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

neela | High School Teacher | (Level 3) Valedictorian

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The given complex number is 3+4i.

To express the number in the polar form.

If z = x+iy,

Then r =  sqrt(x^2+y^2)

x = rccost and y = rsint

y/x = tant, or t = arctan (y/x)

So  z = r(cost + isint ) is the trigonometric or polar form of representing z.

z = 3+4i.

r = (3^2+4^2)^(1/2) = (9+16)^(1/2) = 5.

Therefore  cost = 3/5 and sint 4/5 , tant = 4/3.

z = 3+4i = 5 {cos(arctan5/4) + i sin(arc tan (4/3) } is the polar form of z.

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

The algebraic 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|

We'll identify x and y for te 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))]