In Exercises 54–60, convert each polar equation to a rectangular equation. Then use your knowledge of the rectangular equation to graph the polar equation in a polar coordinate system. θ = 3π/4

Verified step by step guidance

1. Recall the relationship between polar and rectangular coordinates: \(x = r \cos{\theta}\) and \(y = r \sin{\theta}\).
2. Given the polar equation \(\theta = \frac{3\pi}{4}\), recognize that this represents all points where the angle from the positive x-axis is \(\frac{3\pi}{4}\) radians.
3. Express \(\tan{\theta}\) in terms of \(x\) and \(y\) using the identity \(\tan{\theta} = \frac{y}{x}\).
4. Substitute \(\theta = \frac{3\pi}{4}\) into the tangent expression to get \(\tan{\frac{3\pi}{4}} = \frac{y}{x}\).
5. Use the known value of \(\tan{\frac{3\pi}{4}}\) to write the rectangular equation relating \(x\) and \(y\), which can then be used to graph the line in the rectangular coordinate system and interpret it in the polar coordinate system.

Key Concepts

Polar and Rectangular Coordinate Systems

Polar coordinates represent points using a radius and an angle (r, θ), while rectangular coordinates use (x, y) positions on a plane. Understanding how these systems relate is essential for converting equations and interpreting graphs in both formats.

Conversion Formulas Between Polar and Rectangular Coordinates

The key formulas for conversion are x = r cos θ and y = r sin θ, which translate polar coordinates into rectangular form. Additionally, r = √(x² + y²) and θ = arctan(y/x) convert rectangular coordinates back to polar.

Graphing Lines in Polar Coordinates

A polar equation like θ = constant represents a line through the origin at a fixed angle. Converting this to rectangular form helps visualize the line as y = mx, where m = tan(θ), facilitating graphing in both coordinate systems.