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Question: How do you convert \((0,\;2)\) from Cartesian to polar coordinates?...

How do you convert (0,  2)(0,\;2) from Cartesian to polar coordinates?

Explanation

Solution

Cartesian coordinates (x,  y)(x,\;y) can be converted into polar coordinates (r,  θ)(r,\;\theta ) by following method: First find the value of “r” which represents the actual distance of the point from the origin, And then find the value of “θ\theta ”, which represents the angle of the point with respect to horizontal axis in the anti clockwise direction having the origin as the vertex.

Complete step by step solution:
In order to convert (0,  2)(0,\;2) from rectangular coordinate system (also known as Cartesian coordinates) to polar coordinates (r,  θ)(r,\;\theta ), we have to first determine the value of “r” which is the distance between the origin and the given point.
We know that the distance between two points having coordinates (a,  b)  and  (c,  d)(a,\;b)\;{\text{and}}\;(c,\;d) can be
given as
r=(a2−c2)(b2−d2)r = \sqrt {({a^2} - {c^2})({b^2} - {d^2})}
So finding the distance between origin (0,  0)(0,\;0) and the given point (0,  2)(0,\;2)
r=(02−02)+(02−22) r=0+(−22) r=4 r=±2  r = \sqrt {({0^2} - {0^2}) + ({0^2} - {2^2})} \\\ r = \sqrt {0 + ( - {2^2})} \\\ r = \sqrt 4 \\\ r = \pm 2 \\\
Since distance is a positive quantity it never takes negative values
∴r=2\therefore r = 2
Now we will find value of θ\theta which gives information about the angle by which the point is raised from the horizontal axis in the anti clockwise direction
We can find the value of angle of a point (a,  b)(a,\;b) with respect to horizontal axis in clockwise direction having the origin as the vertex as follows
tan⁡θ=ba ∴θ=tan⁡−1(ba)  \tan \theta = \dfrac{b}{a} \\\ \therefore \theta = {\tan ^{ - 1}}\left( {\dfrac{b}{a}} \right) \\\
Now finding the value of θ\theta for the point (0,  2)(0,\;2)
⇒θ=tan⁡−1(20)\Rightarrow \theta = {\tan ^{ - 1}}\left( {\dfrac{2}{0}} \right)
Since the value (20)\left( {\dfrac{2}{0}} \right) is defined, but tangent function is always not defined at π2\dfrac{\pi }{2}
∴θ=π2\therefore \theta = \dfrac{\pi }{2}
Then the required polar coordinates will be (r,  θ)≡(2,  π2)(r,\;\theta ) \equiv \left( {2,\;\dfrac{\pi }{2}} \right)

Note: We have learnt how to convert Cartesian coordinates into polar coordinates in this problem. But for vice-versa, that is for converting polar coordinates (r,  θ)(r,\;\theta ) into Cartesian coordinates (x,  y)(x,\;y) there is a direct formula given following:
x=rcos⁡θ  and  y=sin⁡θx = r\cos \theta \;{\text{and}}\;y = \sin \theta