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

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### 2 Answers

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.

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))]**