How do you convert the polar coordinate ( 4.5541 , 1.2352 ) into cartesian coordinates?

Jan 29, 2016

Polar coordinates are in the form $\left(r , \theta\right)$, Cartesian (also called rectangular) coordinates are in the form $\left(x , y\right)$. In this case the Cartesian coordinates are $\left(1.50 , 4.30\right)$.

Explanation:

Polar coordinates are stated in the form $\left(r , \theta\right)$, a distance from the origin and an angle, in radians, counterclockwise from the positive $x$ axis. In this case, we have a point $4.5541$ units from the origin, at an angle of $1.2352$ $r a d$.

Cartesian coordinates, sometimes also called 'rectangular coordinates', are expressed in the form $\left(x , y\right)$ where $x$ is the distance along the $x$ axis and $y$ is the distance up the $y$ axis.

To convert from one to the other, we use trigonometry. The $r$ value from the polar coordinates is the hypotenuse of a right-angled triangle, and the $x$ and $y$ coordinates are the adjacent and opposite sides respectively, from the perspective of the angle $\theta$.

Knowing the angle and a side allows us to use the definitions of sine and cosine to find the $x$ and $y$ coordinates:

$\sin \theta = \frac{o p p o s i t e}{\text{hypotenuse}} \to$ $o p p o s i t e = \text{hypotenuse} \cdot \sin \theta$

In this case, $y = 4.5541 \cdot \sin \left(1.2352\right) = 4.5541 \cdot 0.9442 = 4.30$

(Be sure to ensure that your calculator is on the 'radians', not 'degrees' setting when calculating the sine and cosine in this context.)

By similar reasoning:

$\cos \theta = \frac{a \mathrm{dj} a c e n t}{\text{hypotenuse}} \to$ $a \mathrm{dj} a c e n t = \text{hypotenuse} \cdot \cos \theta$

In this case, $x = 4.5541 \cdot \cos \left(1.2352\right) = 4.5541 \cdot 0.3293 = 1.50$

So that you don't have to remember all this discussion every time, you can memorize (but should understand):

$x = r \cos \theta$

$y = r \sin \theta$

Combine these, and the Cartesian coordinates for the point are $\left(1.50 , 4.30\right)$.