Problem 7 - Thomas Calculus

Question

Find the polar coordinates, \(0 \leq \theta<2 \pi\) and \(r \geq 0,\) of the following points given in Cartesian coordinates. $$\begin{array}{ll}{\text { a. }(1,1)} & {\text { b. }(-3,0)} \\ {\text { c. }(\sqrt{3},-1)} & {\text { d. }(-3,4)}\end{array}$$

Step-by-Step Solution

Answer

a) \((\sqrt{2}, \frac{\pi}{4})\); b) \((3, \pi)\); c) \((2, \frac{11\pi}{6})\); d) \((5, \pi - \tan^{-1}(\frac{4}{3}))\)."

Step 1: Convert Cartesian to Polar - Point (1,1)

For point \((1,1)\), begin by finding the radius \(r\) using the formula \(r = \sqrt{x^2 + y^2}\). Here, \(x = 1\) and \(y = 1\). Thus, \(r = \sqrt{1^2 + 1^2} = \sqrt{2}\). Next, find the angle \(\theta\) using \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\). Hence, \(\theta = \tan^{-1}(1) = \frac{\pi}{4}\). The polar coordinates are \((\sqrt{2}, \frac{\pi}{4})\).

Step 2: Convert Cartesian to Polar - Point (-3,0)

For point \((-3,0)\), the radius \(r\) is found by \(r = \sqrt{x^2 + y^2} = \sqrt{(-3)^2 + 0^2} = 3\). The angle \(\theta\) is determined by the fact that the point lies on the negative x-axis, so \(\theta = \pi\). Thus, the polar coordinates are \((3, \pi)\).

Step 3: Convert Cartesian to Polar - Point (\(\sqrt{3},-1\))

For point \((\sqrt{3},-1)\), calculate the radius \(r\) as \(r = \sqrt{(\sqrt{3})^2 + (-1)^2} = 2\). To find \(\theta\), use \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\). So, \(\theta = \tan^{-1}\left(\frac{-1}{\sqrt{3}}\right) = -\frac{\pi}{6}\). To adjust for the angle range, add \(2\pi\), making \(\theta = \frac{11\pi}{6}\). The polar coordinates are \((2, \frac{11\pi}{6})\).

Step 4: Convert Cartesian to Polar - Point (-3,4)

For point \((-3,4)\), find \(r\) using \(r = \sqrt{(-3)^2 + 4^2} = 5\). The angle \(\theta\) is found by using \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\). Thus, \(\theta = \tan^{-1}\left(\frac{4}{-3}\right)\). Since the point is in the second quadrant, add \(\pi\) to \(\theta\), yielding \(\theta = \pi - \tan^{-1}\left(\frac{4}{3}\right)\). The polar coordinates are \((5, \pi - \tan^{-1}(\frac{4}{3}))\).

Key Concepts

Cartesian Coordinates

Cartesian Coordinates are a system that defines every point uniquely on a plane using two numerical values. These values are typically written as \((x, y)\), where *x* is the horizontal distance from the origin, and *y* is the vertical distance. This system is also known as the rectangular coordinate system.

• **Origin**: The Fixed Point (0, 0) formed by the intersection of the *x* and *y* axes.
• **Axes**: The horizontal **x-axis** and the vertical **y-axis** form a perpendicular coordinate system.
• **Quadrants**: The plane is divided into four quadrants determined by the signs of *x* and *y*:
• First Quadrant (+, +)
• Second Quadrant (-, +)
• Third Quadrant (-, -)
• Fourth Quadrant (+, -)

Understanding Cartesian Coordinates provides the foundation needed to convert points to Polar Coordinates.

Radius Calculation

The Radius is crucial in the conversion from Cartesian to Polar Coordinates. It represents the distance of a point from the origin in the polar plane. Calculating the radius involves using the Pythagorean theorem since it is the hypotenuse of a right triangle formed by the perpendicular coordinates *x* and *y*.

• The formula for radius \(r\) is: \[ r = \sqrt{x^2 + y^2}\]
• Always results in a non-negative value as it is a distance.
• Uses simple arithmetic and square root for calculation.

In our example, the radius was different for each point, but calculating it is straightforward using the above formula.

Angle Determination

Angle Determination is the process of finding the angle \(\theta\) a point makes with the positive x-axis in Polar Coordinates. Unlike radius, angles can be positive or negative, based on direction.

• To find \(\theta\), use the inverse tangent function: \[ \theta = \tan^{-1}\left(\frac{y}{x}\right)\]
• This function calculates angle relative to the x-axis. It is important to consider the quadrant when using this formula, since it affects the sign and value of \(\theta\).
• In cases where \(\theta\) may be negative or does not fall within the desired range of \(0 \leq \theta < 2\pi\), adjustments such as adding \(2\pi\) might be necessary.

Understanding Angle Determination helps ensure that the angle correctly corresponds to its position in the Cartesian plane.

Coordinate Conversion

Coordinate Conversion is the process used to transform Cartesian Coordinates into Polar Coordinates. This involves using trigonometric functions and formulas to derive a radius and angle that describe a point's position on the plane.

• **From Cartesian \((x, y)\) to Polar \((r, \theta)\):**
  • Calculate the radius using \(r = \sqrt{x^2 + y^2}\)
  • Determine the angle with \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\)
• Ensure the values meet conditions such as \(r \geq 0\) and \(0 \leq \theta < 2 \pi\).
• Convert angles to the proper direction and quadrant-based range.Understanding Coordinate Conversion allows for the accurate translation of points into a system used widely in fields like physics and engineering.