Convert the Cartesian equation x^2 - y^2 = 16 to a polar equation.
r² = 16/cos2è
r = -4
r² = 8
help please im stuck between 2 answers.
x = rcosθ
y = rsinθ
x^2 - y^2 = r^2 cos^2 θ - r^2 sin^2 θ
= r^2(cos^2 θ - sin^2 θ) = r^2 cos2θ
so, we have r^2 cos2θ = 16
r^2 = 16/cos2θ
Thank you.
To convert the Cartesian equation x^2 - y^2 = 16 to a polar equation, we can use the following steps:
1. Replace x with r * cos(θ) and y with r * sin(θ), where r is the radial distance and θ is the angle.
2. Rewrite the equation using the trigonometric identities cos^2(θ) - sin^2(θ) = cos(2θ).
3. Simplify and rearrange the equation to isolate r.
Let's go through these steps in detail:
1. Replace x with r * cos(θ) and y with r * sin(θ):
(r * cos(θ))^2 - (r * sin(θ))^2 = 16
2. Apply the trigonometric identity cos^2(θ) - sin^2(θ) = cos(2θ):
r^2 * cos^2(θ) - r^2 * sin^2(θ) = 16
r^2 * (cos^2(θ) - sin^2(θ)) = 16
r^2 * cos(2θ) = 16
3. Rearrange the equation to isolate r:
r^2 = 16 / cos(2θ)
Now, the conversion of the Cartesian equation x^2 - y^2 = 16 to a polar equation is given by the equation r^2 = 16 / cos(2θ).
It seems like you have additional answers of r = -4 and r^2 = 8. However, these answers are not correct for this particular conversion.
I hope this explanation helps! Let me know if you have further questions.
To convert the Cartesian equation x^2 - y^2 = 16 to a polar equation, we can use the following polar coordinate conversions:
x = r * cos(theta)
y = r * sin(theta)
Substituting these values into the Cartesian equation, we get:
(r * cos(theta))^2 - (r * sin(theta))^2 = 16
Simplifying the equation further, we have:
r^2 * cos^2(theta) - r^2 * sin^2(theta) = 16
Using the trigonometric identity cos^2(theta) - sin^2(theta) = cos(2theta), we can rewrite the equation as:
r^2 * cos(2theta) = 16
Finally, by dividing both sides of the equation by cos(2theta), we obtain the polar equation:
r^2 = 16 / cos(2theta)
So, the correct conversion of the Cartesian equation x^2 - y^2 = 16 to a polar equation is r^2 = 16 / cos(2theta).