Textbook Question
In Exercises 41–48, the rectangular coordinates of a point are given. Find polar coordinates of each point. Express θ in radians. _(2,−2√3)
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9. Polar Equations
Polar Coordinate System
3:47 minutesProblem 49
Textbook Question
In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ.3x + y = 7
Verified step by step guidance1
Start by recalling the relationships between rectangular coordinates \((x, y)\) and polar coordinates \((r, \theta)\): \(x = r \cos \theta\) and \(y = r \sin \theta\).
Substitute \(x = r \cos \theta\) and \(y = r \sin \theta\) into the given equation: \(3x + y = 7\).
This substitution gives: \(3(r \cos \theta) + (r \sin \theta) = 7\).
Factor out \(r\) from the left side of the equation: \(r(3 \cos \theta + \sin \theta) = 7\).
Solve for \(r\) by dividing both sides by \(3 \cos \theta + \sin \theta\): \(r = \frac{7}{3 \cos \theta + \sin \theta}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Rectangular Coordinates
Rectangular coordinates, also known as Cartesian coordinates, represent points in a two-dimensional space using an ordered pair (x, y). In this system, 'x' denotes the horizontal distance from the origin, while 'y' indicates the vertical distance. Understanding how to manipulate these coordinates is essential for converting equations from rectangular to polar form.
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Polar Coordinates
Polar coordinates describe points in a plane using a distance 'r' from the origin and an angle 'θ' from the positive x-axis. The relationship between rectangular and polar coordinates is defined by the equations x = r cos(θ) and y = r sin(θ). This conversion is crucial for expressing rectangular equations in polar form, as required in the given problem.
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Conversion Formulas
To convert a rectangular equation to polar form, one must apply the conversion formulas that relate x and y to r and θ. Specifically, substituting x with r cos(θ) and y with r sin(θ) allows for the transformation of the equation. Mastery of these formulas is vital for successfully expressing the given rectangular equation in terms of polar coordinates.
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