# What is the trigonometric form of  (7-i) ?

Aug 18, 2016

$5 \sqrt{2} \left(\cos \left(0.142\right) - i \sin \left(0.142\right)\right)$

#### Explanation:

To convert from $\textcolor{b l u e}{\text{complex to trigonometric form}}$

That is $\left(x + y i\right) \to \left[r \left(\cos \theta + i \sin \theta\right)\right] \text{ where}$

$\textcolor{\mathmr{and} a n \ge}{\text{Reminder " color(red)(|bar(ul(color(white)(a/a)color(black)(r=sqrt(x^2+y^2))color(white)(a/a)|)))" and}}$

$\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{\theta = {\tan}^{-} 1 \left(\frac{y}{x}\right)} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

For (7 - i) , x = 7 and y = - 1

$\Rightarrow r = \sqrt{{7}^{2} + {\left(- 1\right)}^{2}} = \sqrt{50} = 5 \sqrt{2}$

Now (7 - i) is in the 4th quadrant, so we must ensure that $\theta$ is in the 4th quadrant.

$\theta = {\tan}^{-} 1 \left(- \frac{1}{7}\right) = - 0.142 \text{ in 4th quadrant}$

$\Rightarrow \left(7 - i\right) = 5 \sqrt{2} \left(\cos \left(- 0.142\right) + i \sin \left(- 0.142\right)\right)$

$\textcolor{\mathmr{and} a n \ge}{\text{Reminder}}$

$\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{\cos \left(- \theta\right) = \cos \theta \text{ and } \sin \left(- \theta\right) = - \sin \theta} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

$\Rightarrow \left(7 - i\right) = 5 \sqrt{2} \left(\cos \left(0.142\right) - i \sin \left(0.142\right)\right)$