Question

Which of the following curves is given by the parametric equations? The arrow on the curve indicates the direction in which the curve is traced as t increases. Consider the following parametric equations. x=−3sin( θ/²​ )−3,y=−3cos( θ/²​ )−3,0≤θ≤2π a) Find 4 unique points on the parametric curve. Enter your answers as an ordered triple (x,y,θ), where x and y are the coordinates of the point and θ is the value that gives that point. c) Eliminate the parameter θ to find a Cartesian equation of the curve in x and y without using inverse trigonometric functions. d) If the Cartesian equation from c) needs restrictions on x and/or y to describe the curve from b), enter these restrictions as inequalities (separated by a comma if necessary). If restrictions are implied in the Cartesian equation, enter 'None'. If you need to enter ≤, type "<=". To enter ≥, type ">=".

Answer 1

The parametric equations are x = -3sin(θ/2) and y = -3cos(θ/2). Four unique points on the parametric curve are (0, -3, 0), (-3/√2, -3/√2, π/2), (0, -3, π), and (3/√2, -3/√2, 3π/2).

Step-by-step explanation:

The parametric equations given are:

x = -3sin(θ/2)

y = -3cos(θ/2)

To find four unique points on the parametric curve, we can substitute different values of θ into the equations. Let's choose θ = 0, π/2, π, and 3π/2:

For θ = 0:

x = -3sin(0/2) = 0

y = -3cos(0/2) = -3

The point is (0, -3, 0).

For θ = π/2:

x = -3sin(π/4) = -3/√2

y = -3cos(π/4) = -3/√2

The point is (-3/√2, -3/√2, π/2).

For θ = π:

x = -3sin(π) = 0

y = -3cos(π) = -3

The point is (0, -3, π).

For θ = 3π/2:

x = -3sin(3π/4) = 3/√2

y = -3cos(3π/4) = -3/√2

The point is (3/√2, -3/√2, 3π/2).