Question

What is the Cartesian form of $\left( {1,{\text{ }}\dfrac{\pi }{4}} \right)$ ?

Answer 1

The given form is in polar form. In order to find the cartesian form we will first compare the polar coordinates to $\left( {r,{\text{ }}\theta } \right)$ and find the values of $r$ and $\theta$ . Then we will use the formula $x = r\cos \theta$ and $y = r\sin \theta$ to find out the values of $x$ coordinate and $y$ coordinate. Then we will write $x$ and $y$ as ordered pair and hence we will get the required cartesian form of $\left( {1,{\text{ }}\dfrac{\pi }{4}} \right)$

Complete step by step solution:
We are given polar coordinates points i.e., $\left( {1,{\text{ }}\dfrac{\pi }{4}} \right)$
And we have to find its cartesian form.
So, first of all on comparing the given point with the standard form of polar coordinates i.e., $\left( {r,{\text{ }}\theta } \right)$
We can say that
$r = 1$ and $\theta = \dfrac{\pi }{4}$
Now we know that the cartesian coordinates of a point in the polar form $\left( {r,{\text{ }}\theta } \right)$ is given by the formulas
$x = r\cos \theta {\text{ }} - - - \left( i \right)$
$y = r\sin \theta {\text{ }} - - - \left( {ii} \right)$
Now on substituting the values of $r$ and $\theta$ in the equation $\left( i \right)$ we get
$x = 1 \cdot \cos \left( {\dfrac{\pi }{4}} \right)$
We know that $\cos \left( {\dfrac{\pi }{4}} \right) = \dfrac{1}{{\sqrt 2 }}$
So, on substituting in the above equation, we get
$\Rightarrow x = \dfrac{1}{{\sqrt 2 }}$
Hence, the $x$ coordinate is $\dfrac{1}{{\sqrt 2 }}$
Now similarly, we will find the $y$ coordinate.
So, on substituting the values of $r$ and $\theta$ in the equation $\left( {ii} \right)$ we get
$y = 1 \cdot \sin \left( {\dfrac{\pi }{4}} \right)$
We know that $\sin \left( {\dfrac{\pi }{4}} \right) = \dfrac{1}{{\sqrt 2 }}$
So, on substituting in the above equation, we get
$\Rightarrow y = \dfrac{1}{{\sqrt 2 }}$
Hence, the $y$ coordinate is $\dfrac{1}{{\sqrt 2 }}$
Therefore, the Cartesian coordinate of the given point is $\left( {\dfrac{1}{{\sqrt 2 }}{\text{ }},\dfrac{1}{{\sqrt 2 }}} \right)$
Hence, the Cartesian form of $\left( {1,{\text{ }}\dfrac{\pi }{4}} \right)$ is $\left( {\dfrac{1}{{\sqrt 2 }}{\text{ }},\dfrac{1}{{\sqrt 2 }}} \right)$.

Note:
Cartesian coordinates are used to mark how far along and how far up a point is while polar coordinates are used to mark how far away and at what angle a point is. Basically, in a cartesian point system, a point is located using the perpendicular distance from the $X$ and $Y$ axes. And in a polar coordinate, a point is located using the distance from the origin and the angle this shortest distance makes with the positive $X$ axis.
Also remember to convert a Cartesian coordinate to polar form the formula used is:
$r = \sqrt {{x^2} + {y^2}}$
$\theta = {\tan ^{ - 1}}\left( {\dfrac{y}{x}} \right)$