Question

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

Answer 1

Hint: 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:
\[r = \sqrt {{x^2} + {y^2}} \]
\[\theta = {\tan ^{ - 1}}\left( {\dfrac{y}{x}} \right)\]