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 guidance

1. Recall the relationships between rectangular coordinates \((x, y)\) and polar coordinates \((r, \theta)\): \(x = r \cos{\theta}\) and \(y = r \sin{\theta}\).
2. Substitute \(x = r \cos{\theta}\) and \(y = r \sin{\theta}\) into the given rectangular equation \(3x + y = 7\) to get \(3(r \cos{\theta}) + r \sin{\theta} = 7\).
3. Factor out \(r\) from the left side: \(r(3 \cos{\theta} + \sin{\theta}) = 7\).
4. Solve for \(r\) by dividing both sides by \((3 \cos{\theta} + \sin{\theta})\): \(r = \frac{7}{3 \cos{\theta} + \sin{\theta}}\).
5. This expression gives the polar form of the equation, expressing \(r\) explicitly in terms of \(\theta\).

Key Concepts

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.