Question

How do you convert $\left( {1,\dfrac{\pi }{2}} \right)$ into rectangular form?

Answer 1

We will use the Cartesian format structure and substitute the values of $1$ and $\dfrac{\pi }{2}$ in the formula, then doing some simplification to get the required answer.

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

Complete step-by-step solution:
We know that for a given polar coordinate system in the form of $(r,\theta )$
The conversion formula to convert this into the rectangular form is: $x = r\cos \theta$and$y = r\sin \theta$.
Now the given coordinates are $\left( {1,\dfrac{\pi }{2}} \right)$ so we can conclude that $r = 1$ and $\theta = \dfrac{\pi }{2}$
On substituting the value in $x = r\cos \theta$we get:
$x = 1\cos \left( {\dfrac{\pi }{2}} \right)$
Now we know that the value of $\cos \left( {\dfrac{\pi }{2}} \right) = 0$
Therefore, $x = 0$.
And, on substituting the value in $y = r\sin \theta$we get:
$y = 1\sin \left( {\dfrac{\pi }{2}} \right)$
Now we know that the value of $\sin \left( {\dfrac{\pi }{2}} \right) = 1$
Therefore, $y = 1$.

Therefore, the coordinates $(x,y)$ for the given coordinates is $\left( {0,1} \right)$ which is in the rectangular format.

Note: It is to be remembered that the value of $r$ can be calculated using the rectangular coordinates with the formula $r = \sqrt {{x^2} + {y^2}}$
In the above question we have been told to convert the polar coordinates into the rectangular format. The rectangular format is also called the Cartesian format.
In the Cartesian coordinate system we define the points $x$ and $y$ from how much distance they are from the origin. The distance $x$ is the horizontal distance from the origin and the distance $y$ is the vertical distance, together they create a coordinate
In the polar coordinate system instead of going out through the origin till we hit the point in space and calculate the angle from the $x$ axis to the line reaching that point
These coordinate systems are used in physics for calculation purposes in the two-dimensional space.
When used in the third dimension, an additional parameter $z$ is used in the rectangular system, and an additional angle $\varphi$ is used in the polar system.