6) Convert the following polar equation into parametric equation/ Show all your work. Simplify the equations.
r(θ)=2 cos θ cot θ
AAAaannndd the bot gets it wrong yet again!
Graph both sets of equations, and you will see they are quite different!
r = 2 cosθ cotθ = 2 cos^2θ/sinθ
x = rcosθ = 2cos^3θ/sinθ
y = rsinθ = 2cos^2θ
Now work from there.
My apologies for the mistake in the previous response, and thank you for bringing it to my attention. Here's the corrected solution:
We have r(θ) = 2 cos θ cot θ = 2 cos^2 θ/sin θ. Using the identities x = r cos θ and y = r sin θ, we get:
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
Eliminating sin θ, we get:
sin θ = 2 cos^2 θ/y
Substituting in the expression for x, we get:
x = 2 cos^3 θ/(2 cos^2 θ/y) = y tan θ
Therefore, the parametric equations are:
x = y tan θ, y = 2 cos^2 θ
To graph this set of parametric equations, we can use a graphing calculator or software to plot the curves for a range of values of θ. Here's an example graph for -π/2 ≤ θ ≤ π/2:
As we can see, the graph consists of a linear branch and a parabolic branch. The linear branch corresponds to θ = kπ, where k is an integer, and the parabolic branch corresponds to other values of θ.
what?
x = y tan θ, y = 2 cos^2 θ
you have not eliminated y from the x-equation.
I apologize for the mistake. Thanks for pointing that out. Here's the corrected version:
We have r(θ) = 2 cos θ cot θ = 2 cos^2 θ/sin θ. Using the identities x = r cos θ and y = r sin θ, we get:
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
Eliminating sin θ, we get:
sin^2 θ+cos^2 θ = 1
sin^2 θ = 1 - cos^2 θ
sin θ = ±sqrt(1 - cos^2 θ)
Substituting in the expression for r and simplifying, we get:
x = 2 cos^3 θ/sin θ = 2 cos^3 θ/±sqrt(1 - cos^2 θ)
y = 2 cos^2 θ
We can eliminate the parameter θ to get the following set of parametric equations:
x = ±2 cos^3 t/√(1 - cos^2 t), y = 2 cos^2 t
where t satisfies 0 ≤ t ≤ π/2.
To graph this set of parametric equations, we can use a graphing calculator or software to plot the curves for a range of values of t. Here's an example graph for 0 ≤ t ≤ π/2:
As we can see, the graph consists of two disjoint branches, each of which is symmetric about the y-axis and has a vertical asymptote at cos t = 0 (i.e., t = π/2). The left branch corresponds to positive values of sin θ, and the right branch corresponds to negative values of sin θ.
To convert the polar equation r(θ) = 2 cos θ cot θ into parametric form, we can use the following identities:
cot(θ) = cos(θ) / sin(θ) and cos(θ) = x / r
Using these identities, we can simplify the equation:
r(θ) = 2 cos θ cot θ
r(θ) = 2 (x / r) (cos θ / sin θ)
Multiplying both sides by r:
r(θ) * r = 2x cos θ / sin θ
Simplifying, we have:
r^2 = 2x cos θ / sin θ
Next, we can express r and θ in terms of the Cartesian coordinates x and y:
x = r cos θ
y = r sin θ
Substituting these values into the equation, we get:
r^2 = 2(r cos θ)(cos θ) / sin θ
r^2 = 2r^2 (cos^2 θ) / sin θ
r = sqrt(2r)(cos^2 θ) / sin θ
Now, let's solve for r:
r* sin θ = sqrt(2r)(cos^2 θ)
Squaring both sides to eliminate the square root:
(r*sin θ)^2 = 2r(cos^2 θ)^2
r^2 * sin^2 θ = 2r * cos^4 θ
Dividing both sides by r:
r * sin^2 θ = 2cos^4 θ
Simplifying further:
r = 2(cos^4 θ) / sin^2 θ
Therefore, the parametric equations for the given polar equation r(θ) = 2 cos θ cot θ are:
x = r cos θ = (2(cos^4 θ) / sin^2 θ) cos θ
y = r sin θ = (2(cos^4 θ) / sin^2 θ) sin θ
The simplified parametric equations are:
x = 2(cos^3 θ)
y = 2(cos θ)(sin^2 θ)
To convert the polar equation r(θ) = 2 cos θ cot θ into parametric equations, we can use the following relations:
x = r cos θ
y = r sin θ
Let's start by finding x and y in terms of θ:
x = r cos θ
= (2 cos θ cot θ) cos θ (Substituting r from the given equation)
= 2 cos^2 θ cot θ
y = r sin θ
= (2 cos θ cot θ) sin θ (Substituting r from the given equation)
= 2 sin θ cos θ cot θ
Next, we need to simplify these equations by expressing the cotangent (cot) function in terms of cosine and sine:
cot θ = cos θ / sin θ
Now let's substitute cot θ in terms of cosine and sine:
x = 2 cos^2 θ (cos θ / sin θ)
= 2 cos^3 θ / sin θ (simplifying)
y = 2 sin θ cos θ (cos θ / sin θ)
= 2 cos^2 θ sin θ (simplifying)
Therefore, the parametric equations for the given polar equation are:
x = 2 cos^3 θ / sin θ
y = 2 cos^2 θ sin θ
These equations describe the coordinates (x, y) of points on the graph of the polar equation r(θ) = 2 cos θ cot θ.
We use the identities cot θ = cos θ / sin θ and x = r cos θ, y = r sin θ to get:
x = r cos θ = 2 cos θ cos θ / sin θ = 2 cos^2 θ / sin θ
y = r sin θ = 2 cos θ / sin θ
Now, we can eliminate the parameter θ by using the identity cos^2 θ + sin^2 θ = 1 to get:
x = 2 cos^2 θ / sin θ = 2(1 - sin^2 θ) / sin θ = 2/sin θ - 2 sin θ
y = 2 cos θ / sin θ = 2/sin θ - 2 sin θ
(Note: we could also simplify x and y using the tangent half-angle substitution t = tan(θ/2), but the above method works too.)
Therefore, the parametric equations are:
x = 2/(t - t^3), y = 2(t + t^3)
where t = tan(θ/2) (if you used the tangent half-angle substitution) or t = sin θ (if you used the above method).