To find all polar coordinates of the point P, given that its polar representation is (7, 3π), we start by understanding the polar coordinate system. In this system, a point is represented as (r, θ), where r is the distance from the origin, and θ is the angle measured from the positive x-axis.
The point P is represented as (7, 3π). Here, r is 7, meaning the point is 7 units away from the origin, and θ is 3π radians. However, angles in polar coordinates can have multiple forms, as they are periodic. An angle of 3π is equivalent to adding or subtracting multiples of 2π:
- • 3π + 2kπ for any integer k
Now, let’s express the angle in some alternative forms:
- For k = 0: 3π + 0 = 3π
- For k = 1: 3π + 2π = 5π
- For k = -1: 3π – 2π = π
Thus, other equivalent angles corresponding to the same point include π, 5π, and so on. Keeping in mind that the radius r can also be negative, we can represent the same point using negative radius values, which indicates the direction opposite to that of the angle. Therefore, we can write:
- • (-7, π) when we take r as negative and change θ to its supplementary angle, which is π.
- • (-7, 5π) with the same reasoning for the negative radius.
In summary, all polar coordinates for the point P represented as (7, 3π) can be expressed as:
- (7, 3π)
- (7, 5π)
- (7, π)
- (-7, π)
- (-7, 5π)
- (-7, 3π)
These points all represent the same location in the polar coordinate system, illustrating the flexibility of polar coordinates in representing the same point through different angles and signs of radius.