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Given the polar coordinates of a point $P$, calculate the equivalent cartesian coordinates

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{question}

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To convert between Polar coordinates $[r, \\theta]$ and Cartesian coordinates $(x,y)$, we want to use the relationships

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\\[ x=r \\cos(\\theta) \\qquad y=r \\sin(\\theta) \\,.\\]

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Therefore,

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{advice}

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\\\\[ \\\\begin{split} x &\\\\,= \\\\var{r}\\\\simplify[all,!trig,simplifyFractions]{cos({d}*pi)} \\\\\\\\ &\\\\,= \\\\simplify[all]{{r*x1/sqrt(3)}sqrt(3)}, \\\\end{split} \\\\]

\\n

and 

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\\\\[ \\\\begin{split} y &\\\\,= \\\\var{r}\\\\simplify[all,!trig,fractionNumbers,simplifyFractions]{sin({d}*pi)} \\\\\\\\ &\\\\,= \\\\simplify[all]{{r*y1}}. \\\\end{split} \\\\]

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\\\\[ \\\\begin{split} x &\\\\,= \\\\var{r}\\\\simplify[all,!trig,fractionNumbers,simplifyFractions]{cos({d}*pi)} \\\\\\\\ &\\\\,= \\\\simplify[all]{{r*x1}}, \\\\end{split} \\\\]

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and 

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\\\\[ \\\\begin{split} y &\\\\,= \\\\var{r}\\\\simplify[all,!trig,fractionNumbers,simplifyFractions]{sin({d}*pi)} \\\\\\\\ &\\\\,= \\\\simplify[all]{{r*y1/sqrt(3)}sqrt(3)}. \\\\end{split} \\\\]

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\\\\[ \\\\begin{split} x &\\\\,= \\\\simplify{{k}sqrt(2)}\\\\simplify[all,!trig,fractionNumbers,simplifyFractions]{cos({d}*pi)} \\\\\\\\ &\\\\,= \\\\simplify[all]{{x1*k/abs(x1)}}, \\\\end{split} \\\\]

\\n

and 

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\\\\[ \\\\begin{split} y &\\\\,= \\\\simplify{{k}sqrt(2)}\\\\simplify[all,!trig,fractionNumbers,simplifyFractions]{sin({d}*pi)} \\\\\\\\ &\\\\,= \\\\simplify[all]{{y1*k/abs(y1)}}. \\\\end{split} \\\\]

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\\\\[ \\\\begin{split} x &\\\\,= \\\\var{r}\\\\simplify[all,!trig,fractionNumbers,simplifyFractions]{cos({d}*pi)} \\\\\\\\ &\\\\,= \\\\simplify[all]{{x}}, \\\\end{split} \\\\]

\\n

and 

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\\\\[ \\\\begin{split} y &\\\\,= \\\\var{r}\\\\simplify[all,!trig,fractionNumbers,simplifyFractions]{sin({d}*pi)} \\\\\\\\ &\\\\,= \\\\simplify[all]{{y}}. \\\\end{split} \\\\]

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The Polar coordinates of $P$ are $\\\\simplify[all,fractionNumbers,simplifyFractions]{[{r},{d}*pi]}$, what are the equivalent Cartesian coordinates?

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The Polar coordinates of $P$ are $\\\\simplify[all,fractionNumbers,simplifyFractions]{[{k}sqrt(2),{d}*pi]}$, what are the equivalent Cartesian coordinates?

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$(x,y)=$ ([[0]] , [[1]] )

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(Give your answers to 2 decimal places or in surd form where necessary)

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