# Write on the trigonometric form the following complex number: -3(Cospi/4 + isinpi/4) ?

Feb 2, 2018

see explanation.

#### Explanation:

Well, I think of $- 3 \left(\cos \left(\frac{\pi}{4}\right) + i \sin \left(\frac{\pi}{4}\right)\right)$, or $- 3 c i s \left(\frac{\pi}{4}\right)$, as trig form. An alternative uses Euler's formula,

$r \cdot {e}^{i \theta} = 4 \left(\cos \left(\theta\right) + i \sin \left(\theta\right)\right)$,

so we can also write it as $- 3 {e}^{\frac{\pi}{4} i}$.

We could also write it in rectangular, or standard, form:

Since $\cos \left(\frac{\pi}{4}\right) = \sin \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$ we have:

$- 3 \left(\cos \left(\frac{\pi}{4}\right) + i \sin \left(\frac{\pi}{4}\right)\right) = - 3 \left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} i\right)$

$= - \frac{3 \sqrt{2}}{2} - \frac{3 \sqrt{2}}{2} i$