Answer with explanation:
let, Z= a + i b,be a complex number.
Where, a = r cos A
b= r sin A
[tex]a^2 + b^2=(r cos A)^2 +(r sin A)^2\\\\ a^2 + b^2=r^2(cos^2 A+sin^2 A)\\\\ a^2 + b^2=r^2\\\\r=\sqrt{a^2 + b^2}[/tex]
[tex]Z=r cos A + i r sin A\\\\Z=r(cos A + i sin A)\\\\ Z=re^{iA},\text{Where}, e^{i\alpha }=cos \alpha +i sin\alpha[/tex]
Now, if we replace A, by, 2 kπ + A,in the above equation,where k is any positive integer,beginning from ,0.k=0,1,2,3,4,....
[tex]Z=r cos (2k\pi +A) + i r sin(2k\pi + A)\\\\Z=r[cos (2k\pi +A) + i sin(2k\pi + A)]\\\\ Z=re^{i(2k\pi +A)}[/tex]
So, for different value of , k ,there will be Different Complex number.
→So,the Statement: The trigonometric form of a complex number is unique = False
