Problem 39 - Precalculus Mathematics for Calculus

Question

Convert the rectangular coordinates to polar coordinates with \(r>0\) and \(0 \leq \theta<2 \pi\). $$(3,4)$$

Step-by-Step Solution

Answer

Convert (3, 4) to polar coordinates as (5, 0.927).

Step 1: Identify Rectangular Coordinates

First, identify the given rectangular coordinates, which are \((x, y) = (3, 4)\).

Step 2: Calculate the Radius \(r\)

The radius in polar coordinates is calculated using the formula \( r = \sqrt{x^2 + y^2} \). Substitute \(x = 3\) and \(y = 4\) to get: \[ r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \] So, the radius \(r = 5\).

Step 3: Calculate the Angle \(\theta\)

The angle \(\theta\) is calculated using the formula \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \). With \(x = 3\) and \(y = 4\), the angle is: \[ \theta = \tan^{-1}\left(\frac{4}{3}\right) \] Since both \(x\) and \(y\) are positive, \(\theta\) is already in the correct quadrant (Quadrant I). Using a calculator, \( \theta \approx 0.927 \) radians.

Step 4: Write the Polar Coordinates

Combine the radius and angle to write the polar coordinates. \((r, \theta) = (5, 0.927)\). Ensure \(r > 0\) and \(0 \leq \theta < 2\pi\), which already hold true here.

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are typically expressed in the form

• (x, y) where
• "x" represents the horizontal position, and
• "y" represents the vertical position on a two-dimensional plane.

These coordinates help us locate the exact position of a point in a simple and intuitive way. For example, the point (3, 4) tells us the point is 3 units to the right of the origin and 4 units up.
Understanding how these coordinates work is crucial when converting them into polar coordinates, as this transformation allows us to express the same point using a different kind of measurement system.
Polar coordinates provide an alternate way to describe the same location using a radius and an angle instead of direct horizontal and vertical measures. The conversion from rectangular to polar involves translating these horizontal and vertical measures into a distance from the origin (radius) and an angle from the positive x-axis (angle).

Radius Calculation

The radius is a key element in polar coordinates and represents the distance of a point from the origin of the coordinate system. Calculating the radius from rectangular coordinates is done using the Pythagorean theorem.
The formula: \[ r = \sqrt{x^2 + y^2} \]instructs us to square the x and y values, sum these values, and compute the square root. In our example with (3, 4):

• Square the x-coordinate, 3: \(3^2 = 9\)
• Square the y-coordinate, 4: \(4^2 = 16\)
• Sum the squared values: \(9 + 16 = 25\)
• Take the square root: \(\sqrt{25} = 5\)

Hence, the radius, r, is 5.
This distance, often considered as the "length" from the origin, must always be a positive value in polar coordinates, ensuring that any point's position is defined unambiguously.

Angle Calculation

The angle, often denoted by \(\theta\) (theta), is essential in defining a point's position in polar coordinates. This angle is measured counterclockwise from the positive x-axis to the line segment connecting the origin and the point. To find \(\theta\), we can use the inverse tangent function:
\[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \]In our coordinate conversion example for point (3, 4):

• Plug in the values: \(\frac{4}{3}\)
• Use a calculator to find \(\theta = \tan^{-1}(\frac{4}{3})\)
• This yields approximately 0.927 radians

Since both x and y are positive, the point lies in the first quadrant and \(\theta\) falls naturally within the required range of 0 to \(2\pi\).
Understanding this conversion helps in visualizing problems from an angle-based perspective, crucial for fields like physics and engineering, where the direction of vectors is often significant.