dictetzqh

2021-11-18

The Cartesian coordinates of a point are given. (i) Find polar coordinates $\left(r,\theta \right)$ of the point, where r > 0 and $0\le \theta <2\pi$ (ii) Find polar coordinates $\left(r,\theta \right)$ of the point, where r < 0 and $0\le \theta <2\pi$ . (2, -2)

Jennifer Hill

Step1
$|r|=\sqrt{{2}^{2}+{\left(-2\right)}^{2}}=\sqrt{4+4}=\sqrt{8}=2\sqrt{2}$

Step2
Recall that:
$r\mathrm{cos}\theta =x$
Therefore
$2\sqrt{2}\mathrm{cos}\theta =2$

$\mathrm{cos}\theta =\frac{1}{\sqrt{2}}$ Equation1
Step3
Recall that:
$r\mathrm{sin}\theta =y$
Therefore
$2\sqrt{2}\mathrm{sin}\theta =-2$

$\mathrm{sin}\theta =-\frac{1}{\sqrt{2}}$ Equation 2
Step4
Equation 1 and 2 are satisfied by $\theta =\frac{7\pi }{4}$
Polar coordinates of the point are
$\left(2\sqrt{2},\frac{7\pi }{4}\right)$

Clara Clark

Step1
$|r|=\sqrt{{2}^{2}+{\left(-2\right)}^{2}}=\sqrt{4+4}=\sqrt{8}=2\sqrt{2}$

Step2
Recall that:
$r\mathrm{cos}\theta =x$
Therefore
$2\sqrt{2}\mathrm{cos}\theta =2$

$\mathrm{cos}\theta =-\frac{1}{\sqrt{2}}$ Equation1
Step3
Recall that:
$r\mathrm{sin}\theta =y$
Therefore
$-2\sqrt{2}\mathrm{sin}\theta =-2$

$\mathrm{sin}\theta =\frac{1}{\sqrt{2}}$ Equation 2
Step4
Equation 1 and 2 are satisfied by $\theta =\frac{7\pi }{4}$
Polar coordinates of the point are
$\left(2\sqrt{2},\frac{7\pi }{4}\right)$
Step 5
Equation 1 and 2 are satisfied by $\theta =\frac{3\pi }{4}$
Polar coordinates of the point are
$\left(-2\sqrt{2},\frac{3\pi }{4}\right)$

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