The parametric form of the circle x² + y² = 25 is:
x(t) = 5 cos(t)
y(t) = 5 sin(t)
This is an exercise in parametric equations, which are a way of representing a curve or surface on the plane or in space using arbitrary values or a constant called a parameter instead of using an independent variable. In the case of circles, the points (x,y) can be expressed from a single variable θ, which is called a parameter. The parametric equations of a circle with radius r ≥ 0 and center (h,k) are given by x = h + rcosθ y = k + rsinθ, where 0 < θ < 2π. The parametric equation of a circle can be expressed from the angle θ of the point through the circle with respect to the x-coordinate axis. The parametric equations of the conics, including the circumference, can be deparameterized to obtain the canonical equation.
The parametric equation of a circle can be expressed from the angle θ of the point through the circle with respect to the x-coordinate axis. The angle can be expressed in radians (θ∈[0,2π]) or sexagesimal degrees (θ∈[0º,360º]). The parametric equation of a circle with center C=(-1,3) and radius r=2 cm is defined by x = -1 + 2cosθ and y = 3 + 2sinθ.
The parametric equations of a curve are expressed from the variable (or parameter) t, such that x = x(t) and y = y(t). Each value of t determines a point (x,y) in the plane. The parametric equations of a circle with radius r ≥ 0 and center (h,k) are given by x = h + rcosθ y = k + rsinθ, where 0 < θ < 2π. The parametric equation of a circle centered at the origin and with radius r is x² + y² = r².
The parametric equations for a curve are not unique. They represent the same circle with center at (0,0) and radius R. The parametric equations of the conics, including the circumference, can be deparameterized to obtain the canonical equation. To go from the parametric to the Cartesian equations, the parameter t must be eliminated. The parametric equations of a point on a circle of radius r at an angle θ are x = rcos(θ) and y = rsin(θ).
To get this parametric form of x² + y² = 25, we can use the trigonometric identities:
cos²(t) + sin²(t) = 1
Multiplying both sides by 25, we get:
25 cos²(t) + 25 sin²(t) = 25
Which is equivalent to:
x² + y² = 25
Therefore, we can write:
x(t) = 5 cos(t), y(t) = 5 sin(t)
The parametric form of the circle x² + y² = 25 is:
x(t) = 5 cos(t)
y(t) = 5 sin(t)
This means that for every value of t in the interval 0≤t≤2π, we can get a point on the circle.
Graphic attached