Problem 34 - Algebra and Trigonometry

Question

Polar coordinates of a point are given. Find the rectangular coordinates of each point. $$ \left(6,180^{\circ}\right) $$

Step-by-Step Solution

Answer

The rectangular coordinates are \((-6, 0)\).

Step 1: Understand the Formulas

The transformation from polar to rectangular coordinates can be obtained through these two formulas: \(x = r\cos\theta\) and \(y = r\sin\theta\), where \(r\) is the radius (distance from the origin) and \(\theta\) is the angle.

Step 2: Plug values into the formulas

With (6,180°) given, \(r = 6\) and \(\theta = 180^\circ\). In radians, \(\theta = \pi\). One can now insert these values into the formulas from step 1: \(x = 6\cos(\pi)\) and \(y = 6\sin(\pi)\).

Step 3: Evaluate the Formulas

Recall the values of cosine and sine at \(\pi\): \(\sin(\pi) = 0\) and \(\cos(\pi) = -1\). Therefore, the result is \(x = 6\times (-1) = -6\) and \(y = 6\times 0 = 0\).

Key Concepts

Polar Coordinates

When we think about polar coordinates, we imagine a way to describe a point in a plane using two values: a distance and an angle. This system is different from the more familiar rectangular (or Cartesian) coordinates, where you use an x and a y value to describe a point. In polar coordinates, you have:

• r: The distance from the origin (or starting point) to the point.
• \(\theta\): The angle measured from the positive x-axis to the line connecting the origin to the point.

Polar coordinates are especially useful in scenarios where the relationship between points involves angles and distances, like in circular motions or waves.

For example, in the problem given, the point (6, 180°) in polar coordinates means the point is at 6 units from the origin and forms an angle of 180 degrees with the positive x-axis.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, describe a point in a plane using two perpendicular axes: x and y. These values tell you how far a point is from the reference axes:

• x: The horizontal distance from the y-axis.
• y: The vertical distance from the x-axis.

Understanding rectangular coordinates is essential when working with straight-line motion and in traditional graphing on a standard x-y plane.

To convert from polar to rectangular coordinates, the main idea is to calculate where the point lies on both the x and y axes. Using the formulas \(x = r\cos\theta\) and \(y = r\sin\theta\), we find our x and y coordinates from our polar coordinates. In the exercise, with the polar coordinates (6, 180°), we transform them into rectangular coordinates by evaluating \(x = 6 \cos(180°)\) and \(y = 6 \sin(180°)\). These calculations give us the coordinates (-6,0).

Trigonometric Functions

Trigonometric functions are mathematical functions that relate angles to distances. These functions are crucial to understanding circles and how angles affect distances in cycles.

• Sine Function (\(\sin\)
• Cosine Function (\(\cos\)

These two functions help bridge the gap between polar and rectangular coordinates.

For instance, when converting polar coordinates to rectangular ones, sine and cosine are used to decompose the distance into x (horizontal) and y (vertical) components. Remember the trigonometric values at certain angles: \(\sin(180°) = 0\) and \(\cos(180°) = -1\), which simplifies our calculations. In this context, \(\cos\) determines the x-value and \(\sin\) determines the y-value. Using these, we turn polar coordinates (6, 180°) into rectangular coordinates: \(x = 6\times (-1) = -6\) and \(y = 6\times 0 = 0\).