Match each equation with its polar graph from choices A–D.
r = 2/(cosθ + sinθ)

Verified step by step guidance

1. Recognize that the given equation is in polar form: \(r = \frac{2}{\cos\theta + \sin\theta}\), where \(r\) is the radius and \(\theta\) is the angle.
2. Multiply both sides of the equation by the denominator to eliminate the fraction: \(r(\cos\theta + \sin\theta) = 2\).
3. Recall that in polar coordinates, \(x = r\cos\theta\) and \(y = r\sin\theta\). Substitute these into the equation to rewrite it in Cartesian form: \(x + y = 2\).
4. Interpret the Cartesian equation \(x + y = 2\) as a straight line in the \(xy\)-plane. This means the polar graph corresponds to a line, but expressed in polar coordinates.
5. Use this understanding to match the given polar equation with the correct graph among choices A–D, identifying the graph that represents the line \(x + y = 2\).

Key Concepts

Polar Coordinates and Graphing

Polar coordinates represent points using a radius r and an angle θ from the positive x-axis. Understanding how to plot equations in polar form is essential, as the graph is drawn by varying θ and calculating corresponding r values, which can produce curves like circles, lines, or more complex shapes.

Conversion Between Polar and Cartesian Coordinates

Converting polar equations to Cartesian form (x = r cosθ, y = r sinθ) helps analyze and identify the graph type. For example, rewriting r = 2/(cosθ + sinθ) in terms of x and y can reveal the geometric nature of the curve, making it easier to match with given graph options.

Trigonometric Identities and Manipulation

Using trigonometric identities, such as expressing cosθ + sinθ in a simplified form, aids in transforming and understanding the equation. Recognizing patterns or applying identities can clarify the shape and properties of the polar graph, facilitating accurate matching with the correct graph.