Question

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

Answer 1

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,

& x=r\cos \theta \\\ & y=r\sin \theta \\\ \end{aligned}$$ We can now substitute the value of r and $$\theta $$ in the above formula, we get $$\begin{aligned} & \Rightarrow x=-2\cos \left( 0.5236 \right) \\\ & \Rightarrow y=-2\sin \left( 0.5236 \right) \\\ \end{aligned}$$ We can calculate the sine and cosine using a calculator. $$\begin{aligned} & \Rightarrow \cos \left( 0.5236 \right)=0.866 \\\ & \Rightarrow \sin \left( 0.5236 \right)=0.5 \\\ \end{aligned}$$ We can now substitute these values in the above step and we can now simplify the above step, $$\begin{aligned} & \Rightarrow x=-2\times 0.866=-1.732 \\\ & \Rightarrow y=0\times 0.5=-1 \\\ \end{aligned}$$ Therefore, the cartesian coordinates are $$\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.