Problem 69 - Precalculus Mathematics for Calculus

Question

(a) Use the Law of cosines to prove that the distance between the polar points \(\left(r_{1}, \theta_{1}\right)\) and \(\left(r_{2}, \theta_{2}\right)\) is $$ d=\sqrt{r_{1}^{2}+r_{2}^{2}-2 r_{1} r_{2} \cos \left(\theta_{2}-\theta_{1}\right)} $$ (b) Find the distance between the points whose polar coordinates are \((3,3 \pi / 4)\) and \((1,7 \pi / 6),\) using the formula from part (a). (c) Now convert the points in part (b) to rectangular coordinates. Find the distance between them using the usual Distance Formula. Do you get the same answer?

Step-by-Step Solution

Answer

Yes, the distances computed are the same.

Step 1: Recall the Law of Cosines

The Law of Cosines for a triangle states: If the sides of the triangle are \(a\), \(b\), and \(c\), and the angle opposite the side \(c\) is \(C\), then the formula is \(c^2 = a^2 + b^2 - 2ab \, \cos(C)\). This will guide us in finding the distance between two points in polar coordinates.

Step 2: Apply the Law of Cosines in Polar Form

For points \((r_1, \theta_1)\) and \((r_2, \theta_2)\), consider a triangle formed with these points and the origin. The angle between the lines connecting each point to the origin is \(\theta_2 - \theta_1\). Apply the Law of Cosines: \[ d^2 = r_1^2 + r_2^2 - 2r_1r_2 \cos (\theta_2 - \theta_1) \] Taking the square root gives \[ d = \sqrt{r_1^2 + r_2^2 - 2r_1r_2 \cos (\theta_2 - \theta_1)} \] which is the required distance formula.

Step 3: Calculate the Distance Using Polar Formula

Given points are \((3, \frac{3\pi}{4})\) and \((1, \frac{7\pi}{6})\). Substitute \(r_1 = 3\), \(\theta_1 = \frac{3\pi}{4}\), \(r_2 = 1\), \(\theta_2 = \frac{7\pi}{6}\) into the formula: \[ d = \sqrt{3^2 + 1^2 - 2 \times 3 \times 1 \times \cos\left(\frac{7\pi}{6} - \frac{3\pi}{4}\right)} \] Calculate \(\frac{7\pi}{6} - \frac{3\pi}{4} = \frac{14\pi}{12} - \frac{9\pi}{12} = \frac{5\pi}{12}\). Calculate \(\cos(\frac{5\pi}{12})\) and proceed.

Step 4: Convert Polar Coordinates to Rectangular Coordinates

Use the formulas: \(x = r \cos \theta\), \(y = r \sin \theta\). For \((3, \frac{3\pi}{4})\), convert to \((x_1, y_1) = (3 \cos(\frac{3\pi}{4}), 3 \sin(\frac{3\pi}{4})) = (-\frac{3\sqrt{2}}{2}, \frac{3\sqrt{2}}{2})\). For \((1, \frac{7\pi}{6})\), convert to \((x_2, y_2) = (1 \cos(\frac{7\pi}{6}), 1 \sin(\frac{7\pi}{6})) = (-\frac{\sqrt{3}}{2}, -\frac{1}{2})\).

Step 5: Calculate Distance Using the Distance Formula

Use the formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). Substitute \(x_1 = -\frac{3\sqrt{2}}{2}\), \(y_1 = \frac{3\sqrt{2}}{2}\), \(x_2 = -\frac{\sqrt{3}}{2}\), \(y_2 = -\frac{1}{2}\). Calculate: \[ d = \sqrt{\left(-\frac{\sqrt{3}}{2} + \frac{3\sqrt{2}}{2}\right)^2 + \left(-\frac{1}{2} - \frac{3\sqrt{2}}{2}\right)^2} \] Simplify and check if the answer matches with the result from Step 3.

Key Concepts

Law of Cosines

The Law of Cosines is a key concept for understanding the relationship between the sides and angles of a triangle. It helps us calculate one side when the other two sides and the enclosed angle are known. Using this law, we can generalize the Pythagorean theorem to non-right triangles. The formula is: \[ c^2 = a^2 + b^2 - 2ab \cos(C) \] where:

• \(a\), \(b\), and \(c\) are the side lengths of the triangle,
• \(C\) is the angle opposite side \(c\).

In polar coordinates, if you have two points, you form a triangle with the origin. The sides are the radii and the angle between them. Using the Law of Cosines helps find the direct distance between these two points. This is crucial because you can calculate the distance without converting to rectangular coordinates first.

Distance Formula

The distance formula is derived directly from the Pythagorean theorem and is used to find the length between two points in a plane. In the rectangular (Cartesian) coordinate system, the formula is: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] where:

• \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.

This formula measures the straight-line distance, or Euclidean distance, between the two points. It is particularly useful in many real-world applications like finding the shortest path in navigation systems or the closest point in data analysis. By substituting polar coordinates into this formula (after converting them), you can verify distances calculated using other methods like the polar distance formula.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, represent points on a plane using two numbers: \((x, y)\). These indicate horizontal and vertical distances from a reference point called the origin. It is named after René Descartes, who contributed significantly to its development in analytical geometry.

• The \(x\)-axis runs horizontally,
• The \(y\)-axis runs vertically,
• Any point \((x, y)\) signifies its position relative to these axes.

Rectangular coordinates are the most common way to represent points in space, making calculations straightforward, like using the distance formula. When dealing with polar coordinates, converting them to rectangular coordinates can simplify some calculations, especially when you are more familiar with this system.

Coordinate Conversion

Coordinate conversion is the process of changing points from one form to another, such as from polar to rectangular coordinates. This is crucial for solving problems that require a particular coordinate system. For example, converting polar coordinates \((r, \theta)\) to rectangular coordinates involves the use of trigonometric functions:

• \(x = r \cos(\theta)\)
• \(y = r \sin(\theta)\)

Conversely, converting from rectangular to polar coordinates requires:

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

Coordinate conversion is key in exercises where initial data is given in a non-desirable form for operations or computations. By using the appropriate formulae, finding direct and meaningful results becomes more accessible in whichever system is chosen.