The equation given is in the Cartesian form of a circle which is described as (x-a)² + (y-b)² = r². In our case, it's given as x² + (y + 5)² = 25.
First, rewrite the equation in the standard form. To do this, notice that a and b are the coordinates of the center of the circle and r is the radius. For our equation, we can rewrite it as x² + (y - (-5))² = 5². From this, we can see that the circle is centered at (0,-5) and its radius is 5. So, (a,b) = (0,-5) and r=5.
The polar form of a circle equation for a circle centered not at the origin is r=rsin(θ-φ), where φ is equal to the inverse tangent of b/a.
To find the value of φ, compute φ = inverse tangent (b/a) = inverse tangent of (-5/0) which is minus-half Pi or about -1.5707963267948966; that's because tangent is negative for second and fourth quadrants and the ratio of -5 to 0 points the direction towards negative y-axis.
Finally, substitute r and φ into the polar equation. So, in this case the polar equation becomes r = 5 sin(θ - φ) which gives as the result r = 5 sin(θ - (-1.5707963267948966)).
We can simplify the operation inside the sin function because subtracting negative number equals adding the absolute value of that number.
So, the final polar form of the equation is: r = 5 sin(θ + 1.5707963267948966).