Questions: Convert the following equation to polar coordinates. y=8/x (Type an expression using θ as the variable.)

Solution Steps

To convert the given equation \( y = \frac{8}{x} \) to polar coordinates, we need to use the relationships between Cartesian coordinates \((x, y)\) and polar coordinates \((r, \theta)\). Specifically, we use \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \). Substitute these into the given equation and solve for \( r \).

Given the equation in Cartesian coordinates: \[ y = \frac{8}{x} \] we use the relationships between Cartesian and polar coordinates: \[ x = r \cos(\theta) \quad \text{and} \quad y = r \sin(\theta) \]

Substitute \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \) into the given equation: \[ r \sin(\theta) = \frac{8}{r \cos(\theta)} \]

Multiply both sides by \( r \cos(\theta) \) to eliminate the fraction: \[ r^2 \sin(\theta) \cos(\theta) = 8 \]

Isolate \( r^2 \) by dividing both sides by \( \sin(\theta) \cos(\theta) \): \[ r^2 = \frac{8}{\sin(\theta) \cos(\theta)} \]

Use the trigonometric identity \( \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \) to further simplify: \[ r^2 = \frac{8}{\frac{1}{2} \sin(2\theta)} = \frac{16}{\sin(2\theta)} \]

Final Answer