Question

What is 10∠ 30 + 10∠ 30? Answer in polar form. Note that the angle is measured in degrees here.

Answer 1

Answer:

$10\angle30+10\angle30= 20\angle30$

Step-by-step explanation:

any polar coordinate expression can be broken into component form (or the Cartesian coordinate form) like this:

$r\angle\theta = \begin{bmatrix}r\cos{\theta}\\r\sin{\theta}\end{bmatrix}$

so,

$10\angle30 = \begin{bmatrix}10\cos{30}\\10\sin{30}\end{bmatrix}$

this simplifies to:

$10\angle30 = \begin{bmatrix}10(\frac{\sqrt{3}}{2})\\10(\frac{1}{2})\end{bmatrix}$

$10\angle30 = \begin{bmatrix}5\sqrt{3}\\5\end{bmatrix}$

the other polar coordinate is the same, hence. we'll just move on to sum them up.

$10\angle30+10\angle30$

$\begin{bmatrix}5\sqrt{3}\\5\end{bmatrix}+\begin{bmatrix}5\sqrt{3}\\5\end{bmatrix}$

$\begin{bmatrix}10\sqrt{3}\\10\end{bmatrix}$

r:

$r = \sqrt{(10\sqrt{3})^2+(10)^2}$

$r = 20$

angle:

$\theta = \arctan{\dfrac{y}{x}}$

in our case that's

$\theta = \arctan{\dfrac{1}{\sqrt{3}}}$

$\theta = 30$

so the answer is:

$r\angle\theta = 20\angle30$

A shorter method is: to just keep in mind that the 'r' adds directly and the 'angle' doesn't.