Question

6) Convert the following polar equation into prametric equation/ Show all your work. Simplify the equations.
r(θ)=2 cos θ cot θ


7) Eliminate the parameter for the following parametric equations. Show all your work. Simplify as far as you can.

x(t) = 2t
y(t) = -4t^2+2

Answer 1

Which direction would the arrows be in a graph to represent the orientation of the following parametric equations?

x(t) = 3 cos 1/2 t
y(t) = -5sin 1/2 t

Answer 2

6) To convert a polar equation into parametric equations, we use the following formulas:

x = r cos θ
y = r sin θ

We have:

r(θ) = 2 cos θ cot θ

Using the formula for cotangent:

cot θ = cos θ / sin θ

We can rewrite the equation as:

r(θ) = 2 cos θ (cos θ / sin θ)

Simplifying:

r(θ) = 2 cos^2 θ / sin θ

Now we can use the formulas to get the parametric equations:

x = r cos θ = (2 cos^2 θ / sin θ) cos θ = 2 cos^3 θ / sin θ
y = r sin θ = (2 cos^2 θ / sin θ) sin θ = 2 cos^2 θ

So the parametric equations are:

x = 2 cos^3 θ / sin θ
y = 2 cos^2 θ

7) To eliminate the parameter, we need to express one variable (x or y) in terms of the other. We have:

x(t) = 2t
y(t) = -4t^2 + 2

We can rearrange the first equation to get t in terms of x:

t = x/2

Substituting into the second equation:

y(x) = -4(x/2)^2 + 2
y(x) = -x^2 + 2

So the equation without parameter is:

y = -x^2 + 2

Answer 3

The orientation of the arrows on a graph depends on the direction of increasing values of the parameter t.

For the given parametric equations:

x(t) = 3 cos (1/2)t
y(t) = -5 sin (1/2)t

Note that the coefficient of t in both equations is positive (1/2), indicating that the parameter increases as the angle increases in a counterclockwise direction. Therefore, on the graph, the arrows would be oriented counterclockwise.