Question

Convert the rectangular coordinate, (-472, -4/2),to a polar coordinate (r, 0).r=[ Select]A= [Select ]

Answer 1

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:

`x^2+y^2\text{ = r}^2`

So, the value of r is (if we replace the values of x and y and compute the square root):

`r\text{ = }\sqrt[]{x^2+y^2}\text{ = }\sqrt[]{64}=\text{ 8}`

So the answer is:

r = 8

θ = 225º