Problem 14 - Calculus Early Transcendentals

Question

Find a formula for the distance between the points with polar coordinates \(\left(r_{1}, \theta_{1}\right)\) and \(\left(r_{2}, \theta_{2}\right)\)

Step-by-Step Solution

Answer

The distance is \(d = \sqrt{r_1^2 + r_2^2 - 2r_1r_2\cos(\theta_1 - \theta_2)}\).

Step 1: Convert Polar Coordinates to Cartesian Coordinates

The first step to finding the distance between two points given in polar coordinates is to convert those coordinates to Cartesian coordinates. For the first point \((r_1, \theta_1)\), its Cartesian coordinates \((x_1, y_1)\) can be computed as:\[x_1 = r_1 \cos(\theta_1)\, \y_1 = r_1 \sin(\theta_1)\.\]For the second point \((r_2, \theta_2)\), its Cartesian coordinates \((x_2, y_2)\) are:\[x_2 = r_2 \cos(\theta_2)\, \y_2 = r_2 \sin(\theta_2)\.\]

Step 2: Apply the Distance Formula in Cartesian Coordinates

Now that we have the Cartesian coordinates for both points, we use the distance formula to find the distance \(d\) between the two points:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\].Substitute the expressions for \(x_1, y_1, x_2,\) and \(y_2\):\[d = \sqrt{\left(r_2 \cos(\theta_2) - r_1 \cos(\theta_1)\right)^2 + \left(r_2 \sin(\theta_2) - r_1 \sin(\theta_1)\right)^2}\].

Step 3: Simplify the Expression

To simplify the expression, apply the trigonometric identity to expand the terms:\[d^2 = \left(r_2 \cos(\theta_2) - r_1 \cos(\theta_1)\right)^2 + \left(r_2 \sin(\theta_2) - r_1 \sin(\theta_1)\right)^2\].Using the Pythagorean identity and some algebraic manipulation, this can be rewritten:\[d^2 = r_1^2 + r_2^2 - 2r_1r_2(\cos(\theta_1)\cos(\theta_2) + \sin(\theta_1)\sin(\theta_2))\].Recognizing the trigonometric identity \( \cos(\theta_1 - \theta_2) = \cos(\theta_1)\cos(\theta_2) + \sin(\theta_1)\sin(\theta_2)\), we have:\[d^2 = r_1^2 + r_2^2 - 2r_1r_2\cos(\theta_1 - \theta_2)\].

Step 4: Write the Distance Formula

Thus, the formula for the distance \(d\) between two points with polar coordinates \((r_1, \theta_1)\) and \((r_2, \theta_2)\) is:\[d = \sqrt{r_1^2 + r_2^2 - 2r_1r_2\cos(\theta_1 - \theta_2)}\].This is the final expression for computing the distance between the points in polar form.

Key Concepts

Distance Formula

The Distance Formula is a fundamental tool in geometry, widely used to determine the distance between two points in a coordinate plane. In the context of polar coordinates, the situation is slightly different from Cartesian coordinates. However, by converting polar coordinates to Cartesian coordinates, the problem simplifies. Understanding this formula is essential as it provides a bridge between different types of coordinate systems and helps solve various geometric problems effectively.

• The distance formula in Cartesian coordinates is \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)\.
• In polar coordinates, the distance depends on both the radial distances and the angles, resulting in the formula \(d = \sqrt{r_1^2 + r_2^2 - 2r_1r_2\cos(\theta_1 - \theta_2)}\).

This adaptation helps us compute the distance even when we start from polar descriptions.

Cartesian Coordinates

Cartesian Coordinates provide a straightforward way to represent points in a plane using two numbers, typically labeled as \(x\) and \(y\). These values correspond to the point's horizontal and vertical positions, respectively. By converting polar coordinates (\(r, \theta\)) to Cartesian coordinates, we can apply classical geometry concepts directly.

• From polar to Cartesian:
  • \(x = r\cos(\theta)\)
  • \(y = r\sin(\theta)\)

This conversion allows for seamless integration of radial and angular components into standard algebraic forms, making it easier to calculate distances using perpendicular relationships.

Trigonometric Identity

Trigonometric Identities are equations involving trigonometric functions that hold true for all values of the involved variables. When simplifying expressions in geometry, especially involving polar coordinates, recognizing and utilizing these identities is critical.

• The Pythagorean identity: \( \sin^2(\theta) + \cos^2(\theta) = 1\).
• The angle difference identity: \( \cos(\alpha - \beta) = \cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta) \).

In this exercise, these identities help simplify the expressions from Cartesian equivalent to polar distance formula, by expressing complex angle relationships in simpler, more manageable terms.

Algebraic Manipulation

Algebraic Manipulation involves adjusting and reworking algebraic expressions to make them easier to solve or to put them in a desired form. In the context of deriving the distance formula from polar coordinates, it's essential for rearranging terms and applying identities effectively.

• Start by substituting the Cartesian expressions derived from polar coordinates.
• Utilize trigonometric identities to simplify the expression, such as
  • \(d^2 = r_1^2 + r_2^2 - 2r_1r_2\cos(\theta_1 - \theta_2)\)

This step-by-step manipulation ensures that each part contributes efficiently towards the final formula, ensuring clarity and correctness in problem-solving.