Problem 27 - Algebra and Trigonometry

Question

A point is graphed in polar form. Find its rectangular coordinates. $$ (\sqrt{2},-\pi / 4) $$

Step-by-Step Solution

Answer

The rectangular coordinates are (1, -1).

Step 1: Understand Polar Coordinates

Polar coordinates are given in the form \(r, \theta\), where \(r\) represents the radius (or distance from the origin) and \(\theta\) represents the angle in radians (or degrees) from the positive x-axis.

Step 2: Recognize the Given Polar Coordinates

The given polar coordinates are \( \left( \sqrt{2}, -\frac{\pi}{4} \right) \). Here, \( r = \sqrt{2} \) and \( \theta = -\frac{\pi}{4} \).

Step 3: Convert Polar to Rectangular Coordinates

To convert from polar to rectangular coordinates, use the formulas \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \).

Step 4: Calculate the x-coordinate

Substitute the values into the formula for \( x \): \[ x = \sqrt{2} \cos\left(-\frac{\pi}{4}\right) \] \[= \sqrt{2} \cdot \frac{\sqrt{2}}{2} = 1 \]

Step 5: Calculate the y-coordinate

Substitute the values into the formula for \( y \): \[ y = \sqrt{2} \sin\left(-\frac{\pi}{4}\right) \] \[= \sqrt{2} \cdot \left(-\frac{\sqrt{2}}{2}\right) = -1 \]

Step 6: Write the Rectangular Coordinates

The rectangular coordinates are \( (x, y) = (1, -1) \).

Key Concepts

Polar Coordinates

Polar coordinates are a way of expressing a point's location on a plane using a combination of a distance and an angle. The point is denoted as \((r, \theta)\).

• \(r\): This stands for the radial distance from the origin. It tells us how far the point is from the center, similar to the radius of a circle.
• \(\theta\): This represents the angle starting from the positive x-axis. It's typically measured in radians. The angle tells you the direction in which the point lies.

For instance, in the given polar coordinates \(\left(\sqrt{2}, -\frac{\pi}{4}\right)\), \(r = \sqrt{2}\) is the distance, and \(\theta = -\frac{\pi}{4}\) is the angle. This systems lets us specify a point's location not just in terms of left/right or up/down but with how far away it is and the direction we aim.

Rectangular Coordinates

Rectangular, or Cartesian coordinates, describe a point on a plane using two numbers positioned in an \(x, y\) format, which tell you the point's horizontal and vertical placement.

• \(x\): This value indicates how far over to the right or left the point is from the origin. It's like giving a street number on an address.
• \(y\): This value shows how far up or down the point is from the origin, similar to a floor number in a building.

For example, in our solution, the rectangular coordinates of the given point are \((x, y) = (1, -1)\). This means the point lies one unit to the right from the origin and one unit downward. These coordinates are familiar due to their straight-line connections, making it easy to visualize the placement of points.

Trigonometric Functions

Trigonometric functions like cosine and sine are pivotal in connecting polar coordinates to rectangular coordinates.To convert, we use:

• \(x = r \cos(\theta)\): Here, \(\cos(\theta)\) helps us find the adjacent side's length in a right-angled triangle formed by our point, which gives us the \(x\)-coordinate.
• \(y = r \sin(\theta)\): \(\sin(\theta)\) helps find the opposite side's length, providing our \(y\)-coordinate.

In the example of \(\left(\sqrt{2}, -\frac{\pi}{4}\right)\), to find \(x\), we calculated \(\cos\left(-\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\), giving \(x = \sqrt{2} \times \frac{\sqrt{2}}{2} = 1\). For \(y\), \(\sin\left(-\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2}\), leading to \(y = \sqrt{2} \times -\frac{\sqrt{2}}{2} = -1\). Thanks to these trigonometric links, transforming challenging polar expressions into more relatable rectangular terms becomes efficient.