In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ.


x² = 6y

Verified step by step guidance

1. Step 1: Recall the relationship between rectangular coordinates $(x,y)$ and polar coordinates $(r,\theta )$. The conversions are given by $x=r\cos \theta$ and $y=r\sin \theta$.
2. Step 2: Substitute the polar coordinate expressions for $x$ and $y$ into the given rectangular equation $x^2=6y$. This gives $(r\cos \theta {)}^2=6(r\sin \theta )$.
3. Step 3: Simplify the equation from Step 2. Expand $(r\cos \theta {)}^2$ to get $r^2{\cos }^2\theta =6r\sin \theta$.
4. Step 4: Factor out $r$ from both sides of the equation. This results in $r(r{\cos }^2\theta -6\sin \theta )=0$.
5. Step 5: Solve for $r$ by setting each factor equal to zero. The solutions are $r=0$ or $r{\cos }^2\theta =6\sin \theta$. The second equation can be rearranged to express $r$ in terms of $\theta$.

Key Concepts

Rectangular to Polar Coordinates

In trigonometry, rectangular coordinates (x, y) can be converted to polar coordinates (r, θ) using the relationships x = r cos(θ) and y = r sin(θ). This conversion is essential for expressing equations in a different coordinate system, allowing for easier analysis of certain geometric properties.

Polar Equation Format

A polar equation typically expresses the radius r as a function of the angle θ. This format is crucial for understanding the behavior of curves in polar coordinates, as it allows for the visualization of how the distance from the origin changes with the angle, which is often more intuitive for circular or spiral shapes.

Graphing Polar Equations

Graphing polar equations involves plotting points based on the values of r and θ. Understanding how to interpret these points is vital, as the graph can reveal symmetries and shapes that are not immediately apparent in rectangular coordinates, such as circles, spirals, and other complex curves.