Problem 29 - Precalculus with Calculus Previews

Question

Find polar coordinates that satisfy (a) \(r>0,-\pi<\theta \leq \pi\) (b) \(r<0,-\pi<\theta \leq \pi\) for each point with the given rectangular coordinates. $$ (7,0) $$

Step-by-Step Solution

Answer

(a) \((7, 0)\), (b) \((-7, \pi)\).

Step 1: Convert Rectangular to Polar Coordinates

Given the rectangular coordinates are (7,0). The polar coordinates are expressed as \((r, \theta)\) where \(r\) is the radius and \(\theta\) is the angle. We calculate \(r = \sqrt{x^2 + y^2}\) and \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\).

Step 2: Calculate the Radius

Using the formula for radius \(r = \sqrt{x^2 + y^2}\), substitute \(x = 7\) and \(y = 0\) to get \(r = \sqrt{7^2 + 0^2} = \sqrt{49} = 7\). Thus, the radius \(r = 7\).

Step 3: Calculate the Angle when \(r > 0\)

For \(-\pi < \theta \leq \pi\), we use \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\). Here, \(\theta = \tan^{-1}(0) = 0\). Thus, when \(r > 0\), the polar coordinates are \((7, 0)\).

Step 4: Determine the Radial Coordinate for \(r < 0\)

Since polar coordinates are periodic with respect to angle, you'd essentially be measuring in the opposite direction for \(r < 0\). We have \((r, \theta) = (-7, 0)\), which requires changing \(\theta\) by adding \(\pi\) to the angle. This gives \((r, \theta) = (-7, \pi)\).

Step 5: Final Polar Coordinates

For (a) \(r > 0\), the polar coordinates are \((7, 0)\). For (b) \(r < 0\), the polar coordinates are \((-7, \pi)\).

Key Concepts

Rectangular to Polar Conversion

Converting rectangular coordinates to polar coordinates involves a fundamental transformation in math. In rectangular coordinates, a point is defined by X and Y values on a Cartesian plane. Polar coordinates, however, define a point based on its distance from the origin and its angle relative to the positive X-axis. This translation of coordinates is frequently required in physics and engineering.

To begin with this conversion, we use the formulas:

• \[ r = \sqrt{x^2 + y^2} \]
• \[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \]

These equations provide the radius \( r \) and angle \( \theta \), which are the polar coordinates of the given rectangular point (X, Y). Understanding these basic equations is vital for successfully bridging between these two coordinate systems.

Radius Calculation

The radius in polar coordinates represents the direct distance from the origin to the point in question. Calculating this radius is straightforward with the distance formula:\[ r = \sqrt{x^2 + y^2} \]
In this exercise, the point given is (7, 0). Substituting these into the formula gives:

• \[ r = \sqrt{7^2 + 0^2} = \sqrt{49} = 7 \]

Therefore, the radius (or distance from the origin) is 7 units. This value remains constant irrespective of the angle because it solely depends on the X and Y distances from the origin. Recognizing this constancy can help clarify why the radius \( r \) maintains a singular value for any given point.

Angle Calculation

The angle \( \theta \) in polar coordinates is crucial as it determines the direction from the positive X-axis to the point. This is calculated using the inverse tangent function, given by:

• \[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \]

This exercise involves the point (7, 0). So when calculating the angle:

• \[ \theta = \tan^{-1}(0) = 0 \]

This calculation results in \( \theta = 0 \), indicating that the point is on the positive X-axis. Understanding angle calculation helps visualize the absolute direction of the point in a polar grid, particularly when it aligns purely along one of the axes.

Negative Radial Coordinate

Handling negative radial coordinates can be tricky, as it implies measuring distance in the opposite direction. In polar coordinate systems, a negative value of \( r \) means the point is in the opposite direction, which is effectively resolved by adjusting the angle \( \theta \) by \( \pi \) radians.

In this problem, for the negative radius case, we initially find that:

• The positive radius polar coordinate is \((7, 0)\).
• For \( r = -7 \), the angle adjustment makes it \( \theta = 0 + \pi = \pi \).

This results in polar coordinates \((-7, \pi)\). Such adjustments keep the entire system consistent and enable representation of all potential directions by manipulating both angle and radial sign.