In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ. 3x + y = 7

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Recall 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 rectangular equation \(3x + y = 7\) to get \(3(r \cos{\theta}) + r \sin{\theta} = 7\).

Factor out \(r\) from the left side: \(r(3 \cos{\theta} + \sin{\theta}) = 7\).

Solve for \(r\) by dividing both sides by \((3 \cos{\theta} + \sin{\theta})\): \(r = \frac{7}{3 \cos{\theta} + \sin{\theta}}\).

This expression gives the polar form of the equation, expressing \(r\) explicitly in terms of \(\theta\).

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주요 개념

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Rectangular and Polar Coordinate Systems

Rectangular coordinates represent points using (x, y) on a plane, while polar coordinates use (r, θ), where r is the distance from the origin and θ is the angle from the positive x-axis. Understanding how these systems relate is essential for converting equations between them.

Conversion Formulas Between Coordinates

The key formulas for conversion are x = r cos(θ) and y = r sin(θ). These allow substitution of rectangular variables with polar expressions, enabling the transformation of equations from rectangular to polar form.

Expressing r in Terms of θ

After substituting x and y with r cos(θ) and r sin(θ), the goal is to isolate r on one side of the equation. This often involves algebraic manipulation to express r explicitly as a function of θ, which is the standard form for polar equations.