Problem 13 - Essential Calculus

Question

The rectangular coordinates of a point are given. Plot the point and find \(t w o\) sets of polar coordinates for the point for \(0 \leq \theta<2 \pi\). $$ (\sqrt{3},-1) $$

Step-by-Step Solution

Answer

The two sets of polar coordinates for the point \((\sqrt{3}, -1)\) in the interval \(0 \leq \theta < 2\pi\) are \( (2, \frac{11\pi}{6})\) and \((2, \frac{23\pi}{6})\).

Step 1: Identify the coordinate given

The given point in rectangular coordinates is \((\sqrt{3}, -1)\). In the coordinate system, this point lies in the 4th quadrant as the x-coordinate is positive and y-coordinate is negative.

Step 2: Convert rectangular to polar coordinates

To convert rectangular to polar coordinates, use the following formulas: \(r=\sqrt{x^2 + y^2}\) and \(\theta=tan^{-1}( \frac{y}{x})\). By plugging the values of the given point into the formulas, we get: \(r = \sqrt{(\sqrt{3})^2 + (-1)^2} = 2\) and \(\theta = tan^{-1}( \frac{-1}{\sqrt{3}}) = -\frac{\pi}{6}\). Because \(\theta\) is negative, we add \(2\pi\) to get the positive angle value, resulting in \(\theta = \frac{11\pi}{6}\). Hence, one set of polar coordinates is \( (2, \frac{11\pi}{6})\).

Step 3: Find second set of polar coordinates

To find the second set of polar coordinates, we realize that when rotating in a circle, adding \(2\pi\) to the angle gives us the same point. Therefore, for our point, we add \(2\pi\) to the theta from our first set of polar coordinates, thus \(\theta = \frac{11\pi}{6} + 2\pi = \frac{23\pi}{6}\). The second set of polar coordinates is \( (2, \frac{23\pi}{6})\).

Key Concepts

Rectangular to Polar Conversion

Understanding rectangular to polar conversion is crucial for working in fields such as physics, engineering, and mathematics. This transformation is used to switch between rectangular (Cartesian) coordinates, which are defined by an x and y-coordinate pair, to polar coordinates, which are defined by a radius and an angle.

The basic formulas for conversion are:

• The radius (r) is found through the equation: \( r = \sqrt{x^2 + y^2} \)
• The angle (\( \theta \)) is calculated using: \( \theta = \tan^{-1}(\frac{y}{x}) \)

To perform this conversion, determine the radius using the Pythagorean Theorem. For the angle, inverse trigonometric functions are used, which take into account the signs of both the x and y coordinates. These signs dictate in which quadrant the angle lies. After obtaining the initial angle, sometimes adjustments are necessary, such as adding \( 2\pi \) to obtain a positive angle in cases where the initially computed angle is negative.

Plotting Points in Polar System

In the polar coordinate system, plotting points can be seen somewhat as a treasure hunt where the map is a circular grid. When you plot a point, you're essentially asked to walk a straight line for a certain distance, and then to turn at a specific angle.

To plot points in this system, start at the origin, move radially outwards the distance of the radius (r), and then rotate through an angle (\( \theta \)) from the positive x-axis. This is akin to following a path straight out from the center of a clock and then turning clockwise or counterclockwise to reach the specific hour associated with the angle.

For instance, when given polar coordinates \( (2, \frac{11\pi}{6}) \), you would start at the center, move 2 units out, and then turn almost a full circle (since \( \frac{11\pi}{6} \) is just short of \( 2\pi \), or a full rotation). Polar plots are especially useful for understanding patterns that are naturally circular or periodic, such as wave forms or celestial orbits.

Trigonometry in Coordinate Conversion

The role of trigonometry in coordinate conversion cannot be overstated. It's the bridge between linear and angular measurements in the conversion process. The sine, cosine, and tangent functions relate the angles to the dimensions of a right triangle, which is essentially what you're working with when converting between rectangular and polar coordinates.

When given a point \((x, y)\) in rectangular coordinates, imagine drawing a straight line from the origin to your point, and then dropping a vertical line down to the x-axis, forming a right triangle. The x-value corresponds to the adjacent side of the angle, the y-value to the opposite side, and the hypotenuse is the line you've drawn from the origin to your point.

To find the angle using tangent, which compares the opposite side to the adjacent side, the formula \( \theta = \tan^{-1}(\frac{y}{x}) \) comes into play. This trigonometric function is particularly useful because it implicitly addresses the quadrant in which the point lies, through the resultant sign and value of the angle. Adjustments using principles of trigonometry and an understanding of the unit circle are then applied to find the correct angle in standard position for polar coordinates.