Polar to Cartesian Equations

Sketch the lines in Exercises 23-28. Also, find a Cartesian equation for each line.

r cos (θ − 3π/4) = (√2)/2

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Recall the polar to Cartesian coordinate conversions: \(x = r \cos \theta\) and \(y = r \sin \theta\).

Use the cosine difference identity to expand \(\cos(\theta - \frac{3\pi}{4})\): \(\cos(\theta - \frac{3\pi}{4}) = \cos \theta \cos \frac{3\pi}{4} + \sin \theta \sin \frac{3\pi}{4}\).

Substitute the known values \(\cos \frac{3\pi}{4} = -\frac{\sqrt{2}}{2}\) and \(\sin \frac{3\pi}{4} = \frac{\sqrt{2}}{2}\) into the expression.

Rewrite the original equation \(r \cos(\theta - \frac{3\pi}{4}) = \frac{\sqrt{2}}{2}\) as: \(r \left( -\frac{\sqrt{2}}{2} \cos \theta + \frac{\sqrt{2}}{2} \sin \theta \right) = \frac{\sqrt{2}}{2}\).

Replace \(r \cos \theta\) with \(x\) and \(r \sin \theta\) with \(y\) to get a Cartesian equation in terms of \(x\) and \(y\). Simplify this equation to find the Cartesian form of the line.

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주요 개념

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Polar Coordinates and Their Components

Polar coordinates represent points in the plane using a radius r and an angle θ from the positive x-axis. Understanding how r and θ define a point's position is essential for converting between polar and Cartesian forms.

Conversion Between Polar and Cartesian Coordinates

The relationships x = r cos θ and y = r sin θ allow conversion from polar to Cartesian coordinates. Manipulating these equations helps rewrite polar equations in terms of x and y, facilitating graphing and analysis.

Using Angle Shift Identities in Polar Equations

Expressions like r cos(θ − α) can be expanded using trigonometric identities: r cos(θ − α) = r cos θ cos α + r sin θ sin α. This expansion is key to expressing the equation in terms of x and y for Cartesian conversion.

Polar to Cartesian EquationsSketch the lines in Exercises 23-28. Also, find a Cartesian equation for each line.r cos (θ − 3π/4) = (√2)/2