convert the equation y^2 = 2x -x^2 into polar form.
convert the equation r = 2tan(theta) into rectangular form.
I'm not really sure what either of these mean. I vaguely understand polar coordinates, but I'm not sure how to convert equations back and forth between polar and rectangular. Please help =(
Sure, I can help you with that!
To convert the equation y^2 = 2x - x^2 into polar form, we need to express the equation in terms of polar coordinates, which are represented by the variables r (radius) and θ (angle).
First, let's express x and y in terms of r and θ using the polar coordinate conversion equations:
x = rcos(θ)
y = rsin(θ)
Replacing x and y in the given equation, we have:
(y^2) = 2x - (x^2)
[(rsin(θ))^2] = 2(rcos(θ)) - [(rcos(θ))^2]
Simplifying further:
r^2sin^2(θ) = 2rcos(θ) - r^2cos^2(θ)
r^2sin^2(θ) + r^2cos^2(θ) = 2rcos(θ)
r^2(sin^2(θ) + cos^2(θ)) = 2rcos(θ)
Since sin^2(θ) + cos^2(θ) = 1, we can replace it in the equation:
r^2 = 2rcos(θ)
This is the equation in polar form.
Now, let's move on to converting the equation r = 2tan(θ) into rectangular form.
In polar coordinates, the equation r = 2tan(θ) represents a straight line passing through the origin.
To convert it into rectangular form, we can use the following relations:
x = rcos(θ)
y = rsin(θ)
Replacing r in terms of x and y, we get:
x^2 + y^2 = (2tan(θ))^2
x^2 + y^2 = 4tan^2(θ)
Since tan^2(θ) = (sin(θ))^2 / (cos(θ))^2, we have:
x^2 + y^2 = 4(sin^2(θ) / cos^2(θ))
Multiplying both sides by cos^2(θ), we get:
x^2cos^2(θ) + y^2cos^2(θ) = 4sin^2(θ)
Using the trigonometric identity sin^2(θ) + cos^2(θ) = 1, we can replace sin^2(θ) in the equation:
x^2cos^2(θ) + y^2cos^2(θ) = 4(1 - cos^2(θ))
Simplifying further:
x^2cos^2(θ) + y^2cos^2(θ) = 4 - 4cos^2(θ)
Now, using the identity cos^2(θ) = 1 - sin^2(θ), we can replace cos^2(θ) in the equation:
x^2(1 - sin^2(θ)) + y^2(1 - sin^2(θ)) = 4 - 4(1 - sin^2(θ))
Simplifying again:
x^2 - x^2sin^2(θ) + y^2 - y^2sin^2(θ) = 4 - 4 + 4sin^2(θ)
Combining like terms:
x^2(1 - sin^2(θ)) + y^2(1 - sin^2(θ)) = 4sin^2(θ)
x^2cos^2(θ) + y^2cos^2(θ) = 4sin^2(θ)
This is the equation in rectangular form.
I hope this explanation helps! Let me know if you have any further questions.
r = tan(θ)
√(x² + y²) = y/x
tan(θ) = y/x
r = √(x² + y²)
x² + y² = 4y²/x²
x^4 + x²y² = 4y²
x^4 + x²y² - 4y² = 0