In Exercises 79–80, convert each polar equation to a rectangular equation. Then determine the graph's slope and y-intercept.
r sin (θ − π/4) = 2

Verified step by step guidance

1. Recall the relationships between polar and rectangular coordinates: \(x = r \cos \theta\) and \(y = r \sin \theta\). Also, \(r = \sqrt{x^2 + y^2}\) and \(\tan \theta = \frac{y}{x}\).
2. Use the angle difference identity for sine: \(\sin(\theta - \frac{\pi}{4}) = \sin \theta \cos \frac{\pi}{4} - \cos \theta \sin \frac{\pi}{4}\). Substitute this into the given equation to get \(r (\sin \theta \cos \frac{\pi}{4} - \cos \theta \sin \frac{\pi}{4}) = 2\).
3. Since \(\cos \frac{\pi}{4} = \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}\), rewrite the equation as \(r \left( \sin \theta \frac{\sqrt{2}}{2} - \cos \theta \frac{\sqrt{2}}{2} \right) = 2\).
4. Distribute \(r\) and replace \(r \sin \theta\) with \(y\) and \(r \cos \theta\) with \(x\), yielding \(\frac{\sqrt{2}}{2} y - \frac{\sqrt{2}}{2} x = 2\).
5. Multiply both sides by \(\sqrt{2}\) to clear the fractions, then rearrange the equation into the slope-intercept form \(y = mx + b\) to identify the slope and y-intercept.

Key Concepts

Conversion between Polar and Rectangular Coordinates

Polar coordinates (r, θ) relate to rectangular coordinates (x, y) through the formulas x = r cos θ and y = r sin θ. Converting a polar equation to rectangular form involves substituting these expressions to rewrite the equation in terms of x and y.

Trigonometric Angle Difference Identity

The angle difference identity for sine states that sin(α − β) = sin α cos β − cos α sin β. Applying this identity to sin(θ − π/4) allows the polar equation to be expanded into terms involving sin θ and cos θ, facilitating conversion to rectangular coordinates.

Slope and Y-Intercept of a Line

Once the equation is in rectangular form (y = mx + b), the slope (m) represents the line's steepness, and the y-intercept (b) is the point where the line crosses the y-axis. Identifying these helps in graphing and understanding the line's behavior.