Problem 25 - College Trigonometry

Question

Convert the point from polar coordinates into rectangular coordinates. $$ \left(42, \frac{13 \pi}{6}\right) $$

Step-by-Step Solution

Answer

The rectangular coordinates are \((21\sqrt{3}, 21)\).

Step 1: Understanding Polar to Rectangular Conversion

To convert a point from polar coordinates \((r, \theta)\) to rectangular coordinates \((x, y)\), we use the formulas: \[x = r \cdot \cos(\theta)\]\[y = r \cdot \sin(\theta)\] where \(r\) is the radial distance and \(\theta\) is the angle in radians.

Step 2: Identifying the Given Values

In this problem, the polar coordinates are given as \[ (r, \theta) = \left( 42, \frac{13 \pi}{6} \right) \]where \(r = 42\) and \( \theta = \frac{13 \pi}{6} \).

Step 3: Converting the Angle to a Simplified Form

The angle \( \theta = \frac{13 \pi}{6} \) can be simplified to a more familiar angle by recognizing that \( \frac{13 \pi}{6} = 2\pi + \frac{\pi}{6} \), which indicates a complete revolution (\(2\pi\)) plus \(\frac{\pi}{6}\), corresponding to the reference angle in the first position.

Step 4: Calculating the X-coordinate

To find \(x\), substitute the values into the formula for \(x\):\[x = 42 \cdot \cos\left(\frac{13 \pi}{6}\right) = 42 \cdot \cos\left(\frac{\pi}{6}\right) \]We know \(\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}\).Thus:\[x = 42 \cdot \frac{\sqrt{3}}{2} = 21\sqrt{3}\]

Step 5: Calculating the Y-coordinate

To find \(y\), substitute the values into the formula for \(y\):\[y = 42 \cdot \sin\left(\frac{13 \pi}{6}\right) = 42 \cdot \sin\left(\frac{\pi}{6}\right) \]We know \(\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}\).Thus:\[y = 42 \cdot \frac{1}{2} = 21\]

Step 6: Combining Results to Form Rectangular Coordinates

With \(x = 21\sqrt{3}\) and \(y = 21\), the rectangular coordinates are given by:\[(21\sqrt{3}, 21)\]

Key Concepts

Understanding Trigonometric Functions for Conversion

Trigonometric functions play a crucial role in converting polar coordinates to rectangular coordinates. They help us relate the angle and radial distance in polar coordinates to the horizontal and vertical distances in rectangular coordinates. When we have an angle measured in radians, such as \(\theta\), we use the cosine and sine functions to find the corresponding \(x\) and \(y\) values respectively.

• For the \(x\)-coordinate: Use \(x = r \cdot \cos(\theta)\).
• For the \(y\)-coordinate: Use \(y = r \cdot \sin(\theta)\).

Trigonometric functions give us the projection of the polar point onto the \(x\) and \(y\) axes. This projection converts the circular information into linear distances. By understanding these functions, we can start converting angles and distances in the polar system into a more familiar Cartesian system, making spatial interpretation much easier.

How Coordinate Conversion Works

The process of coordinate conversion translates information from one form to another while maintaining the relationships represented in the original form. In the context of polar to rectangular conversion, we translate a point given by radius \(r\) and angle \(\theta\) into Cartesian \(x\) and \(y\) coordinates.
Ponder these forms:

• Polar coordinates: Consist of \((r, \theta)\), where \(r\) is the distance from the origin and \(\theta\) is the angle from the positive x-axis.
• Rectangular coordinates: Expressed as \((x, y)\), which represent direct Eastern (horizontal) and Northern (vertical) displacements from the origin.

In converting, these relationships hold:
\[x = r \cdot \cos(\theta)\]
\[y = r \cdot \sin(\theta)\]
By carefully substituting \(r\) and \(\theta\) into these equations, we can solve for \(x\) and \(y\). This conversion is step by step: simplify the angle when necessary, calculate \(x\), then calculate \(y\), and finally, combine them into rectangular coordinates. This systematic approach ensures accuracy and aids in the understanding of point positioning between the two coordinate systems.

The Polar Coordinate System

Polar coordinates provide a unique way to describe a point's position, especially useful in circular or rotational contexts. Instead of starting from a grid, this system uses a circle's center and a rotational angle past the x-axis.

• The radius \(r\) defines distance from the central point, called the pole (similar to the origin in Cartesian coordinates).
• The angle \(\theta\) is measured from the positive x-axis and can be given in degrees or radians, with radians being more common in physics and mathematics.

The angle indicates direction, and the distance \(r\) gives magnitude, perfectly framing rotational dynamics.
Often, the angle might need simplifying, as seen when \(\theta = \frac{13 \pi}{6}\) translates to a full rotation plus an angle often found in trigonometric tables. Recognizing this can simplify computations, as in the exercise with \(\cos\left(\frac{13 \pi}{6}\right)\) and \(\sin\left(\frac{13 \pi}{6}\right)\), which simplify to familiar values due to one full circle or \(2\pi\), making it equivalent to \(\frac{\pi}{6}\). Understanding this system aids significantly when translating points for various mathematical and practical applications.