A plane is 100 miles from a radar station at an angle of 135^∘ clockwise from north. Find polar

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A plane is 100 miles from a radar station at an angle of...

A plane is 100 miles from a radar station at an angle of $135^{\circ}$ clockwise from north. Find polar coordinates for the location of the plane. A plane is 50 miles from a radar station at an angle of $210^{\circ}$ clockwise from north. Find polar coordinates for the location of the plane.

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Key Concepts

Polar Coordinates

Polar coordinates represent points in the plane using a radius and an angle. The radius measures the distance from a fixed origin, while the angle specifies the direction of the point relative to a reference line. This coordinate system is especially useful in contexts where positions are naturally expressed in terms of distance and direction.

Angle Conversion Between Navigational and Mathematical Conventions

In many applications, angles are given as measurements clockwise from north, which differs from the standard mathematical convention of measuring angles counterclockwise from the positive x?axis. To reconcile these conventions, one typically converts the navigational angle to the standard polar angle by subtracting the navigational angle from 90°, or by performing an equivalent transformation, ensuring consistency in calculations.

Conversion Between Polar and Rectangular Coordinates

Understanding how to transition between polar and rectangular coordinate systems is crucial. A point given in polar coordinates (r, ?) can be converted to rectangular coordinates using the relationships x = r cos ? and y = r sin ?. This conversion is fundamental in many problems, as it allows one to leverage the strengths of both coordinate systems for analysis and computation.

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