Problem 140 - Calculus Volume 3

Question

The rectangular coordinates of a point are given. Find two sets of polar coordinates for the point in (0, 2?]. Round to three decimal places. $$ (4,3) $$

Step-by-Step Solution

Answer

Polar coordinates: \((5, 0.644)\) and \((5, 6.927)\).

Step 1: Understanding Rectangular and Polar Coordinates

Rectangular coordinates are given as \((x, y)\). To convert this to polar coordinates \((r, \theta)\), use the relationships: \(r = \sqrt{x^2 + y^2}\) and \(\theta = \tan^{-1}(\frac{y}{x})\).

Step 2: Calculate Radius r

Using the formula \(r = \sqrt{x^2 + y^2}\), substitute \(x = 4\) and \(y = 3\): \[ r = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \] Thus, \(r = 5\).

Step 3: Calculate Angle θ in radians

Use the arctangent function to find the angle in radians: \[ \theta = \tan^{-1}\left(\frac{3}{4}\right) \approx 0.6435 \] Now, \(\theta \approx 0.644\) radians after rounding to three decimal places.

Step 4: Determine Two Sets of Polar Coordinates

Polar coordinates can be expressed as \((r, \theta + 2\pi k)\) for any integer \(k\). First set is \((5, 0.644)\). For a second set, add \(2\pi\) to \(\theta\): \[ \theta = 0.644 + 2\pi \approx 0.644 + 6.283 = 6.927 \] So, the second set is \((5, 6.927)\).

Step 5: Verify Range of θ

Both \(0.644\) and \(6.927\) are within the desired range \((0, 2\pi]\), ensuring our polar coordinates are correct.

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a way to describe the position of a point in a plane using a pair of numbers:

• The first number, typically called x, represents the horizontal position.
• The second number, called y, represents the vertical position.

For example, the point (4, 3) is a location on the Cartesian plane where:
- You move 4 units to the right along the x-axis from the origin. - Then move 3 units up along the y-axis.
Rectangular coordinates are straightforward but sometimes another system, like polar coordinates, is more convenient depending on the context.

Conversion Between Coordinate Systems

Converting between rectangular and polar coordinates involves understanding the relationship between distance and direction.

• The distance from the origin to the point is known as the radius (r).
• The direction from the positive x-axis to the point is described by the angle (\( \theta \)).

To find the polar coordinates (\( r, \theta \)) from rectangular coordinates (\( x, y \)), you can use these formulas:
- The radius (r) is calculated as \[ r = \sqrt{x^2 + y^2} \]- The angle (\( \theta \)) is found using the arctangent function: \[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \]These conversions allow you to switch between the systems and make use of whichever is more useful for solving a particular problem.

Radius and Angle Calculation

Calculating the radius and angle involves using trigonometric identities and formulas:
- For radius calculation, we utilize the Pythagorean theorem. For a point (4, 3), calculate \( r = \sqrt{4^2 + 3^2} \), yielding \( r = 5 \).
- For angle calculation, use the arctangent function, derived from trigonometric identities: \( \theta = \tan^{-1}\left(\frac{3}{4}\right) \). After calculation, \( \theta \approx 0.644 \) radians.
These calculations are essential to convert a point from rectangular form to polar form effectively.

Trigonometry in Polar Coordinates

Trigonometry plays a crucial role when dealing with polar coordinates, especially in finding and validating angles.

• The tangent function helps to calculate the angle by relating the ratio of the lengths of the opposite side to the adjacent side of a right triangle, linking (\( y/x \)) to \( \theta \).
• Verifying the angle's range ensures that it lies between 0 and 2\( \pi \), suitable for polar coordinate representation.

Often, the same point can be represented by multiple sets of polar coordinates.
For example, adding 2\( \pi \) to the angle provides an alternative position that is effectively the same, confirming the cyclic nature of angles in polar coordinates.
This understanding of trigonometry supports the transition between different representations of coordinates.