sub>2</sub>) + i sin(θ<sub>1</sub> - θ<sub>2</sub>)]
. . . . . . . . . . z<sub>2</sub> . . . . r<sub>2</sub>
.
 
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