To write the given equation in polar coordinates, we need to substitute the rectangular coordinates (x, y) with their equivalent in polar coordinates (r, θ). In polar coordinates, r represents the distance from the origin, and θ represents the angle with respect to the positive x-axis.
To convert from rectangular to polar coordinates, we use the following equations:
r = √(x^2 + y^2)
θ = arctan(y / x)
Applying these conversions, we can rewrite the equation (x-4)^2 + y^2 = 16 as:
(r cos θ - 4)^2 + (r sin θ)^2 = 16
Now, let's solve this equation for r.
Expanding the equation, we get:
(r^2 cos^2 θ - 8r cos θ + 16) + (r^2 sin^2 θ) = 16
Combining like terms:
r^2 cos^2 θ + r^2 sin^2 θ - 8r cos θ = 0
Using the trigonometric identity cos^2 θ + sin^2 θ = 1, we have:
r^2 - 8r cos θ = 0
Factoring out r, we obtain:
r(r - 8 cos θ) = 0
To solve for r, we have two possibilities:
1. r = 0
2. r - 8 cos θ = 0
If r = 0, it represents the origin in polar coordinates. So, let's solve the second equation for r:
r - 8 cos θ = 0
r = 8 cos θ
Therefore, the equivalent equation in polar coordinates is r = 8 cos θ.