Problem 82 - Precalculus

Question

Convert \((-a, b)\) to polar coordinates. Assume that \(a>0, b>0.\)

Step-by-Step Solution

Answer

The polar coordinates are \((r, θ) = \left( \sqrt{a^2 + b^2}, \pi - \tan^{-1} \left( \frac{b}{a} \right) \right)\).

Step 1: Understand Cartesian Coordinates

The point (-a, b) in Cartesian coordinates is based on the x-coordinate being -a and the y-coordinate being b. Our goal is to convert this point to polar coordinates.

Step 2: Compute the Radius

In polar coordinates, the radius r is equal to the distance from the origin to the point.You can calculate it using the formula:\[r = \sqrt{x^2 + y^2}\]Plugging in our x and y values:\[r = \sqrt{(-a)^2 + b^2} = \sqrt{a^2 + b^2}\]

Step 3: Determine the Angle

The angle θ in polar coordinates is found using the \( ext{atan2}(y, x)\) function, which calculates the angle based on the quadrants of x and y:\[θ = ext{atan2}(b, -a)\]Since x = -a is negative and y = b is positive, the angle is in the second quadrant. Therefore:\[θ = ext{atan2}(b, -a) = an^{-1} \left( \frac{b}{-a} \right) + \pi\]

Key Concepts

Cartesian Coordinates

When we talk about Cartesian coordinates, we're referring to a system that describes a point in a plane using two values: the x-coordinate and the y-coordinate. This coordinate system is straightforward and involves plotting the values on a horizontal and vertical axis. For the point \((-a, b)\), the x-coordinate is \(-a\) and the y-coordinate is \(b\).
- The x-coordinate \(-a\) tells us how far left or right the point is from the y-axis.- The y-coordinate \(b\) tells us how far up or down the point is from the x-axis.To convert from Cartesian to polar coordinates, we must focus on two primary components: the radius and the angle. Moving from a linear grid to a circular one involves computing these values based on the given coordinates.

radius calculation

The radius in polar coordinates signifies the straight-line distance from the origin \((0, 0)\) to the point in question. It's like drawing a circle from the origin to the point. The formula for calculating the radius \(r\) from Cartesian coordinates \((x, y)\) is: \[r = \sqrt{x^2 + y^2}\]
Applying this to our point \((-a, b)\), we substitute the values into the formula:- Replace \(x\) with \(-a\)- Replace \(y\) with \(b\)
The calculation then becomes:\[r = \sqrt{(-a)^2 + b^2} = \sqrt{a^2 + b^2}\]Calculated as such, the radius \(r\) gives us a positive value because it reflects a distance, and distances are inherently positive.

angle determination

Determining the angle in polar coordinates is slightly more complex. It involves finding how far the point \((-a, b)\) is from the positive x-axis. We use the \(\text{atan2}(y, x)\) function, which determines the angle accounting for the correct quadrant. For the point \((-a, b)\) :- \(x = -a\) is negative, and- \(y = b\) is positive.
This indicates we're in the second quadrant.Using the formula:\[θ = \text{atan2}(b, -a)\]takes into account the specific signs of \(b\) and \(-a\).In the second quadrant, the angle \(θ\) relative to the positive x-axis can be calculated by:\[θ = \tan^{-1} \left( \frac{b}{-a} \right) + \pi\]Adding \(\pi\) adjusts the angle to reflect its correct position in the second quadrant. This step ensures you get the proper \(θ\) representing both the direction and orientation of the initial point in polar coordinates.