Problem 56 - College Trigonometry

Question

Convert the point from rectangular coordinates into polar coordinates with \(r \geq 0\) and \(0 \leq \theta<2 \pi\). $$ \left(-\frac{\sqrt{65}}{5}, \frac{2 \sqrt{65}}{5}\right) $$

Step-by-Step Solution

Answer

The polar coordinates are \((\sqrt{13}, 2.034)\).

Step 1: Calculate the Radius

The formula for the radius \( r \) in polar coordinates is \( r = \sqrt{x^2 + y^2} \). Substitute \( x = -\frac{\sqrt{65}}{5} \) and \( y = \frac{2\sqrt{65}}{5} \) into the equation:\[r = \sqrt{\left(-\frac{\sqrt{65}}{5}\right)^2 + \left(\frac{2\sqrt{65}}{5}\right)^2} = \sqrt{\frac{65}{25} + \frac{4 \times 65}{25}} = \sqrt{\frac{5 \times 65}{25}} = \sqrt{13}\]

Step 2: Calculate the Angle

The angle \( \theta \) in polar coordinates can be found using \( \tan \theta = \frac{y}{x} \). First calculate \( \frac{y}{x} \):\[\frac{y}{x} = \frac{\frac{2\sqrt{65}}{5}}{-\frac{\sqrt{65}}{5}} = -2\]Since the point is in the second quadrant (negative \( x \), positive \( y \)), \( \theta = \pi - \tan^{-1}(2) \). Thus:\[\theta \approx \pi - 1.107 \approx 2.034 \text{ radians}\]

Step 3: Write the Polar Coordinates

The polar coordinates \((r, \theta)\) are found using the calculated radius and angle. So, we have:\[ (\sqrt{13}, 2.034) \]

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, denote the position of a point in a two-dimensional plane using an ordered pair \(x, y\). This coordinate system is based on perpendicular axes, typically labeled as the x-axis (horizontal) and the y-axis (vertical).These coordinates are essential for pinpointing exact locations in various mathematical contexts. If you imagine graphing a point, you start from the origin (0,0) and move horizontally to reach x, and then vertically to reach the y point. The given point \(-\frac{\sqrt{65}}{5}, \frac{2\sqrt{65}}{5}\) means that from the origin, you'd move to the left, a bit over two units considering the scale, and then up about four units.

• Understanding Quadrants: This system divides the plane into four quadrants. This practice helps identify the sign (positive/negative) of coordinates based on their quadrant.
• Use in Conversion: Rectangular coordinates are the starting point for conversion into polar coordinates. These forms help in simplifying complex calculations, especially involving rotations and trigonometry.

Radius Calculation

The radius in polar coordinates represents the direct distance from the origin to a specific point on the plane. It is a crucial component, as it determines the position's distance regardless of its direction.To find the radius \((r)\), you employ the Pythagorean theorem. Specifically, you apply the formula \(r = \sqrt{x^2 + y^2}\). This equation sums up the squares of the coordinates and takes the square root of this sum.Substituting the provided point \((x = -\frac{\sqrt{65}}{5}, y = \frac{2\sqrt{65}}{5})\) into this formula simplifies to \(\sqrt{13}\).

• Why Use the Pythagorean Theorem?: Since you're finding a hypotenuse (radius) in a right triangle formed by the x and y axes, the Pythagorean theorem becomes invaluable.
• Ensures Non-Negative Values: The radius is always non-negative, representing a distance, making any symbolic negativity irrelevant.

Think of this as measuring the straight line needed to reach the point from the origin, akin to measuring the length of an arrow from its center to its tip.

Angle Conversion

When converting to polar coordinates, establishing the correct angle is imperative. The angle \(\theta\) reflects the direction between the positive x-axis and the point.To find the angle, use \(\tan \theta = \frac{y}{x}\), where you compute the arctangent of \(\frac{y}{x}\).For the coordinates given, this results in \(\tan \theta = -2\). Because the point lies in the second quadrant (x is negative, y is positive), adjust the angle by using \(\theta = \pi - \tan^{-1}(2)\), approximating to 2.034 radians.

• Acknowledging Quadrants: Recognizing which quadrant the point is in adjusts the base angle derived from \(\tan^{-1}\). This adjustment ensures accurate direction representation.
• Choosing the Correct Range: Keeping \(\theta\) within the range \(0 \leq \theta < 2\pi\) ensures it properly describes all possible point directions in the plane.

This concept, crucial in trigonometry and physics, captures how far a point rotates around the origin, forming the foundational step in transforming between coordinate systems.