Polar conversion Consider the equation r=4/(sinθ+cosθ). 


a. Convert the equation to Cartesian coordinates and identify the curve it describes.  

Verified step by step guidance

1. Recall the relationships between polar and Cartesian coordinates: \(x = r \cos\theta\), \(y = r \sin\theta\), and \(r = \sqrt{x^2 + y^2}\).
2. Start with the given polar equation: \(r = \frac{4}{\sin\theta + \cos\theta}\).
3. Multiply both sides of the equation by \((\sin\theta + \cos\theta)\) to get: \(r (\sin\theta + \cos\theta) = 4\).
4. Substitute \(r \sin\theta = y\) and \(r \cos\theta = x\) into the equation, yielding \(y + x = 4\).
5. Recognize that the equation \(x + y = 4\) is a linear equation representing a straight line in Cartesian coordinates.

Key Concepts

Polar to Cartesian Coordinate Conversion

This involves translating equations from polar form (r, θ) to Cartesian form (x, y) using the relationships x = r cosθ and y = r sinθ. Understanding these conversions allows one to rewrite polar equations in terms of x and y, facilitating analysis using familiar Cartesian methods.

Trigonometric Identities and Manipulation

Trigonometric identities, such as expressing sinθ and cosθ in terms of x and y, or using sum formulas, are essential for simplifying and rearranging equations during conversion. Mastery of these identities helps in isolating variables and recognizing standard curve forms.

Identification of Conic Sections

After converting to Cartesian form, recognizing the resulting equation as a conic section (circle, ellipse, parabola, or hyperbola) is crucial. This involves comparing the equation to standard forms and understanding geometric properties to classify the curve accurately.