Problem 29 - Precalculus Mathematics for Calculus

Question

Find the rectangular coordinates for the point whose polar coordinates are given. $$(\sqrt{2},-\pi / 4)$$

Step-by-Step Solution

Answer

The rectangular coordinates are (1, -1).

Step 1: Understand Polar Coordinates

In polar coordinates, a point is represented as \((r, \theta)\), where \(r\) is the radial distance from the origin and \(\theta\) is the angle from the positive x-axis, measured in radians. For this exercise, \(r = \sqrt{2}\) and \(\theta = -\frac{\pi}{4}\).

Step 2: Recall the Conversion Formulas

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

Step 3: Calculate the x-coordinate

Substitute \(r = \sqrt{2}\) and \(\theta = -\frac{\pi}{4}\) into the conversion formula for \(x\): \[x = \sqrt{2} \cdot \cos\left(-\frac{\pi}{4}\right)\]Since \(\cos\left(-\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}\), this becomes: \[x = \sqrt{2} \cdot \frac{1}{\sqrt{2}} = 1\]

Step 4: Calculate the y-coordinate

Substitute \(r = \sqrt{2}\) and \(\theta = -\frac{\pi}{4}\) into the conversion formula for \(y\):\[y = \sqrt{2} \cdot \sin\left(-\frac{\pi}{4}\right)\]Since \(\sin\left(-\frac{\pi}{4}\right) = -\frac{1}{\sqrt{2}}\), this becomes:\[y = \sqrt{2} \cdot \left(-\frac{1}{\sqrt{2}}\right) = -1\]

Step 5: State the Rectangular Coordinates

Now that you have calculated both coordinates, the rectangular coordinates for the point are \((x, y) = (1, -1)\).

Key Concepts

Coordinate Conversion

In the vast realm of mathematics, converting between different coordinate systems is a frequently encountered challenge. When dealing with polar coordinates, the process of moving to rectangular coordinates involves specific conversions that allow clear representation in a Cartesian plane. Polar coordinates are uniquely defined by

• the radial distance from the origin, denoted as \( r \)
• the angle \( \theta \) which measures clockwise or counter-clockwise from the positive x-axis

This conversion can be incredibly practical for applications in physics and engineering where directions and magnitudes are key. To perform conversion from polar to rectangular coordinates, two simple formulas are used:

• \( x = r \cdot \cos(\theta) \)
• \( y = r \cdot \sin(\theta) \)

These equations derive from trigonometry, allowing us to transpose a point positioned in terms of a circle into its equivalent position on the x-y grid, offering a more intuitive perception of spatial relationships.

Trigonometric Functions

Trigonometric functions are essential for the coordinate conversion process as they provide a way to relate angles and distances. In particular, cosine and sine functions are crucial for transforming polar coordinates into rectangular ones.
The cosine function, denoted as \( \cos(\theta) \), connects the angle to the x-coordinate. It helps determine how far left or right a point is from the origin based on the angle from the positive x-axis. Meanwhile, the sine function, \( \sin(\theta) \), links the angle to the y-coordinate, supplying details regarding whether the point is above or below the x-axis.

• For our specific point with \( \theta = -\frac{\pi}{4} \), \( \cos(-\frac{\pi}{4}) = \frac{1}{\sqrt{2}} \) and \( \sin(-\frac{\pi}{4}) = -\frac{1}{\sqrt{2}} \).

These functions are pivotal in understanding movement around a circle, where both sine and cosine oscillate between -1 and 1 as \( \theta \) varies, reflecting symmetry and periodicity that are valuable across many domains in science and engineering.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a fundamental part of coordinate geometry. They represent points on a plane using two values:

• \( x \) coordinate: horizontal position from the y-axis
• \( y \) coordinate: vertical position from the x-axis

After performing the conversion from polar coordinates, for example, the point \((\sqrt{2}, -\frac{\pi}{4})\), you end up with the rectangular coordinates \((1, -1)\). This means the point is one unit to the right and one unit down from the origin.
Using a Cartesian plane lets us easily visualize and understand the geometry involved. It gives an intuitive view of distances and locations, allowing direct comparison and calculations involving lines, slopes, and more. This coordinate system is universally used in mathematics, providing a foundation for much of the geometric, algebraic, and calculus applications we encounter.