# How do you express the complex number in trigonometric form: 5-5i?

Apr 6, 2018

$5 \sqrt{2} \left(\cos \left(- \frac{\pi}{4}\right) + i \sin \left(- \frac{\pi}{4}\right)\right)$

#### Explanation:

To convert from $z = x + i y$ form to $z = r \left(\cos \theta + i \sin \theta\right)$ form, you need to find $r$ and $\theta$.

$r = \sqrt{{x}^{2} + {y}^{2}}$ and $\tan \theta = \frac{y}{x}$

So, $r = \sqrt{{5}^{2} + {\left(- 5\right)}^{2}} = \sqrt{50} = 5 \sqrt{2}$

$\tan \theta = \frac{5}{-} 5 \implies \tan \theta = - 1 \implies \theta = - \frac{\pi}{4}$

$\therefore 5 - 5 i = 5 \sqrt{2} \left(\cos \left(- \frac{\pi}{4}\right) + i \sin \left(- \frac{\pi}{4}\right)\right)$

Apr 6, 2018

In trigonometric form: $7.07 \left(\cos 45 - i \sin 45\right)$

#### Explanation:

$Z = a + i b = 5 - 5 i$. Modulus:$| Z | = \sqrt{{a}^{2} + {b}^{2}}$

Modulus:$| Z | = \sqrt{{5}^{2} + {\left(- 5\right)}^{2}}$

$= \sqrt{50} \approx 7.07$ Argument: $\tan \alpha = | \frac{b}{a} | = | \frac{5}{-} 5 | = 1$

:.alpha =tan^-1 (1) = 45^0 ; Z lies on fourth

quadrant.$\therefore \theta = 360 - \alpha = {315}^{0} \mathmr{and} {\left(- 45\right)}^{0}$

In trigonometric form expressed as

$Z = | Z | \left(\cos \theta + i \sin \theta\right)$

$\therefore Z = 7.07 \left(\cos 315 + i \sin 315\right)$ or

$Z = 7.07 \left(\cos 45 - i \sin 45\right)$ [Ans]

: