Question

Plot the point given in polar coordinates and find the corresponding rectangular coordinates for the point. $$ (-2,7 \pi / 4) $$

Short answer

The rectangular coordinates corresponding to the polar coordinates (-2,7π/4) are approximately (1.41, -1.41). This is calculated by finding x = r*cos(θ) and y = r*sin(θ), respectively for the given polar coordinates after understanding their meaning and plotting the point.

Step-by-step solution

Understanding Polar Coordinates

Polar coordinates are often denoted as (r, θ), where r represents the distance from the origin and θ is the angle in standard position measured from the positive x-axis. A point in polar coordinates is represented as (r,θ). It differs from the rectangular coordinates, where a point is given in terms of x and y, representing its distances from the X and Y axes respectively. A positive value of r indicates that it's measured in the direction of θ, while a negative value indicates it's measured in the opposite direction.

Plotting the Point in Polar Coordinates

Given point in polar coordinates is (-2, 7π/4). A point with a negative radius is plotted just like a point with a positive radius, but in the opposite direction. So, we move to the angle 7π/4 and then move 2 units in the opposite direction. Meaning instead of moving 'outwards' from the origin, we now move 'inwards'.

Conversion to Rectangular Coordinates

To convert polar coordinates (r, θ) to rectangular coordinates (x,y), we can use the following mathematical relations: \(x = r*cos(θ)\) and \(y = r*sin(θ)\). So, using the given polar coordinates (-2, 7π/4), we have: \(x = -2*cos(7π/4)\) and \(y = -2*sin(7π/4)\). After calculating these, we obtain the rectangular coordinates.