Question

z=3 – 43Find the angle 8 (in radians) that makes in the complex plane.Round your answer, if necessary, to the nearest thousandth. Express between - and T.Show Calculator

Answer 1

Given:-

$z=3-4i$

To find:-

The angle.

Complex number will be denoted as,

$z=x+iy$

So the value of x and y are,

$x=3,y=-4$

At first we need to find the value of r.

The formula to calculate r is,

$r=\sqrt[]{x^2+y^2}$

Substituting the values. we get,

$\begin{gathered} r=\sqrt[]{3^2+(-4)^2} \\ r=\sqrt[]{9+16} \\ r=\sqrt[]{25} \\ r=5 \end{gathered}$

Now we use the angle formula,

$\cos \theta=\frac{x}{r}$

Substituting the value. we get,

$\begin{gathered} \cos \theta=\frac{3}{5} \\ \end{gathered}$

So the value of theta is,

$\theta=\cos ^{-1}(\frac{3}{5})$