# Complex numbers.Write the trigonometric form of the complex number z = 3 + 4i.

*print*Print*list*Cite

### 1 Answer

We'll write the rectangular form of any complex number is z = x + y*i.

The trigonometric form of a complex number is:

z = |z|(cos a + i*sin a)

|z| = sqrt(x^2 + y^2)

cos a = x/|z|

sin a = y/|z|

Comparing, we'll identify x and y for the given complex number:

x = Re(z) = 3

y = Im(z) = 4 (only the coefficient of i)

|z| = sqrt(3^2 + 4^2) => |z| = sqrt(9 + 16) => |z| = sqrt25 => |z| = 5

cos a = 3/5

sin a = 4/5

tan a = y/x

tan a = 4/3

a = arctan (4/3)

The trigonometric form of the complex number z is:

z = 5*[cos (arctan (4/3)) + i*sin (arctan (4/3))]