Answer:
Well, when we have a point in rectangular coordinates (x, y) and we want to write it in polar coordinates (r, θ) the rule we use is:
r = √(x^2 + y^2)
θ = Atg(Iy/xI) + 90°*(n - 1)
where n is number of the quadrant where our point is.
For example, if we are on the second quadrant, we use n = 2.
Now, if instead, we have the point (r, θ) and we want to rewrite it in rectangular coordinates, then the transformation is:
x = r*cos(θ)
y = r*sin(θ)
Now we have the point (- 2, -(5pi)/4) because the first part is negative, this number can not be in polar coordinates (r can not be negative) then we have:
x = -2
y = -(5*pi)/4
Using the first relation, we can find that:
r = √( (-2)^2 + (-5*3.14/4)^2) = 4.4
(rememer that the point (- 2, -(5pi)/4) is on the third quadrant, then we will use n = 3.)
θ = Atg(-(5*pi)/4/-2) + 90°*(3 - 1)= Atg((5*pi)/2) + 180° = 82.7° + 180° = 262.7°
Then the point that represents (- 2, - (5pi)/4) in polar coordinates is the point:
(4.4, 262.7°)
If instead of degrees, the angle part is written in radians, we have:
180° = 3.14 radians.
262.7° = x radians.
then:
x/3.14 = (262.7)/180
x = 3.14*(262.7)/180 = 4.58 radians.
then the point will be:
(4.4, 4.58 radians)