Textbook Question
25–30. Converting coordinates Express the following polar coordinates in Cartesian coordinates.
(2, 7π/4)
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16. Parametric Equations & Polar Coordinates
Polar Coordinates
1:48 minutesProblem 12.2.6
Textbook Question
What is the polar equation of the horizontal line y = 5?
Verified step by step guidance1
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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1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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Intro to Polar 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.
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05:32Intro to Polar Coordinates
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.
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