In Exercises 59–74, convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. r = 4 csc θ

Verified step by step guidance

1. Start with the given polar equation: $r = 4 \csc \theta$.
2. Recall that $\csc \theta = \frac{1}{\sin \theta}$, so rewrite the equation as $r = \frac{4}{\sin \theta}$.
3. Multiply both sides by $\sin \theta$ to eliminate the fraction: $r \sin \theta = 4$.
4. Use the polar to rectangular conversion $r \sin \theta = y$ to substitute for $y$: $y = 4$.
5. The rectangular equation is $y = 4$, which is a horizontal line in the rectangular coordinate system.

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 equations, 'r' denotes the radius (distance from the origin), and 'θ' represents the angle. Understanding how to interpret and manipulate these coordinates is essential for converting polar equations to rectangular form.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, use two perpendicular axes (x and y) to define the position of points in a plane. The conversion from polar to rectangular coordinates involves using the relationships x = r cos(θ) and y = r sin(θ). This understanding is crucial for graphing equations in the rectangular coordinate system.

Trigonometric Functions

Trigonometric functions, such as sine and cosecant, relate angles to the ratios of sides in right triangles. In the given polar equation, 'csc θ' is the cosecant function, which is the reciprocal of sine (csc θ = 1/sin θ). Recognizing how these functions interact with polar coordinates is vital for accurately converting and graphing the equation.