What is the polar equation of the horizontal line y = 5?

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

Since the line is given by \(y = 5\), substitute \(y\) with \(r \sin{\theta}\) to get the equation \(r \sin{\theta} = 5\).

To express \(r\) in terms of \(\theta\), solve the equation for \(r\): \(r = \frac{5}{\sin{\theta}}\).

Note that this equation is valid for values of \(\theta\) where \(\sin{\theta} \neq 0\), which corresponds to angles where the line \(y=5\) exists in polar coordinates.

Thus, the polar equation of the horizontal line \(y=5\) is \(r = \frac{5}{\sin{\theta}}\).

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주요 개념

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Polar Coordinates

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 Between Cartesian and Polar Coordinates

To convert between Cartesian (x, y) and polar (r, θ) coordinates, use the formulas x = r cos θ and y = r sin θ. Conversely, r = √(x² + y²) and θ = arctan(y/x). These relationships allow expressing Cartesian equations in terms of r and θ.

Equation of a Horizontal Line in Polar Form

A horizontal line y = c can be expressed in polar coordinates by substituting y = r sin θ. Thus, the equation becomes r sin θ = c. This form relates the radius and angle for all points on the horizontal line.