Question

Change the Cartesian coordinate ($-1,1$) to polar coordinate system.

• A. $(\sqrt{2},\frac{3\pi }{4})$
• B. $(1,\frac{3\pi }{4})$
• C. $(\sqrt{2},\frac{7\pi }{4})$
• D. $(1,\frac{\pi }{4})$

Answer 1

The correct option is A $(\sqrt{2},\frac{3\pi }{4})$

Given Cartesian coordinate ($-1,1$)

For polar coordinates,
radius $r=\sqrt{x^2+y^2}$
$r=\sqrt{(-1{)}^2+(1{)}^2}=\sqrt{2}$

Reference angle, $\tan \theta =\frac{y}{x}$
$\tan \theta =\frac{1}{-1}$

$\theta =\frac{3\pi }{4}$ or $\frac{7\pi }{4}$

As we see, given Cartesian coordinate $-1,1$ is in the second quadrant. Hence,
$\theta =\frac{3\pi }{4}$

So, polar coordinate will be ($\sqrt{2},\frac{3\pi }{4}$)

Hence, option $\left(\text{A}\right)$ is the right choice.