Question

How do you change the polar coordinates $(4,{45^ \circ })$ into rectangular coordinates?

Answer 1

Polar coordinates are those coordinates which are used to denote a particular graph or a line in the coordinate plane but instead of using Cartesian coordinates which we normally use for the purpose which are denoted by $(x,y)$ we denote the polar coordinates of the curve or a graph by $(r,\theta )$ where $r$ is the distance of any given point from the origin and the $\theta$i s the angle the given point makes with the $x$ axis in particular. These coordinates are interchangeable and a curve or a line can easily be represented by both of them. For converting the given polar coordinates into rectangular coordinates (also called as ‘Cartesian coordinates’) we use the formula for conversion
$x = r\cos \theta$ and
$y = r\sin \theta$which will give us the values of $x$ and $y$ for the given coordinates. Here $r = 4$ and $\theta = {45^ \circ }$.

Complete step by step solution:
The polar coordinates can be easily converted into the rectangular coordinates by using the above formula
$x = r\cos \theta$ and
$y = r\sin \theta$which will give us the values of $x$ and $y$ for the given coordinates. Here $r = 4$ and $\theta = {45^ \circ }$.
The value of $x$can be calculated as:
$x = 4\cos {45^ \circ }$
Solving which we get
$x = 2\sqrt 2$
And similarly solving for $y$we get
$y = 4\sin {45^ \circ }$
Solving which we get
$y = 2\sqrt 2$
Thus the rectangular coordinates for the given polar coordinates are written as $(2\sqrt 2 ,2\sqrt 2 )$.

Note: The polar coordinates use both $r$ and $\theta$ . we know that the angle the given point makes with the $x$ axis is calculated as $\theta$ but we should remember that distance of the point from origin is calculated by the distance formula and is written as $r$.