Problem 87 - Contemporary Precalculus

Question

Distance Formula for Polar Coordinates: Prove that the distance from \((r, \theta)\) to \((s, \beta)\) is $$ \sqrt{r^{2}+s^{2}-2 r s \cos (\theta-\beta)} $$ [Hint: If \(r > 0, s > 0,\) and \(\theta > \beta,\) then the triangle with vertices \((r, \theta),(s, \beta),(0,0)\) has an angle of \(\theta-\beta,\) whose sides have lengths \(r\) and \(s .\) Use the Law of cosines.]

Step-by-Step Solution

Answer

Question: Prove the distance formula in polar coordinates as: \(\sqrt{r^{2}+s^{2}-2 r s \cos (\theta-\beta)}\) Answer: To prove the polar distance formula, we first converted the polar coordinates to cartesian coordinates. Then, we formed a triangle with vertices A, B, and origin O and applied the law of cosines. By substituting coordinates and using the trigonometric identities, we manipulated the equation which led to the desired polar distance formula. Thus, the distance formula in polar coordinates is proven to be: \(\sqrt{r^{2}+s^{2}-2 r s \cos (\theta-\beta)}\).

Step 1: 1. Convert coordinates to Cartesian

First, let's convert given polar coordinates to cartesian coordinates as: A(\((r, \theta)\)): \((x_{1}, y_{1}) = (r\cos(\theta), r\sin(\theta))\) B(\((s, \beta)\)): \((x_{2}, y_{2}) = (s\cos(\beta), s\sin(\beta))\)

Step 2: 2. Determine the third point

We have been given a hint that a triangle can be formed with vertices A, B, and origin O(0,0).

Step 3: 3. Apply Law of Cosines

Using the Law of Cosines on triangle ABO, \(AB^2 = AO^2 + BO^2 - 2(AO)(BO)\cos(\angle AOB)\) where AB, AO and BO are distances between A and B, A and O, and B and O respectively.

Step 4: 4. Substitute coordinates

Now substitute the coordinates: \(AB^2 = (x_{1}-x_{2})^2 + (y_{1}-y_{2})^2 = r^2 + s^2 - 2rs\cos(\theta - \beta)\)

Step 5: 5. Manipulate the equation

Expanding the left side of the equation: \((x_{1}-x_{2})^2 + (y_{1}-y_{2})^2 = (r\cos(\theta)-s\cos(\beta))^2 + (r\sin(\theta)-s\sin(\beta))^2\) Simplify: \(r^2\cos^2(\theta) + s^2\cos^2(\beta) - 2rs\cos(\theta)\cos(\beta) + r^2\sin^2(\theta) + s^2\sin^2(\beta) - 2rs\sin(\theta)\sin(\beta) = r^2 + s^2 - 2rs\cos(\theta - \beta)\) Using Trigonometric Identity: \(\cos^2(\theta) + \sin^2(\theta) = 1\), \(r^2 + s^2 - 2rs(\cos(\theta)\cos(\beta) + \sin(\theta)\sin(\beta)) = r^2 + s^2 - 2rs\cos(\theta - \beta)\) Using Trigonometric Identity: \(\cos(\theta)\cos(\beta) + \sin(\theta)\sin(\beta) = \cos(\theta - \beta)\), \(r^2 + s^2 - 2rs\cdot(cos(\theta-\beta)) = r^2 + s^2 - 2rs\cdot\cos(\theta - \beta)\) Since both sides of the equation are equal, the distance formula in polar coordinates is proven to be: \(\sqrt{r^{2}+s^{2}-2 r s \cos (\theta-\beta)}\)

Key Concepts

Distance Formula

When working with polar coordinates, it's essential to understand how to calculate the distance between two points. The distance formula in Cartesian coordinates, which utilizes the square root of the sum of squared differences, is familiar to many. However, polar coordinates require a different approach.

To find the distance between two points in polar form:

• Consider two points \( (r, \theta) \) and \( (s, \beta) \), where each point is represented by its radius and angle.
• Form a triangle with these two points and the origin (0,0), noting that the angle between the radii is \( \theta - \beta \).
• Utilize the distance formula for polar coordinates: \[ \ \sqrt{r^{2}+s^{2}-2rs \cos(\theta-\beta)} \ \]

This formula derives from recognizing the relationship between the polar and Cartesian systems, and especially the Law of Cosines. Understanding it allows you to easily find the distance between any two points in polar coordinates.

Law of Cosines

The Law of Cosines is a crucial tool in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. In our specific problem, it forms the bridge between Cartesian and polar coordinates.

Here's how it functions in our context:

• The triangle is formed by the two points and the origin in the polar coordinate system.
• The sides of the triangle are the radii from the origin to the points, denoted as \( r \) and \( s \).
• The angle \( \theta - \beta \) is between these two radii.
• The Law of Cosines states: \[ AB^2 = r^2 + s^2 - 2rs \cos(\theta-\beta) \]

This identity allows you to compute the length of the side opposite the angle \( \theta - \beta \), which is essentially the distance between the two points calculated in polar coordinates.

Trigonometric Identities

Trigonometric identities are powerful in simplifying expressions and calculations in various mathematical contexts, including our work with polar coordinates.

In this situation, several key identities were used:

• \( \cos^2(\theta) + \sin^2(\theta) = 1 \), which helps in verifying parts of the calculations when converting between systems.
• \( \cos(\theta-\beta) = \cos(\theta)\cos(\beta) + \sin(\theta)\sin(\beta) \), a critical tool for substituting expressions within the Law of Cosines to relate angles in the polar coordinate system.

By applying these identities, it's possible to derive consistent and simplified expressions through relationships that hold true across different trigonometric contexts. Mastering them aids greatly in transitions between polar and Cartesian coordinates.

Conversion to Cartesian Coordinates

To better understand polar coordinates and perform calculations, converting them to Cartesian coordinates can be very helpful. This conversion translates the radial distance and angle into the familiar x and y coordinates:

For a given point \( A(r, \theta) \):

• The Cartesian coordinates are \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \).

Similarly, for \( B(s, \beta) \):

• Convert to \( x = s \cos(\beta) \) and \( y = s \sin(\beta) \).

This conversion allows for straightforward application of the Cartesian distance formula, reaffirming our derived polar distance formula once reconverted. Understanding both polar and Cartesian viewpoints provides a versatile skill set for dealing with various mathematical problems.