Question

find the cartesian coordinates of the point whose polar coordinates are 1 by 2 ,7 pi by 3​

Answer 1

SOLUTION

TO DETERMINE

The cartesian coordinates of the point whose polar coordinates are

$\displaystyle \sf{ \bigg( \frac{1}{2} \: ,\: \frac{7\pi}{3} \bigg)}$

EVALUATION

Let ( r , θ ) be the polar coordinates of a point and ( x, y) be the coordinates of the same point in Cartesian Coordinate system

Then

$\displaystyle \sf{r = \frac{1}{2} \: \: \: and \: \: \: \theta = \frac{7\pi}{3} }$

Now the relation between cartesian coordinate system and polar coordinate system is

$\sf{x = r \cos \theta \: \: and \: \: y = r \sin \theta}$

$\displaystyle \sf{x = r \cos \theta}$

$\displaystyle \sf{ \implies \: x = \frac{1}{2} \cos \frac{7\pi}{3} }$

$\displaystyle \sf{ \implies \: x = \frac{1}{2} \times \frac{1}{2} }$

$\displaystyle \sf{ \implies \: x = \frac{1}{4} }$

Again

$\displaystyle \sf{y = r \sin \theta}$

$\displaystyle \sf{ \implies \: y = \frac{1}{2} \sin \frac{7\pi}{3} }$

$\displaystyle \sf{ \implies \: y = \frac{1}{2} \times \frac{ \sqrt{3} }{2} }$

$\displaystyle \sf{ \implies \: y = \frac{ \sqrt{3} }{4} }$

Hence the required cartesian coordinates

$\displaystyle \sf{ \bigg( \: \frac{1}{4} \: ,\: \frac{ \sqrt{3} }{4} \: \bigg)}$

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