Problem 40 - Precalculus Mathematics for Calculus

Question

Convert the rectangular coordinates to polar coordinates with \(r>0\) and \(0 \leq \theta<2 \pi\). $$(1,-2)$$

Step-by-Step Solution

Answer

The polar coordinates for the point are \((\sqrt{5}, 5.176)\).

Step 1: Calculate the Radius r

Polar coordinates are defined as \((r, \theta)\). The radius \(r\) is the distance from the origin to the point, calculated using the Pythagorean theorem. The formula is \(r = \sqrt{x^2 + y^2}\). Here, \(x = 1\) and \(y = -2\), so:\[r = \sqrt{1^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5}\]Thus, the radius is \(r = \sqrt{5}\).

Step 2: Calculate the Angle \(\theta\) in Radians

The angle \(\theta\) is found using the tangent function, where \(\tan(\theta) = \frac{y}{x}\). So for our coordinates, \(\tan(\theta) = \frac{-2}{1} = -2\).We find \(\theta\) using the arctangent function: \(\theta = \tan^{-1}(-2)\). Most calculators return the angle \(\theta\) in the range \(-\frac{\pi}{2}, \frac{\pi}{2}\), which gives us an angle in Quadrant IV. Calculating this gives:\[\theta = \tan^{-1}(-2) \approx -1.107 ext{ radians}\]Since \(\theta\) must be between \(0\) and \(2\pi\), add \(2\pi\) to \(-1.107\) radians:\[\theta = -1.107 + 2\pi \approx 5.176 ext{ radians}\]

Step 3: Write the Polar Coordinates

Using the calculated values from Steps 1 and 2, the polar coordinates for the point \((1, -2)\) are:\[(\sqrt{5}, 5.176)\] This places the point in Polar coordinates with \( r > 0 \) and \( 0 \leq \theta < 2\pi \).

Key Concepts

Rectangular Coordinates

Rectangular coordinates, often called Cartesian coordinates, are used to specify the position of points in a plane using a pair of numbers. These numbers represent distances along perpendicular axes, usually labeled as the x-axis and y-axis. The origin of this coordinate system is the point where both axes intersect, denoted as (0,0).
A point like (1, -2) in rectangular coordinates translates to:

• An x-coordinate of 1, which means a step of 1 unit to the right of the origin along the x-axis.
• A y-coordinate of -2, which indicates a step of 2 units downward from the x-axis.

In this manner, rectangular coordinates allow us to easily locate any position in a 2D space.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry used to determine the distance between points in a plane. This theorem is particularly useful in converting rectangular coordinates to polar coordinates.
The theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. It can be expressed as:
\[c^2 = a^2 + b^2\]
In the context of polar coordinates, the hypotenuse is the distance from the origin to the point (called the radius \(r\)), and the legs of the triangle are the x and y coordinates of the point. Thus, for converting (1, -2):

• 
  The hypotenuse or radius is: \(r = \sqrt{1^2 + (-2)^2} = \sqrt{5}\).

This calculation helps us find the magnitude of the point’s distance from the origin.

Tangent Function

The tangent function is one of the basic trigonometric functions and is commonly used in the process of converting rectangular coordinates to polar coordinates.
The function is defined as the ratio of the y-coordinate to the x-coordinate of a given point. It can be written as:
\[\tan(\theta) = \frac{y}{x}\]
For the point (1, -2), the tangent function is:

• \(\tan(\theta) = \frac{-2}{1} = -2\)

This function gives us the slope of the line connecting the origin to the point, providing insight into the angle \(\theta\) relative to the x-axis. Knowing \(\tan(\theta)\) forms the basis for finding the exact angle in radians, which is crucial in determining the polar coordinates.

Arctangent Function

The arctangent function, denoted as \(\tan^{-1}\), is used to find an angle when given the tangent of that angle. It's particularly useful when determining the angle \(\theta\) in polar coordinates.
When you have the ratio \(\frac{y}{x}\), you apply the arctangent function to find the angle \(\theta\). For instance, given \(\tan(\theta) = -2\), the angle is found by:

• \(\theta = \tan^{-1}(-2)\)

This typically results in an angle in radians, and calculators often provide the result in the range of \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).
For the coordinates (1, -2), the initial result is approximately \(-1.107\) radians, which refers to an angle in the 4th quadrant. However, since polar angles should be positive and within \(0\) and \(2\pi\), we adjust by adding \(2\pi\), resulting in approximately \(5.176\) radians.
The arctangent function thus plays a crucial role in aligning the calculated angle with conventional polar coordinate formats.