Problem 63 - Precalculus with Limits

Question

In Exercises 55-64, use a graphing utility to find one set of polar coordinates for the point given in rectangular coordinates. \(\left(\frac{7}{4}, \frac{3}{2}\right)\)

Step-by-Step Solution

Answer

\( (r, θ) \) are polar coordinates for the given point, with r and θ calculated in steps 1 and 2 respectively.

Step 1: Compute Radial Distance r

Calculate the radial distance r using the Pythagorean theorem: \( r = \sqrt{x^2 + y^2} \). Substituting x and y with the given rectangular coordinates, \( r = \sqrt{(7/4)^2 + (3/2)^2} \).

Step 2: Compute Angle θ

Calculate the angle θ with the formula: \( θ = atan(y/x) \). However, this will give the angle between the line connecting the origin and the point and the positive x-axis. The result needs to be adjusted based on the quadrant of the rectangular coordinate. For the given values of x , y (i.e., x is positive, y is positive), the point lies in the first quadrant. Thus, no adjustment is necessary. Plug the values into the formula to get \( θ = atan(3/2 / 7/4) \).

Step 3: Graphing and Conversion

Using a graphing utility, graph the point with the derived polar coordinates. Make sure that the point aligns with the given rectangular coordinates on the graph. If it does, the conversion is correct. If not, recheck the calculations.

Key Concepts

Graphing Utility

A graphing utility is an invaluable tool in visualizing mathematical concepts, particularly when converting between coordinate systems. When working with conversions from rectangular to polar coordinates, graphing utilities help verify the accuracy of the manual calculations. By plotting the derived polar coordinates, one can instantly see if the location matches the original rectangular coordinates. Most modern graphing utilities also have built-in functions to perform this conversion automatically, providing an excellent way to check one's work.

Pythagorean Theorem

The Pythagorean theorem is central to converting rectangular coordinates to polar coordinates. It is used to calculate the radial distance from the origin to a point in a plane. According to this theorem, in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In rectangular to polar coordinate conversion, the hypotenuse is the radial distance, and the other two sides correspond to the x and y coordinates of the point.

Polar Coordinates

Polar coordinates represent points in a plane using a distance and an angle from a fixed point called the pole, usually the origin of a coordinate system. The distance from the pole is referred to as the radial distance (or radius) and denoted by 'r', while the angle is denoted by the Greek letter theta (θ). Polar coordinates are particularly useful in situations where relationships are more naturally expressed in terms of angles and distances from a central point, such as in circular and spiral patterns.

Rectangular Coordinates

Rectangular (or Cartesian) coordinates are the most common way to represent points in a plane. This system uses two perpendicular coordinate axes: the x-axis (horizontal) and the y-axis (vertical), which intersect at the origin. The position of any point is given by an ordered pair (x, y), representing horizontal and vertical displacements from the origin. Rectangular coordinates make it straightforward to represent points on a grid and perform algebraic calculations.

Radial Distance

In the context of polar coordinates, the radial distance, 'r', is the straight line distance from the origin to a point in the plane. It is equivalent to the hypotenuse of a right triangle formed by the x and y coordinates as its legs. The radial distance is always a non-negative number and can be calculated using the Pythagorean theorem given the rectangular coordinates (x, y):
\( r = \( \-sqrt{x^2 + y^2} \) \). This formula is crucial for transforming rectangular coordinates into polar ones.

Angle θ Calculation

The angle θ (theta) in polar coordinates indicates the direction of the radial line from the origin to the point and is measured from the positive x-axis. To calculate θ, one can use the arctangent function on the ratio of the y-coordinate to the x-coordinate (y/x). However, attention must be paid to the quadrant in which the point lies, as this affects the angle's calculation and interpretation. In the first quadrant, where both x and y are positive, the angle θ can be calculated directly from the arctangent without adjustment.