Question

How do you find the polar coordinates of the point?

Answer 1

Hint:in this question, we would convert the Cartesian point into polar coordinates of the point. The final answer would be in the form of \[\left( {r,\theta } \right)\]. This question also involves the operation of addition/ subtraction/ multiplication/ division. We need to know Pythagoras#39;s theorem
for finding the formula to find the value of \[r\] from Cartesian points.

Complete step by step solution:
In this question, we need to convert the Cartesian points \[\left( {x,y} \right)\] into polar
points \[\left( {r,\theta } \right)\]. For that, we assume the following Cartesian point,
\[\left( {x,y} \right) = \left( {0,2} \right)\]
First, we have to find the value of \[r\]in\[\left( {r,\theta } \right)\].
We know that,
\[r = \sqrt {{x^2} + {y^2}} \] (By using Pythagoras theorem)
Here, \[x = 0\]and\[y = 2\]
\[
r = \sqrt {\left( {{0^2}} \right) + \left( {{2^2}} \right)} \\
r = \sqrt 4 \\
\]
We know that, the square root value of \[4\]is\[2\]
So, we get
\[r = 2\]
Next, we have to find the value of \[\theta \]in\[\left( {r,\theta } \right)\]
We know that,
\[\tan \theta = \left( {\dfrac{y}{x}} \right)\]
Here, \[x = 0\]and\[y = 2\]
So, we get
\[\tan \theta = \left( {\dfrac{2}{0}} \right)\]
We know that anything divided by zero is equal to infinity.
So, we get
\[
\tan \theta = \infty \\
\\
\]
\[\theta = arc\tan (\infty )\]
(When \[\tan \]it moves from left side to right side of the equation it converts into\[\arctan \].)
\[\theta = \left( {{{90}^ \circ }} \right)\]
The above equation can also be written as,
\[\theta = \left( {\dfrac{\pi }{2}} \right)\]
So, the final answer is, \[\left( {r,\theta } \right) = \left( {2,\dfrac{\pi }{2}} \right)\]

Note: In this type of question remember that the polar coordinates are \[\left( {r,\theta } \right)\].
Note that, the formula for finding the values of \[r\]and\[\theta \](\[r = \sqrt {{x^2} + {y^2}}
\],\[\tan \theta = \left( {\dfrac{y}{x}} \right)\]). So, the final answer would be in the form of\[\left( {r,\theta } \right)\]. Also, remember the trigonometric table values for finding the value of \[\theta \]from\[\tan \theta \]. Note that when \[\tan \]it moves from the left side to the right side of the equation it converts into \[\arctan \]. Also, remember that anything divided by zero becomes infinity.