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