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

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Recall the relationship between polar and rectangular coordinates: \(x = r \cos{\theta}\) and \(y = r \sin{\theta}\).

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.

Express \(\tan{\theta}\) in terms of \(x\) and \(y\) using the identity \(\tan{\theta} = \frac{y}{x}\).

Substitute \(\theta = \frac{3\pi}{4}\) into the tangent expression to get \(\tan{\frac{3\pi}{4}} = \frac{y}{x}\).

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.

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주요 개념

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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.