Problem 26 - College Trigonometry

Question

Convert the point from polar coordinates into rectangular coordinates. $$ (-117,117 \pi) $$

Step-by-Step Solution

Answer

The rectangular coordinates are (117, 0).

Step 1: Understand the Conversion Formulas

To convert from polar coordinates \((r, \theta)\) to rectangular coordinates \((x, y)\), we use the formulas:\[ x = r \cdot \cos(\theta) \]\[ y = r \cdot \sin(\theta) \]

Step 2: Analyze the Given Polar Coordinates

The given polar coordinates are \((-117, 117 \pi)\). In this case, \(r = -117\) and \(\theta = 117\pi\).

Step 3: Calculate the X-coordinate

Substitute \(r = -117\) and \(\theta = 117\pi\) into the formula for \(x\):\[ x = -117 \cdot \cos(117\pi) \]Knowing that \( \cos(\pi) = -1\) and it repeats every \(2\pi\), the value of \(\cos(117\pi) = \cos(\pi) = -1\). Therefore,\[ x = -117 \times -1 = 117 \]

Step 4: Calculate the Y-coordinate

Substitute \(r = -117\) and \(\theta = 117\pi\) into the formula for \(y\):\[ y = -117 \cdot \sin(117\pi) \]Knowing that \(\sin(\pi) = 0\) and it repeats every \(2\pi\), the value of \(\sin(117\pi) = \sin(\pi) = 0\). Therefore, \[ y = -117 \times 0 = 0 \]

Step 5: Write the Rectangular Coordinates

The rectangular coordinates are \((x, y) = (117, 0)\).

Key Concepts

Rectangular Coordinates

Rectangular coordinates help describe a point's location on a two-dimensional plane using a pair of numbers. This coordinate system is also known as Cartesian coordinates due to its origin from the mathematician René Descartes. Each point in this system is described by an ordered pair

• The first number, known as the x-coordinate (or abscissa), tells how far along the horizontal axis (x-axis) the point is.
• The second number, referred to as the y-coordinate (or ordinate), reflects how far along the vertical axis (y-axis) the point is.

It's like using grid squares on a city map: one number tells you the street, and the other tells you the avenue. By specifying these two numbers, we can pinpoint any exact location on the grid. The origin of the system, where both axes meet, is denoted as (0, 0). Whenever converting from another coordinate system into rectangular coordinates, as in the exercise, this basic understanding helps you visualize the transition and place the point correctly on the grid.

Coordinate Conversion

Coordinate conversion is the process of transforming points expressed in one coordinate system into another. In this exercise, the task is to convert polar coordinates into rectangular coordinates. Polar coordinates consist of:

• A radius, \( r \), which specifies the distance from a central point or origin.
• An angle, \( \theta \), which indicates the direction from the origin's reference line.

To make this conversion, the formulas \( x = r \cdot \cos(\theta) \) and \( y = r \cdot \sin(\theta) \) are employed.
These utilize trigonometric functions to translate the distance and angle into horizontal (x) and vertical (y) coordinates.
Understanding how these formulas function is fundamental. They allow us to accurately place the polar point into the rectangular coordinate plane.
Visualizing how the radius and angle influence a point's position in space enhances comprehension of moving between these two uniquely useful systems.

Trigonometric Functions

Trigonometric functions relate the angles of triangles to the lengths of their sides, but they also have broader applications such as converting coordinates. In this exercise:

• The cosine function \( \cos(\theta) \), related to the x-axis, calculates the horizontal weight of the point's position.
• The sine function \( \sin(\theta) \), which aligns with the y-axis, determines the vertical component.

Knowing periodic identities is crucial. For instance, \( \cos(\pi) = -1 \) and \( \sin(\pi) = 0 \). These values repeat every complete rotation or \( 2\pi \), making them predictable and useful for conversion.
In the exercise, using the angle \( 117 \pi \), we see this repetition applies directly in simplifying to standard unit circle values.
Through practice, recognizing these fundamental trigonometric values helps efficiently tackle angle conversions in more complex scenarios.