How do you convert the Cartesian coordinates (10,10) to polar coordinates?

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How do you convert the Cartesian coordinates (10,10) to polar coordinates?

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Answer 1 · 2015-08-06T21:47:03

Cartesian: $(10; 10)$ $(10;10)$

Polar: $(10 \sqrt{2}; \frac{π}{4})$ $(10sqrt2;pi/4)$

Explanation:

The problem is represented by the graph below:

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In a 2D space, a point is found with two coordinates:

The cartesian coordinates are vertical and horizontal positions $(x; y)$ $(x;y)$ .

The polar coordinates are distance from origin and inclination with horizontal $(R, α)$ $(R,alpha)$ .

The three vectors $\to x, \to y and \to R$ $vecx, vecy and vecR$ create a right triangle in which you can apply the pythagorean theorem and the trigonometric properties. Thus, you find:

$R = \sqrt{x^{2} + y^{2}}$ $R=sqrt(x^2+y^2)$

$α = {cos}^{- 1} (\frac{x}{R}) = {sin}^{- 1} (\frac{y}{R})$ $alpha=cos^(-1)(x/R)=sin^(-1)(y/R)$

In your case, that is:

$R = \sqrt{10^{2} + 10^{2}} = \sqrt{100 + 100} = \sqrt{200} = 10 \sqrt{2}$ $R=sqrt(10^2+10^2)=sqrt(100+100)=sqrt200=10sqrt2$

$alpha=sin^(-1)(10/(10sqrt2))=sin^(-1)(1/sqrt2)=45°=pi/4$

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How do you convert the Cartesian coordinates (10,10) to polar coordinates?

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