Every point in the plane has exactly two polar coordinates.

A)The statement is true because every point in the plane has a unique polar representation.

B)The statement is false because left parenthesis r comma theta right parenthesis equals left parenthesis r comma theta plus 2 n pi right parenthesis(r,θ)=(r,θ+2nπ) for any integer n. These are all distinct polar coordinates.

C)The statement is false because left parenthesis r comma theta right parenthesis equals left parenthesis r comma theta plus n pi right parenthesis(r,θ)=(r,θ+nπ) for any integer n. These are all distinct polar coordinates.

D)The statement is false because left parenthesis r comma theta right parenthesis equals left parenthesis r comma 2 n pi theta right parenthesis(r,θ)=(r,2nπθ) for any integer n. These are all distinct polar coordinates.

The correct answer is B) The statement is false because (r, θ) = (r, θ + 2nπ) for any integer n. These are all distinct polar coordinates.

To understand why this is the correct answer, let's first review what polar coordinates are. In polar coordinates, a point in the plane is represented by its distance from the origin (r) and the angle it makes with the positive x-axis (θ).

Now, let's consider the statement that every point in the plane has exactly two polar coordinates. This implies that for any given point, there are only two possible values for r and θ that uniquely represent the point.

However, this is not true because the angle θ can be expressed in terms of θ + 2nπ, where n is an integer. Each value of n gives us a different angle that points to the same location. In other words, the point (r, θ) and the point (r, θ + 2nπ) represent the same point in the plane.

Therefore, there are infinitely many possible values for θ that represent the same point, disproving the statement. So, option B) correctly states that the statement is false.

B) The statement is false because left parenthesis r, theta right parenthesis equals left parenthesis r, theta plus 2n pi right parenthesis (r,θ)=(r,θ+2nπ) for any integer n. These are all distinct polar coordinates.

well, every 2pi brings theta back to the same point, so ...