Question

Convert the rectangular coordinates (√3, √3) to polar form. Letr>0 and 0 ≤ 0 < 2.

Answer 1

Answer:
`(r,\theta) = \left(√(6), (\pi)/(4)\right)`

In other words,
`r = √(6) \ \text{ and } \ \theta = (\pi)/(4)` where theta is in radians.

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Work Shown:

`(\text{x},\text{y}) = (√(3),√(3))\\\\r = \sqrt{\text{x}^2+\text{y}^2}\\\\r = \sqrt{(√(3))^2+(√(3))^2}\\\\r = √(3+3)\\\\r = √(6)\\\\`

and,

`\theta = \tan^(-1)\left(\frac{\text{y}}{\text{x}}\right)\\\\\theta = \tan^(-1)\left((√(3))/(√(3))\right)\\\\\theta = \tan^(-1)\left(1\right)\\\\\theta = (\pi)/(4) \text{ radians}\\\\`

Use a calculator or the unit circle to arrive at the last step. Keep in mind that (√3, √3) is in the first quadrant where
`0 < \theta < (\pi)/(2)` since both x and y are positive.