Question

How do you convert polar coordinate \[\left( -2,0.5236 \right)\] into cartesian coordinate?

Answer 1

Hint: In this problem, we have to convert the given polar form to the rectangular form. We are given a polar coordinate to convert them into a rectangular coordinate. We should know that the polar coordinates are given as \[\left( r,\theta \right)\] format and the rectangular coordinate is in the form of \[\left( x,y \right)\] . We can use the conversion formula to convert from polar coordinates to rectangular coordinates. We can substitute the value of r and \[\theta \] in the conversion formula which is given in this problem as polar coordinate, to find the rectangular coordinates.

Complete step by step solution:
We know that the given polar coordinates are,
\[\left( -2,0.5236 \right)\]
Where, r = -2, \[\theta =0.5236\]
We know that the conversion formula to convert from polar coordinates to rectangular coordinates is,
\[\begin{align}
  & x=r\cos \theta \\
 & y=r\sin \theta \\
\end{align}\]
We can now substitute the value of r and \[\theta \] in the above formula, we get
\[\begin{align}
  & \Rightarrow x=-2\cos \left( 0.5236 \right) \\
 & \Rightarrow y=-2\sin \left( 0.5236 \right) \\
\end{align}\]
We can calculate the sine and cosine using a calculator.
\[\begin{align}
  & \Rightarrow \cos \left( 0.5236 \right)=0.866 \\
 & \Rightarrow \sin \left( 0.5236 \right)=0.5 \\
\end{align}\]
We can now substitute these values in the above step and we can now simplify the above step,
\[\begin{align}
  & \Rightarrow x=-2\times 0.866=-1.732 \\
 & \Rightarrow y=0\times 0.5=-1 \\
\end{align}\]
Therefore, the cartesian coordinates are \[\left( -1.732,-1 \right)\].

Note: Students make mistakes while writing the conversion formula to convert from polar form to a rectangular form, we should know that the polar coordinates are given as \[\left( r,\theta \right)\] format and the rectangular coordinate is in the form of \[\left( x,y \right)\] . We can calculate the trigonometric values using the calculator to get an exact answer and to simplify it.