In Exercises 11-20 the rectangular coordinates of a point are given. Plot the point and find two

step 1 In this problem, we wish to plot the given point in rectangular coordinates, and then we want to

In Exercises 11-20 the rectangular coordinates of a point...

Key Concepts

Multiple Representations in Polar Coordinates

A single point can have multiple polar coordinate representations due to the periodic nature of the angle (? + 2?k, where k is an integer) and the possibility of using a negative radius paired with an angle adjustment. These properties allow for different, yet equivalent, polar coordinate sets.

Quadrant Considerations

When converting coordinates, it’s crucial to consider the quadrant in which the point is located since the signs of x and y affect the angle ?. Adjusting the computed angle ensures that the polar coordinate representation accurately reflects the true position of the point.

Rectangular Coordinates

Rectangular coordinates represent a point in the plane by an ordered pair (x, y), where x indicates the horizontal position and y indicates the vertical position. This system is fundamental in geometry and serves as a basis for many conversions and analyses.

Polar Coordinates

Polar coordinates describe a point in the plane using a distance from a reference point (called the radius, r) and an angle (?) from a reference direction. This alternative coordinate system simplifies problems with circular or rotational symmetry.

Conversion Formulas

The relationship between rectangular and polar coordinates is established via the equations x = r cos ? and y = r sin ?, as well as the formula r = ?(x² + y²) to compute the radius and the use of the tangent function (tan ? = y/x) to determine the angle, with adjustments made for the correct quadrant.

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