Question

Convert the rectangular coordinates to polar coordinates with
r > 0 and 0 ≤ < 2.

Answer 1

,

`(6, (11\pi)/(6) )`

Explanation:

Note I'm using a, instead of theta to represent angles.

To convert rectangular to polar, apply these formulas

`r = \sqrt{ {x}^(2) + {y}^(2) }`

`\alpha = \tan {}^( - 1) ( (y)/(x) )`

Note : Rectangular coordinates are the coordinates you were learning since elementary school or middle school.

The first number is x, and the second is y.

So

`r = \sqrt{(3 √(3) ) {}^(2) + ( - 3) {}^(2) }`

`r = √(36)`

`r = 6`

`\alpha = \tan {}^( - 1) ( ( - 3)/(3 √(3) ) )`

Since our y coordinate is negative and x coordinate is Positve , on the unit circle, the angle must be in the fourth quadrant.

So the angle must be in between

`(3\pi)/(2) < \alpha < 2\pi`

`\alpha = \tan {}^( - 1) ( ( - 1)/( √(3) ) )`

`\alpha = (11\pi)/(16)`

So our answer is

(6, 11pi/6).