Consider the relation between the rectangular and polar coordinates

.

Here and .

and

and .

As the inverse cosine cannot be negative and is defined on .

So a second expression is needed when is negative.

Consider the points and with radius .

and .

and .

For , .

Observe the graph,

is located in the third quadrant.

Inorder to obtain  the correct directed angle, subtract from .

Thus,

Thus, which is in the third quadrant.

For , .

Observe the graph,

is located in the fourth quadrant.

In order to obtain  the correct directed angle, subtract from .

Thus,

Therefore, 

when is positive and when is negative.

As the inverse sine cannot be negative and is defined on .

So a second expression is needed when is negative.

Consider the points and with radius .

and .

and .

For , .

Observe the graph:

is located in the second quadrant.

In order to obtain the correct directed angle, subtract from .

Thus,

.

Thus, which is in the second quadrant.

For , .

Observe the graph:

is located in the third quadrant.

In order to obtain the correct directed angle, subtract from .

Thus,

.

Thus, which is in the third quadrant.

Therefore, 

when is positive and when is negative.

when is positive and when is negative.

when is positive and when is negative.