Convert x² + (y + 8)² = 64 to a polar equation that expresses r in terms of θ.

Verified step by step guidance

1. Recognize that the given equation is in the form of a circle: $x^2+(y+8{)}^2=64$.
2. Recall the polar coordinate transformations: $x=r\cos (\theta )$ and $y=r\sin (\theta )$.
3. Substitute $x=r\cos (\theta )$ and $y=r\sin (\theta )$ into the equation: $(r\cos (\theta ){)}^2+(r\sin (\theta )+8{)}^2=64$.
4. Simplify the equation: $r^2{\cos }^2(\theta )+(r\sin (\theta )+8{)}^2=64$.
5. Further simplify and solve for $r$ in terms of $\theta$ by expanding and combining like terms.

Key Concepts

Polar Coordinates

Polar coordinates represent points in a plane using a distance from a reference point (the origin) and an angle from a reference direction (usually the positive x-axis). In polar coordinates, a point is expressed as (r, θ), where r is the radial distance and θ is the angle. Understanding this system is crucial for converting Cartesian equations to polar form.

Conversion from Cartesian to Polar

To convert a Cartesian equation to polar form, we use the relationships x = r cos(θ) and y = r sin(θ). By substituting these expressions into the Cartesian equation, we can express the equation in terms of r and θ. This process is essential for solving problems that require the analysis of curves in polar coordinates.

Circle Equation in Cartesian Coordinates

The given equation x² + (y + 8)² = 64 represents a circle in Cartesian coordinates, centered at (0, -8) with a radius of 8. Recognizing the standard form of a circle is important for identifying its properties and for correctly transforming it into polar coordinates. This understanding aids in visualizing the geometric implications of the equation.