49–52. Cartesian-to-polar coordinates Convert the following equations to polar coordinates.

y = 3

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Recall the relationships between Cartesian coordinates \((x, y)\) and polar coordinates \((r, \theta)\): \(x = r \cos{\theta}\) and \(y = r \sin{\theta}\).

Given the equation \(y = 3\), substitute \(y\) with its polar form: \(r \sin{\theta} = 3\).

Isolate \(r\) to express it in terms of \(\theta\): \(r = \frac{3}{\sin{\theta}}\).

Note the domain restrictions where \(\sin{\theta} \neq 0\) to avoid division by zero.

The polar form of the equation \(y = 3\) is therefore \(r = \frac{3}{\sin{\theta}}\).

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Polar Coordinates System

Polar coordinates represent points in the plane using a radius and an angle, denoted as (r, θ), where r is the distance from the origin and θ is the angle measured from the positive x-axis. This system is useful for describing curves that are difficult to express in Cartesian coordinates.

Conversion Formulas Between Cartesian and Polar Coordinates

To convert from Cartesian (x, y) to polar (r, θ), use r = √(x² + y²) and θ = arctan(y/x). Conversely, x = r cos(θ) and y = r sin(θ). These formulas allow rewriting equations from one coordinate system to the other.

Expressing Cartesian Equations in Polar Form

To convert a Cartesian equation like y = 3 into polar form, substitute y with r sin(θ). This transforms the equation into r sin(θ) = 3, which can then be analyzed or manipulated using polar coordinates.

49–52. Cartesian-to-polar coordinates Convert the following equations to polar coordinates.y = 3