We have two formulas to convert from polar to cartesian coordinates:
x = r cos θ
y = r sin θ
We need to find the values of r and θ, in terms of x = -4 √2 and y = -4 √2
If we divide:
y/x = sin θ / cos θ = tan θ
Replacing x and y we find that:
tan θ = ( -4 √2 ) / ( -4 √2 ) = 1
We can calculate the inverse of the tan function:
θ = arc tan (1) = 225º,
Sorry, but the inverse of the tan function has many results. We can see that the results are two, 45º and also 45º+180º = 225º. The first one give us a point in the first quadrant (positive values for x and y); and the second one give us a point in the third quadrant (negative values for x and y). So we must select the second value in order to have negative values of x and y.
Now we must calculate the value of r. In order to calculate the value of r we can use the following formula:
[tex]x^2+y^2\text{ = r}^2[/tex]
So, the value of r is (if we replace the values of x and y and compute the square root):
[tex]r\text{ = }\sqrt[]{x^2+y^2}\text{ = }\sqrt[]{64}=\text{ 8}[/tex]
So the answer is:
r = 8
θ = 225º
