Problem 21 - Algebra and Trigonometry Real Mathematics, Real People

Question

Plot the point given in polar coordinates and find the corresponding rectangular coordinates for the point. $$\left(1,-\frac{3 \pi}{4}\right)$$

Step-by-Step Solution

Answer

The rectangular coordinates for the polar point \((1,-\frac{3\pi}{4})\) are \(-\frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2})\)

Step 1: Identify the polar coordinates

The given polar coordinates are \(r = 1\) and \(\theta = -\frac{3\pi}{4}\).

Step 2: Calculate the rectangular coordinates

Apply the formulas to convert polar coordinates to rectangular coordinates. \(x = r \cdot \cos \theta\) and \(y = r \cdot \sin \theta\). So, \(x=1\cdot \cos(-\frac{3\pi}{4})\) and \(y=1\cdot \sin(-\frac{3\pi}{4})\).

Step 3: Simplify the equations

Once all operations have been carried out, we find that \(x = -\frac{1}{\sqrt{2}}\) and \(y = -\frac{1}{\sqrt{2}}\)

Step 4: Express the answer in rectangular coordinates

The final step is to present the answers as a rectangular coordinate. The rectangular coordinates for the given polar coordinates are (-1/sqrt(2), -1/sqrt(2))

Key Concepts

Polar Coordinates

Polar coordinates are a way of locating a point in a plane using a radius and an angle with respect to a reference direction. Think of it like pointing a laser from the origin and defining the point where the laser hits a circle.

• The first component, \( r \), is the radius or distance from the origin to the point. In this case, \( r = 1 \).
• The second component, \( \theta \), is the angle measured from the positive x-axis. Here, \( \theta = -\frac{3\pi}{4} \). The negative angle indicates a clockwise rotation.

One important aspect of polar coordinates is that a single point can be represented by multiple pairs of \( r \) and \( \theta \). This is because adding multiples of \( 2\pi \) (a full rotation) to \( \theta \) will not change the position of the point. This flexibility makes polar coordinates particularly useful in fields like navigation and physics where direction and distance from a central point are key.

Rectangular Coordinates

Rectangular coordinates, often referred to as Cartesian coordinates, consist of two values \( x \) and \( y \). These values are distances along the horizontal and vertical axes respectively. They are used widely thanks to their straightforward and intuitive nature.

• The \( x \)-coordinate describes the horizontal position of the point. A positive \( x \) indicates a position to the right of the origin, while a negative \( x \) indicates a position to the left.
• The \( y \)-coordinate describes the vertical position. A positive \( y \) indicates a position above the origin, and a negative \( y \) indicates a position below.

In practical terms, rectangular coordinates are used extensively in engineering and everyday map reading. They represent points unambiguously, unlike polar coordinates, by using the perpendicular landmarks of the x and y axes. Each point has only one set of \( x \) and \( y \), avoiding any ambiguity.

Coordinate Conversion

To move between polar and rectangular coordinates, we use trigonometric functions. This conversion allows us to take advantage of the strengths of each system.
For our given example, \( r = 1 \) and \( \theta = -\frac{3\pi}{4} \), we apply these steps for conversion:

• For the \( x \) coordinate, use the formula: \( x = r \cdot \cos(\theta) \). Here, \( x = 1 \cdot \cos(-\frac{3\pi}{4}) \). Calculating the cosine of \(-\frac{3\pi}{4}\), the result is \( x = -\frac{1}{\sqrt{2}} \).
• For the \( y \) coordinate, use the formula: \( y = r \cdot \sin(\theta) \). Here, \( y = 1 \cdot \sin(-\frac{3\pi}{4}) \). This gives us \( y = -\frac{1}{\sqrt{2}} \).

Conversion formulas build a bridge between two coordinate systems, enhancing our ability to model and analyze phenomena across different settings. Accurate conversion ensures correct positioning in applications like computer graphics and navigation systems. Through careful calculation, we keep spatial coherence across various frameworks.