Final answer:
Parametric equations A and B both represent a circle with the equation x^2 + y^2 = 1. Equations C and D both describe a hyperbola, with C resulting in y^2 - x^2 = 1 and D in x^2 - y^2 = 1.
Step-by-step explanation:
To write the given parametric equations in rectangular form, we utilize the trigonometric identities. Let's examine each pair:
- A. x = sin(t), y = cos(t): We can apply the Pythagorean identity sin2(t) + cos2(t) = 1. Substituting x and y into the identity gives x2 + y2 = 1, which is the equation of a circle with radius 1.
- B. x = cos(t), y = sin(t): Similar to A, using the Pythagorean identity gives us x2 + y2 = 1, which is also an equation of a circle with radius 1.
- C. x = tan(t), y = sec(t): Using the identity 1 + tan2(t) = sec2(t), we substitute x and y to get 1 + x2 = y2, which can be rewritten as y2 - x2 = 1, representing a hyperbola.
- D. x = sec(t), y = tan(t): Similar to C but arranging the identity sec2(t) - 1 = tan2(t), substituting gives us x2 - 1 = y2, which also represents a hyperbola.
In summary, options A and B result in the equation of a circle, while options C and D give us hyperbolic equations.