To find the polar coordinates of the point where the angle is given as θ = 9π/5, we first need to understand the basic concept of polar coordinates. In polar coordinates, a point in a plane is represented by two values: the radius (r) and the angle (θ).
The angle 9π/5 is more than 2π (which is the equivalent of 10π/5), so we can find an equivalent angle by subtracting 2π:
θ = 9π/5 – 2π = 9π/5 – 10π/5 = -π/5.
This means that 9π/5 radians can be simplified to an equivalent angle of -π/5. However, angles in polar coordinates can be expressed as multiple rotations around the circle. Thus, we can say the angle can be adjusted by adding or subtracting multiples of 2π:
θ = -π/5 + 2kπ, where k is any integer.
The radius (r) can be any positive number or can also be negative. If the radius is positive, the point will be at that angle from the origin; if the radius is negative, the point will be in the opposite direction. Therefore, the general form of the polar coordinates for point p can be written as:
(r, θ) = (r, 9π/5) and (r, θ) = (r, -π/5) for any r > 0, plus considering negative values:
(|r|, 9π/5 + π) for r < 0, which is equivalent to (|r|, 14π/5).
In summary, the polar coordinates for point p can be expressed as:
- (r, 9π/5)
- (r, -π/5)
- (|r|, 14π/5) where r < 0