Problem 137 - Calculus Volume 3

Question

The rectangular coordinates of a point are given. Find two sets of polar coordinates for the point in (0, 2?]. Round to three decimal places. $$ (3,-4)(3,-4) $$

Step-by-Step Solution

Answer

Two polar coordinates are \((5, 5.356)\) and \((-5, 2.214)\).

Step 1: Understand the Conversion

Polar coordinates \((r, \theta)\) are related to rectangular coordinates \((x, y)\) by the formulas: \[r = \sqrt{x^2 + y^2}\] and \[\theta = \tan^{-1}\left(\frac{y}{x}\right)\].For the point \((3, -4)\), these formulas will help convert it to polar form.

Step 2: Calculate the Distance (r)

Using the formula for \(r\): \[r = \sqrt{x^2 + y^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5\].Thus, the distance \(r\) is 5.

Step 3: Calculate the Angle (θ)

Using the formula for \(\theta\): \[\theta = \tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}\left(\frac{-4}{3}\right)\].Determine \(\theta\) using a calculator, which gives \(\theta \approx -0.927\) radians. Since this is negative and not in the interval \((0, 2\pi]\), we need to adjust it.

Step 4: Adjust Angle to Standard Range

Since the angle must be positive, add \(2\pi\) to \(-0.927\) radians: \[-0.927 + 2\pi \approx 5.356\].Thus, one polar coordinate is \((5, 5.356)\).

Step 5: Find a Second Set of Polar Coordinates

The polar angle can also be given with a different equivalent angle by using the formula \[\theta + \pi\] instead of \(\theta\), changing the sign of \(r\): \(-5, 5.356 - \pi\) or \(-5, 2.214\).Thus, another set of polar coordinates is \((-5, 2.214)\).

Key Concepts

Rectangular Coordinates

Rectangular coordinates are a way to represent a point in a 2D plane using two values, typically written as (x, y). These correspond to the horizontal and vertical positions of the point from a fixed origin, usually the (0, 0) point.
Understanding rectangular coordinates helps in visualizing mathematical problems and converting them into other forms, like polar coordinates.

• The first value, x, represents how far the point is horizontally from the origin. Positive x values usually indicate rightward direction, while negative values indicate left.
• The second value, y, indicates the vertical distance. Positive y values indicate upward direction, whereas negative values represent downward movement.

When working with problems involving points like (3, -4), we're dealing with a point three units right and four units down from the origin.

Conversion Formulas

Converting rectangular coordinates to polar coordinates involves a couple of key formulas.
These transformations are essential to switch from Cartesian to polar systems, allowing us to interpret points based on direction and distance from a reference point.

• The first formula is for the radial distance, given by: \[ r = \sqrt{x^2 + y^2} \]. This formula calculates the straight-line distance or "radius" from the origin to the point.
• The second formula calculates the angle with respect to the positive x-axis, using: \[ \theta = \tan^{-1} \left(\frac{y}{x}\right) \]. This inverse tangent (or arctan) function helps determine the angle based on the ratio of y to x.

These formulas are foundational in converting coordinates accurately, such as changing (3, -4) into polar coordinates.

Angle Adjustment

Angles in polar coordinates are traditionally expressed in the range from 0 to \( 2\pi \) radians. However, the direct calculation might result in angles outside this range.

To adjust angles properly:

• Check if the computed angle is negative or outside \( (0, 2\pi] \). If it is negative, you'd add \( 2\pi \) to transform it into a positive equivalent angle within the standard range.
• For example, an angle of \(-0.927\) radians becomes \( 5.356 \) radians upon adding \( 2\pi \). This ensures that the angle correctly resides within the accepted interval.
• Additionally, determining an equivalent angle by adding \( \pi \), especially when seeking a second solution, helps to express the point differently by alternating the sign of the distance.

Such adjustments are vital for meeting the requirements of polar coordinate conventions and ensuring the solutions are accurate.

Distance Calculation

Distance in polar coordinates refers to how far a point is from the origin. This is one component of the polar form, expressed as the radius or "r."
The formula used to compute the distance, \[ r = \sqrt{x^2 + y^2} \], is straightforward. It involves squaring the x and y values, adding them together, and then taking the square root of the result.
For example, for the point (3, -4), the distance would be calculated as follows:

• First, compute the squares: \( 3^2 = 9 \) and \((-4)^2 = 16 \).
• Sum these squares: \( 9 + 16 = 25 \).
• Finally, take the square root of 25 to find the distance: \( \sqrt{25} = 5 \).

Thus, the distance "r" from the origin to the point (3, -4) is 5. This radial distance is crucial in defining a point's position in polar coordinates.