The letters r and θ represent polar coordinates. Write the following equation using rectangular coordinates (x, y).
r=1+sinθ
I chose: x²+y²=√(x²+y²+y)
(I am not completely sure about this)
r=1+sinθ
√(x^2 + y^2) = 1 + y/√(x^2 + y^2)
x^2 + y^2 = √(x^2 + y^2) + y
confirmation:
www.wolframalpha.com/input/?i=plot+x%5E2+%2B+y%5E2+%3D+%E2%88%9A(x%5E2+%2B+y%5E2)+%2B+y
www.wolframalpha.com/input/?i=polar+plot+r+%3D+1+%2B+sin%CE%B8
To write the equation in rectangular coordinates (x, y), we can use the relationship between polar coordinates and rectangular coordinates. This involves converting the polar coordinates (r, θ) into rectangular coordinates (x, y) using the following formulas:
x = r * cos(θ)
y = r * sin(θ)
In this case, the polar equation is given as r = 1 + sin(θ). To convert it to rectangular coordinates, we substitute the values of r and θ using the formulas mentioned earlier:
x = (1 + sin(θ)) * cos(θ)
y = (1 + sin(θ)) * sin(θ)
Now we can simplify these equations to obtain the equation in rectangular coordinates (x, y):
x = cos(θ) + sin(θ) * cos(θ)
y = sin(θ) + sin²(θ)
Thus, the equation in rectangular coordinates is:
x = cos(θ) + sin(θ) * cos(θ)
y = sin(θ) + sin²(θ)
Please note that the equation you provided, x² + y² = √(x² + y² + y), doesn't represent the correct conversion from polar to rectangular coordinates for the given polar equation.