# What are the approximate rectangular coordinates for the point with polar coordinates (5, 30°)?

Oct 5, 2016

$\left(\frac{5 \sqrt{3}}{2} , \frac{5}{2}\right)$

#### Explanation:

$\left(5 , 30\right) \implies \left(r , \theta\right)$

A triangle with a 30 degree angle is a special triangle.

90 degrees corresponds with the side of length $2 x$
60 degrees corresponds with the side of length $x \sqrt{3}$
30 degrees corresponds with the side of length $1 x$

90 also corresponds to the length of $r$ which is 5.

$\frac{\cancel{2} x}{\cancel{2}} = \frac{5}{2}$

Which now shows that ...

90 degrees corresponds with the side of length $5$
60 degrees corresponds with the side of length $\frac{5 \sqrt{3}}{2}$
30 degrees corresponds with the side of length $\frac{5}{2}$

$x = 5 \cos \left(30\right) = 5 \left(\frac{\cancel{5} \frac{\sqrt{3}}{2}}{\cancel{5}}\right) = \frac{5 \sqrt{3}}{2}$

$y = 5 \sin \left(30\right) = \cancel{5} \left(\frac{\frac{5}{2}}{\cancel{5}}\right) = \frac{5}{2}$

$\left(\frac{5 \sqrt{3}}{2} , \frac{5}{2}\right)$

View this tutorial for another explanation of a similar problem.