Final answer:
To convert the equation ρ = 1 to rectangular coordinates and write it in standard form, use the following transformations from polar to rectangular coordinates: x = ρ • cos(θ) and y = ρ • sin(θ). Since ρ = 1, the equations become x = cos(θ) and y = sin(θ), resulting in the standard form of the equation of a circle: x² + y² = 1.
Step-by-step explanation:
To convert the equation ρ = 1 to rectangular coordinates and write it in standard form, we can recall the relationship between polar coordinates and rectangular coordinates. In polar coordinates, a point P is represented by (ρ, θ), where ρ is the radial distance from the origin and θ is the angle measured counterclockwise from the positive x-axis. The rectangular coordinates (x, y) can be found using the equations:
x = ρ • cos(θ)
y = ρ • sin(θ)
Since ρ = 1 for our specific equation, these equations simplify to:
x = cos(θ)
y = sin(θ)
The equation ρ = 1 represents a circle with a radius of 1 centered at the origin in polar coordinates. Therefore, in rectangular coordinates, the equation of this circle is:
x² + y² = 1
This is the standard form for the equation of a circle with a radius of one unit in rectangular coordinates.