# What is the trigonometric form of  (2-i)*(1+i) ?

Sep 15, 2017

See explanation.

#### Explanation:

First we have to multiply the numbers:

$\left(2 - i\right) \cdot \left(1 + i\right) = 2 + 2 i - i - {i}^{2} = 2 + i + 1 = 3 + i$

Now we have to write the result $3 + i$ in trigonometric form:

$| z | = \sqrt{{3}^{2} + {1}^{2}} = \sqrt{10}$

Now we have to find the angle using:

$\cos \varphi = \frac{r e \left(z\right)}{|} z |$

$\cos \varphi = \frac{3}{\sqrt{10}} = \frac{3 \sqrt{10}}{10} \approx 0.949$

From this we have $\varphi \approx {18.43}^{o}$.

Finally we can write the result as:

$z = \sqrt{10} \cdot \left(\cos {18.43}^{o} + i \sin {18.43}^{o}\right)$