Question

How do you find the polar coordinates of the point?

Answer 1

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.