Question

How do you convert $\left( {2,\dfrac{\pi }{4}} \right)$ into rectangular coordinates?

Answer 1

The rectangular coordinate system consists of two real number lines that intersect at a right angle. The first number is called the x-coordinate, and the second number is called the y-coordinate. Here, to find rectangular coordinates we need to find the distance of the projection along the x-axis for the first point and along the y-axis for the second. Using the formulae that links Polar and Cartesian coordinates we need to find the value of x and y to get the rectangular coordinates.

Formula used:
$x = r\cos \theta$
$y = r\sin \theta$

Complete step by step solution:
Given,
Polar coordinates: $\left( {2,\dfrac{\pi }{4}} \right)$, in which we need to convert it into rectangular coordinates.
These coordinates describe a line 2 units long, starting at the origin, $\left( {0,0} \right)$, at an angle of $\dfrac{\pi }{4}$ radians anticlockwise (counter clockwise) from the positive axis.
For rectangular coordinates we need to find the distance of the projection along the x-axis for the first point and along the y-axis.
Using the formulae that links Polar and Cartesian coordinates as:
$x = r\cos \theta$
$y = r\sin \theta$
Here, as given we have $r = 2$ and $\theta = \dfrac{\pi }{4}$. Hence, substitute the values in the formulas as:
$x = r\cos \theta$
$\Rightarrow x = 2\cos \left( {\dfrac{\pi }{4}} \right)$
We, know that the value of $\cos \left( {\dfrac{\pi }{4}} \right)$ is $\dfrac{1}{{\sqrt 2 }}$, hence we get:
$\Rightarrow x = 2 \cdot \dfrac{1}{{\sqrt 2 }}$
$\Rightarrow x = \dfrac{2}{{\sqrt 2 }} \times \dfrac{{\sqrt 2 }}{{\sqrt 2 }}$
$\Rightarrow x = \sqrt 2$
Hence, the value of x is:
$\Rightarrow x = 1.414$
Hence, now substitute the values in the formulas as:
$y = r\sin \theta$
$\Rightarrow y = 2\sin \left( {\dfrac{\pi }{4}} \right)$
We, know that the value of $\sin \left( {\dfrac{\pi }{4}} \right)$ is $\dfrac{1}{{\sqrt 2 }}$, hence we get:
$\Rightarrow y = 2 \cdot \dfrac{1}{{\sqrt 2 }}$
$\Rightarrow y = \dfrac{2}{{\sqrt 2 }} \times \dfrac{{\sqrt 2 }}{{\sqrt 2 }}$
$\Rightarrow y = \sqrt 2$
Hence, the value of y is:
$\Rightarrow y = 1.414$
Therefore,
$\left( {2,\dfrac{\pi }{4}} \right) \to \left( {\sqrt 2 ,\sqrt 2 } \right)$

Note: The key point to convert the given points into rectangular coordinates is that we must know the formulae that links Polar and Cartesian coordinates. And we can also use given polar coordinate; $\dfrac{\pi }{4}$as$45^\circ$, as some find it easier to work in radians, some in degrees, we have taken it as given, i.e., in radians.