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. Start by using the angle subtraction identity for sine: $\sin (\theta -\frac{\pi }{4})=\sin (\theta )\cos (\frac{\pi }{4})-\cos (\theta )\sin (\frac{\pi }{4})$.
2. Substitute the values of $\cos (\frac{\pi }{4})$ and $\sin (\frac{\pi }{4})$, both of which are $\frac{\sqrt{2}}{2}$, into the equation: $r(\sin (\theta )\frac{\sqrt{2}}{2}-\cos (\theta )\frac{\sqrt{2}}{2})=2$.
3. Multiply through by $r$ to get: $r\sin (\theta )\frac{\sqrt{2}}{2}-r\cos (\theta )\frac{\sqrt{2}}{2}=2$.
4. Recognize that $r\sin (\theta )=y$ and $r\cos (\theta )=x$, so substitute these into the equation: $y\frac{\sqrt{2}}{2}-x\frac{\sqrt{2}}{2}=2$.
5. Multiply the entire equation by $\sqrt{2}$ to simplify: $y-x=2\sqrt{2}$. This is the rectangular form of the equation.

Key Concepts

Polar Coordinates

Polar coordinates represent points in a plane using a distance from a reference point (the origin) and an angle from a reference direction. In polar equations, 'r' denotes the radius (distance from the origin), and 'θ' represents the angle. Understanding how to interpret and manipulate these coordinates is essential for converting polar equations to rectangular form.

Rectangular Coordinates

Rectangular coordinates, or Cartesian coordinates, use the x and y axes to define a point in a plane. The conversion from polar to rectangular coordinates involves using the relationships x = r cos(θ) and y = r sin(θ). This transformation is crucial for analyzing the properties of the graph, such as slope and y-intercept, in a familiar coordinate system.

Slope and Y-Intercept

The slope of a line in a rectangular coordinate system indicates its steepness and direction, while the y-intercept is the point where the line crosses the y-axis. To find these values from a rectangular equation, one typically rearranges the equation into slope-intercept form (y = mx + b), where 'm' represents the slope and 'b' the y-intercept. This understanding is vital for interpreting the graph of the equation derived from the polar form.