Problem 20 - Precalculus

Question

Convert each point to exact polar coordinates. Assume that \(0 \leq \theta<2 \pi.\) $$(2 \sqrt{3},-2)$$

Step-by-Step Solution

Answer

The polar coordinates are \( (4, \frac{11\pi}{6}) \).

Step 1: Understand the Cartesian Point

Given the Cartesian coordinates \((x, y) = (2 \sqrt{3}, -2)\). In polar coordinates, we need to find \(r\) (the distance from the origin) and \(\theta\) (the angle from the positive x-axis).

Step 2: Calculate the Radius (r)

The radius \(r\) in polar coordinates is given by the formula \(r = \sqrt{x^2 + y^2}\). Substitute \(x = 2 \sqrt{3}\) and \(y = -2\):\[r = \sqrt{(2 \sqrt{3})^2 + (-2)^2} = \sqrt{12 + 4} = \sqrt{16} = 4\]Thus, \(r = 4\).

Step 3: Calculate the Angle (θ)

The angle \(\theta\) is calculated using \(\tan \theta = \frac{y}{x}\). Here, \(\tan \theta = \frac{-2}{2\sqrt{3}} = -\frac{1}{\sqrt{3}}\). The reference angle for \(\tan \theta = -\frac{1}{\sqrt{3}}\) is \(\frac{\pi}{6}\). Since the point is in the fourth quadrant, the angle \(\theta = 2\pi - \frac{\pi}{6} = \frac{11\pi}{6}\).

Step 4: Write the Polar Coordinates

Now that we have both \(r\) and \(\theta\), the polar coordinates are given by \((r, \theta) = (4, \frac{11\pi}{6})\).

Key Concepts

Cartesian coordinates

Cartesian coordinates are used to define a point in a plane by utilizing two perpendicular axes, often labeled as the x-axis and the y-axis. Each point on this plane is represented by a pair of numerical coordinates, written as \((x, y)\). Here, \(x\) and \(y\) are the distances from the point to the y-axis and x-axis, respectively. This system forms the basis for graphing and understanding two-dimensional spaces in mathematics and various applications.
In our exercise, we start with the Cartesian coordinates \((2 \sqrt{3}, -2)\). These values tell us that the point is located 2 units below the x-axis and roughly 3.46 units (which is \(2 \sqrt{3}\)) along the positive x-axis. Understanding these coordinates is crucial as it sets the foundation before converting them into polar coordinates.

radius calculation

The radius in polar coordinates is like a long arm reaching out from the origin to the point. It represents the straight-line distance from the origin of the coordinate system to the point itself. Calculating this distance involves using the Pythagorean theorem in the form of \(r = \sqrt{x^2 + y^2}\), where \(x\) and \(y\) are the Cartesian coordinates.
Applying this to our specific problem, we substitute \(x = 2 \sqrt{3}\) and \(y = -2\) into the formula.
- First, we calculate \((2 \sqrt{3})^2\) which equals 12.- Then, \((-2)^2\) which equals 4.

• Add these results to get 16.
• Finally, take the square root of 16 to find \(r = 4\).

This radius denotes how far the point is from the origin, which is exactly 4 units, regardless of direction.

angle calculation

Calculating the angle, \(\theta\), in polar coordinates indicates the direction of the point from the positive x-axis. This angle accounts for which quadrant of the Cartesian plane the point is located in.
To find \(\theta\), use the trigonometric function tangent, which is given by \(\tan \theta = \frac{y}{x}\). For our coordinates:

• \(y = -2\) and \(x = 2 \sqrt{3}\), resulting in \(\tan \theta = -\frac{1}{\sqrt{3}}\).
• The reference angle where the tangent is \(-\frac{1}{\sqrt{3}}\) is \(\frac{\pi}{6}\).

However, our point is in the fourth quadrant (since the x-value is positive and the y-value is negative), so we adjust our angle by subtracting the reference angle from \(2\pi\):

• \(\theta = 2\pi - \frac{\pi}{6} = \frac{11\pi}{6}\).

This angle measurement reflects the direction towards the point in a counterclockwise direction from the positive x-axis, making it a pivotal part of polar coordinate representation.

trigonometric functions

Trigonometric functions like sine, cosine, and tangent are essential tools in converting Cartesian coordinates to polar coordinates. These functions relate the angles and sides of right triangles, making them invaluable for calculations involving circles and rotational movement, which is exactly what polar coordinates capture.
In our specific conversion:

• We primarily use the tangent function to find the angle \(\theta\), through the expression \(\tan \theta = \frac{y}{x}\).
• The sine and cosine functions could similarly be employed if needed for further calculations, as they define other relationships with \(r\):
• \(\sin \theta = \frac{y}{r}\)
• \(\cos \theta = \frac{x}{r}\)

These trigonometric relationships help not only in understanding the angles involved but also in verifying or simplifying calculations. By mastering trigonometric functions, the conversion process between Cartesian and polar systems becomes more intuitive and approachable.