Chicago In Chicago, the road system is set up like a...

Chicago In Chicago, the road system is set up like a Cartesian plane, where streets are indicated by the number of blocks they are from Madison Street and State Street. For example, Wrigley Field in Chicago is located at 1060 West Addison, which is 10 blocks west of State Street and 36 blocks north of Madison Street. Treat the intersection of Madison Street and State Street as the origin of a coordinate system, with east being the positive $x$-axis. (a) Write the location of Wrigley Field using rectangular coordinates. (b) Write the location of Wrigley Field using polar coordinates. Use the east direction for the polar axis. Express $\theta$ in degrees. (c) U.S. Cellular Field, home of the White Sox, is located at 35 th and Princeton, which is 3 blocks west of State Street and 35 blocks south of Madison. Write the location of U.S. Cellular Field using rectangular coordinates. (d) Write the location of U.S. Cellular Field using polar coordinates. Use the east direction for the polar axis. Express $\theta$ in degrees.

Chicago In Chicago, the road system is set up like a Cartesian plane, where streets are indicated by the number of blocks they are from Madison Street and State Street. For example, Wrigley Field in Chicago is located at 1060 West Addison, which is 10 blocks west of State Street and 36 blocks north of Madison Street. Treat the intersection of Madison Street and State Street as the origin of a coordinate system, with east being the positive $x$-axis.
(a) Write the location of Wrigley Field using rectangular coordinates.
(b) Write the location of Wrigley Field using polar coordinates. Use the east direction for the polar axis. Express $\theta$ in degrees.
(c) U.S. Cellular Field, home of the White Sox, is located at 35 th and Princeton, which is 3 blocks west of State Street and 35 blocks south of Madison. Write the location of U.S. Cellular Field using rectangular coordinates.
(d) Write the location of U.S. Cellular Field using polar coordinates. Use the east direction for the polar axis. Express $\theta$ in degrees.

Key Concepts

Cartesian Coordinate System

This system uses two perpendicular axes to define the position of a point in a plane by an ordered pair (x, y). The horizontal axis (x-axis) typically represents east-west positions, and the vertical axis (y-axis) represents north-south positions. It is fundamental in coordinate geometry and is used extensively to model relationships in two dimensions.

Polar Coordinate System

The polar coordinate system represents points in a plane using a distance from a fixed reference point (called the origin) and an angle measured from a fixed direction, usually the positive x-axis (east). This system is especially useful in situations where relationships are more naturally expressed in terms of angles and radii rather than x and y coordinates.

Conversion between Rectangular and Polar Coordinates

Converting between rectangular (Cartesian) and polar coordinates involves using the relationships x = r cos(?) and y = r sin(?) for the conversion from polar to rectangular, and r = ?(x² + y²) with ? = arctan(y/x) for converting from rectangular to polar. Understanding these conversions is crucial when interpreting or translating problems between coordinate systems.

Angle Measurement in Degrees

In many problems, especially those involving real-world directions, angles are measured in degrees rather than radians. This convention is important for clarity and comprehension when dealing with directional information, such as designating east as 0° and then measuring counterclockwise from that reference direction.