Explanation:
To convert equations from polar form to rectangular form, we can use the following conversions:
1. For the equation r = tan(θ):
In rectangular form, we can express r in terms of x and y using the relationships:
r = √(x² + y²)
tan(θ) = y / x
Substituting these values into the equation r = tan(θ), we get:
√(x² + y²) = y / x
Squaring both sides of the equation, we have:
x² + y² = y² / x²
Multiplying both sides by x², we get:
x⁴ + x²y² = y²
Therefore, the rectangular form of the equation r = tan(θ) is:
x⁴ + x²y² - y² = 0
2. For the equation r = 2 / (1 - sin(θ)):
Using the same conversions as above, we have:
r = √(x² + y²)
1 - sin(θ) = 1 - y / r
Substituting these values into the equation r = 2 / (1 - sin(θ)), we get:
√(x² + y²) = 2 / (1 - y / √(x² + y²))
Squaring both sides of the equation, we have:
x² + y² = 4 / (1 - y / √(x² + y²))
Multiplying both sides by (1 - y / √(x² + y²)), we get:
(x² + y²)(1 - y / √(x² + y²)) = 4
Expanding and simplifying the equation, we have:
x² + y² - y = 4 - 4y / √(x² + y²)
Multiplying both sides by √(x² + y²), we get:
(x² + y² - y)√(x² + y²) = 4√(x² + y²) - 4y
Therefore, the rectangular form of the equation r = 2 / (1 - sin(θ)) is:
(x² + y² - y)√(x² + y²) = 4√(x² + y²) - 4y