To convert the polar equation r=8cos(θ)+2sin(θ) to Cartesian coordinates, we use x=r cos(θ) and y=r sin(θ), apply trigonometric identities, and simplify. This results in the equations x=4+4cos(2θ)+sin(2θ) and y=4sin(2θ)+1-cos(2θ), indicating a periodic curve in Cartesian coordinates.
To convert the polar equation r = 8 cos(θ) + 2 sin(θ) to Cartesian coordinates, we use the relationships x = r cos(θ) and y = r sin(θ). We aim to express r in terms of x and y to describe the resulting curve.
Substituting the polar to Cartesian relationships, we get:
- x = (8 cos(θ) + 2 sin(θ)) cos(θ) = 8 cos2(θ) + 2 sin(θ) cos(θ)
- y = (8 cos(θ) + 2 sin(θ)) sin(θ) = 8 sin(θ) cos(θ) + 2 sin2(θ)
Now, we can use trigonometric identities to simplify the equations:
- sin(2θ) = 2 sin(θ) cos(θ)
- cos2(θ) = (1 + cos(2θ)) / 2
- sin2(θ) = (1 - cos(2θ)) / 2
Applying these identities gives us:
- x = 8 *(1 + cos(2θ)) / 2 + 2 * sin(2θ)/2 = 4 + 4 cos(2θ) + sin(2θ)
- y = 8 * sin(2θ)/2 + 2 *(1 - cos(2θ)) / 2 = 4 sin(2θ) + 1 - cos(2θ)
Describing the resulting curve: The presence of sin(2θ) and cos(2θ) suggests that the curve will have a periodic nature and may form a closed loop or a more complex pattern depending on how these terms combine in the Cartesian plane. Visualizing or plotting the curve might be necessary to further understand its shape.