Question

Convert the rectangular coordinates (3,4) into polar coordinates ​

Answer 1

(5,53°) or (5,53.13010235°) if you want an exact value.

Explanation:

Polar coordinate is in
`\displaystyle{(r,\theta)}` form and rectangular coordinate is in
`\displaystyle{(x,y)}` form.

To convert from rectangular coordinate to polar coordinate, you’ll have to know that:

`\displaystyle{x=r\cos \theta}\\\displaystyle{y=r\sin \theta}`

Above is crucial when it comes to converting from rectangular coordinate to polar coordinate, or other way around. The thing is to convert from rectangular to polar, you’ll also have to know radius too which you can find by:

`\displaystyle{r=√(x^2+y^2)}`

We know that x = 3 and y = 4, substitute in the formula:

`\displaystyle{r=√(3^2+4^2)}\\\\\displaystyle{r=√(9+16)}\\\\\displaystyle{r=√(25)}\\\\\displaystyle{r=5}`

We know that radius is 5. We will have to find the theta next, first substitute in the conversion:

`\displaystyle{3=5\cos \theta}\\\displaystyle{4=5\sin \theta}`

Since this can generally be found through calculator as it’s not a part of special triangle - can be found using hand method - The approximation of both theta’s values are 53°.

Hence, the polar coordinate is (5,53°) or (5,53.13010235°) if you want an exact value.