# Example Problems Involving Complex Numbers

### Ex 1) Evaluate the following complex number and express the result in polar notation

$$(40 \angle 50^{\circ} + 20 \angle(-30^{\circ}))^{\frac{1}{2}}$$ Converting to rectangular form: $$= \Big[ 40cos50^{\circ} + j40sin50^{\circ} + 20cos(-30^{\circ}) + j20sin(-30^{\circ}) \Big]^{\frac{1}{2}}$$ Numerically evaluating the trig terms: $$= [43.03 + j20.64]^{\frac{1}{2}}$$ Recall that to convert to polar form: $$r = \sqrt{x^2 + y^2}$$ and: $$\phi = tan^{-1} \Big( \frac{y}{x} \Big)$$ Therefore we get: $$= \Big[ \sqrt{43.03^2 + 20.64^2} \angle \Big( tan^{-1} \Big( \frac{20.64}{43.03} \Big) \Big) \Big]^{\frac{1}{2}}$$ $$= [47.72 \angle 25.63^{\circ}]^{\frac{1}{2}}$$ Using the rule for complex numbers that involves square roots: $$=\sqrt{47.72} \angle \Big(\frac{25.63}{2} \Big)$$

$$= 6.91 \angle 12.82^{\circ}$$

Now that we have a decent understanding of sinusoids and complex numbers, let's take a look at phasor notation.

Continue on to Phasor Notation....