Problem 52 - College Trigonometry

Question

Convert the point from rectangular coordinates into polar coordinates with \(r \geq 0\) and \(0 \leq \theta<2 \pi\). $$ \left(-\frac{\sqrt{5}}{15},-\frac{2 \sqrt{5}}{15}\right) $$

Step-by-Step Solution

Answer

The polar coordinates are \(\left(\frac{1}{3}, 4.25\right)\).

Step 1: Identify Rectangular Coordinates

The rectangular coordinates given are \(x = -\frac{\sqrt{5}}{15}\) and \(y = -\frac{2\sqrt{5}}{15}\).

Step 2: Calculate Radius \(r\)

Use the formula for the radius in polar coordinates, \(r = \sqrt{x^2 + y^2}\). Substituting the given \(x\) and \(y\) values:\[r = \sqrt{\left(-\frac{\sqrt{5}}{15}\right)^2 + \left(-\frac{2\sqrt{5}}{15}\right)^2}= \sqrt{\frac{5}{225} + \frac{20}{225}}= \sqrt{\frac{25}{225}} = \frac{1}{3}.\]Thus, the radius \(r\) is \(\frac{1}{3}\).

Step 3: Calculate Angle \(\theta\)

To find \(\theta\), use the formula \(\theta = \arctan\left(\frac{y}{x}\right)\). Substituting \(x\) and \(y\):\[\theta = \arctan\left(\frac{-\frac{2\sqrt{5}}{15}}{-\frac{\sqrt{5}}{15}}\right) = \arctan(2)\approx 1.107\text{ radians}.\]Since \(x < 0\) and \(y < 0\), the point is in the third quadrant, so we need to add \(\pi\) to the angle:\[\theta = 1.107 + \pi \approx 1.107 + 3.142 \approx 4.249 \text{ radians}.\]Thus, \(\theta \approx 4.25\) radians.

Step 4: Express in Polar Coordinates

The polar coordinates are expressed as \((r, \theta)\). After calculating, we have:\[(r, \theta) = \left(\frac{1}{3}, 4.25 \right).\]These are the polar coordinates: radius \(r = \frac{1}{3}\) and angle \(\theta \approx 4.25\) radians.

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, describe a point in a plane using two values usually represented by \(x\) for the horizontal axis and \(y\) for the vertical axis. They are one of the most common ways to pinpoint a location in a two-dimensional plane.

In this exercise, the given rectangular coordinates are \(-\frac{\sqrt{5}}{15}\) for \(x\) and \(-\frac{2\sqrt{5}}{15}\) for \(y\). This negative value in both directions indicates that the point is located in the third quadrant of the plane.

• Quadrant I: positive \(x\) and positive \(y\)
• Quadrant II: negative \(x\) and positive \(y\)
• Quadrant III: negative \(x\) and negative \(y\)
• Quadrant IV: positive \(x\) and negative \(y\)

Understanding which quadrant the point is in is crucial for correctly calculating the angle when converting to polar coordinates. The rectangular coordinate system is perfect for easily visualizing and plotting any point on a graph.

Radius Calculation

The radius in polar coordinates is a measure of the distance from the origin (0,0) to the specified point. It is similar to finding the hypotenuse of a right triangle formed by the \(x\) and \(y\) values.

To calculate the radius \(r\), we use the formula:

• \(r = \sqrt{x^2 + y^2}\)

Plug in the given values:\[r = \sqrt{\left(-\frac{\sqrt{5}}{15}\right)^2 + \left(-\frac{2\sqrt{5}}{15}\right)^2}\]After performing the square calculations, you end up with:\[r = \sqrt{\frac{5}{225} + \frac{20}{225}} = \sqrt{\frac{25}{225}} = \frac{1}{3}\]This indicates that the point is one-third of a unit away from the origin. Radius provides one half of the information for converting to polar coordinates and is essential in determining the position of the point in polar form.

Angle Calculation

Calculating the angle \(\theta\) is crucial to determine the direction of the point from the origin in polar coordinates. The standard technique is using the arctangent function, which gives the angle whose tangent is the quotient of the \(y\) and \(x\) coordinates.

For this calculation:

• \(\theta = \arctan\left(\frac{y}{x}\right)\)

Substitute \(x\) and \(y\): \[\theta = \arctan\left(\frac{-\frac{2\sqrt{5}}{15}}{-\frac{\sqrt{5}}{15}}\right)\]Simplify this expression to:\[\theta = \arctan(2) \approx 1.107\text{ radians}\]Given that both \(x\) and \(y\) are negative, the point is located in the third quadrant. This requires an adjustment by adding \(\pi\) (approximately 3.142 radians) to the initial angle:\[\theta = 1.107 + \pi \approx 4.249\text{ radians}\]Thus, in polar coordinates, the angle \(\theta\) provides the direction necessary to orient the radius accordingly from the origin. Calculating \(\theta\) accurately is fundamental to converting points from rectangular to polar coordinates.