Problem 36 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The polar coordinates are \((6, \frac{11\pi}{6})\).

Step 1: Identify the rectangular coordinates

The given rectangular coordinates are \((x, y) = (3\sqrt{3}, -3)\).

Step 2: Calculate the radius \(r\)

The polar coordinate radius \(r\) can be calculated using the formula \(r = \sqrt{x^2 + y^2}\). Plugging in the given values: \[r = \sqrt{(3\sqrt{3})^2 + (-3)^2} = \sqrt{27 + 9} = \sqrt{36} = 6.\]

Step 3: Calculate the angle \(\theta\) using tangent

The angle \(\theta\) is found using \(\tan\theta = \frac{y}{x}\). So, \(\tan\theta = \frac{-3}{3\sqrt{3}} = -\frac{1}{\sqrt{3}}\). The reference angle where \(\tan\theta = -\frac{1}{\sqrt{3}}\) is \(\theta = \frac{\pi}{6}\). However, since the point is in the fourth quadrant (positive \(x\), negative \(y\)), \(\theta = 2\pi - \frac{\pi}{6} = \frac{11\pi}{6}\).

Step 4: Write the polar coordinates

With the computed values, the polar coordinates are \((r, \theta) = (6, \frac{11\pi}{6})\).

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are an essential part of coordinate geometry. These coordinates are defined by a pair of numerical values, typically labeled as \((x, y)\). This system uses two perpendicular axes: The horizontal \(x\)-axis and the vertical \(y\)-axis.

Every point on a plane can be described by its distance from these axes, creating a unique coordinate pair. For example, the point \((3\sqrt{3}, -3)\) can be accurately found on a graph using this method.

• The \(x\) coordinate indicates how far to move along the horizontal axis.
• The \(y\) coordinate shows how far to move along the vertical axis.

Understanding rectangular coordinates is crucial, as they serve as the starting point for converting to polar coordinates.

Coordinate Conversion

Coordinate conversion is the process of translating between different coordinate systems, often from rectangular to polar coordinates or vice versa. This conversion is integral to solving problems in mathematics and physics.

When converting rectangular coordinates \((x, y)\) to polar coordinates \((r, \theta)\), two main aspects are calculated: the radius \(r\) and the angle \(\theta\). Conversion requires understanding of both algebra and trigonometry.

• Find the radius \(r\): This is done using the equation \(r = \sqrt{x^2 + y^2}\).
• Determine the angle \(\theta\): Use trigonometric relationships, particularly tangent, to find \(\theta\).

This change in perspective can often simplify complex problems, as polar coordinates explicitly detail direction and distance from a point.

Angle Calculation

In converting coordinates, the calculation of the angle \(\theta\) is a key step. This angle represents the rotation from the positive \(x\)-axis to the line connecting the point to the origin. To find \(\theta\), the formula \(\tan\theta = \frac{y}{x}\) is used.

This determines the tangent of the angle, a fundamental trigonometric function. From there, use the inverse tangent function to find the reference angle.

• Consider the signs of \(x\) and \(y\) to determine the correct quadrant for \(\theta\).
• Adjust the reference angle accordingly, ensuring \(0 \leq \theta < 2\pi\).

For instance, with the coordinates \((3\sqrt{3}, -3)\), the angle in the fourth quadrant results in \(\theta = \frac{11\pi}{6}\). Identifying the correct quadrant and adjusting \(\theta\) is crucial for accuracy.

Radius Calculation

Calculating the radius \(r\) is an essential part of converting to polar coordinates. The radius represents the distance from the origin \((0, 0)\) to the point. This is derived from the Pythagorean theorem, creating the equation \(r = \sqrt{x^2 + y^2}\).

To calculate \(r\):

• Square both the \(x\) and \(y\) values.
• Add the squared values together.
• Take the square root of the result to obtain \(r\).

For example, with the coordinates \((3\sqrt{3}, -3)\), applying these calculations gives a radius of \(r = 6\). This step provides the necessary distance for defining the location in polar coordinates, anchoring all other calculations.