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. Start by recalling 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 equation: \(3x + y = 7\).
3. This substitution gives: \(3(r \cos \theta) + (r \sin \theta) = 7\).
4. Factor out \(r\) from the left side of the equation: \(r(3 \cos \theta + \sin \theta) = 7\).
5. Solve for \(r\) by dividing both sides by \(3 \cos \theta + \sin \theta\): \(r = \frac{7}{3 \cos \theta + \sin \theta}\).

Key Concepts

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.

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.

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.