Problem 18 - Precalculus with Calculus Previews

Question

Find the rectangular coordinates for each point with the given polar coordinates. $$ (\sqrt{2}, 11 \pi / 6) $$

Step-by-Step Solution

Answer

The rectangular coordinates are \(\left(\frac{\sqrt{6}}{2}, -\frac{\sqrt{2}}{2}\right)\).

Step 1: Understand Polar Coordinates

The polar coordinates of a point consist of two values: the radius \( r \) and the angle \( \theta \). The radius corresponds to the distance from the origin, while the angle indicates the direction with respect to the positive x-axis.

Step 2: Recall Conversion Formulas

To convert polar coordinates \((r, \theta)\) to rectangular coordinates \((x, y)\), use the formulas: \(x = r \cos \theta\) and \(y = r \sin \theta\). These formulas translate the polar point to its equivalent Cartesian point by breaking down the distance and angle into x and y components.

Step 3: Calculate the x-coordinate

Substitute \(r = \sqrt{2}\) and \(\theta = \frac{11\pi}{6}\) into the formula for \(x\):\[x = \sqrt{2} \cos \frac{11\pi}{6}\]Since \(\cos \frac{11\pi}{6} = \frac{\sqrt{3}}{2}\):\[x = \sqrt{2} \times \frac{\sqrt{3}}{2} = \frac{\sqrt{6}}{2}\]

Step 4: Calculate the y-coordinate

Substitute \(r = \sqrt{2}\) and \(\theta = \frac{11\pi}{6}\) into the formula for \(y\):\[y = \sqrt{2} \sin \frac{11\pi}{6}\]Recall that \(\sin \frac{11\pi}{6} = -\frac{1}{2}\):\[y = \sqrt{2} \times -\frac{1}{2} = -\frac{\sqrt{2}}{2}\]

Step 5: Conclusion

Combine the calculated x and y values to form the rectangular coordinates. The rectangular coordinates corresponding to the polar point \((\sqrt{2}, \frac{11\pi}{6})\) are \(\left(\frac{\sqrt{6}}{2}, -\frac{\sqrt{2}}{2}\right)\).

Key Concepts

Polar Coordinates

Polar coordinates are a way to express the location of a point in a plane using two values. These points are represented as

• a radius (\(r\)) - this tells you how far away the point is from a central point, called the origin.
• an angle (\(\theta\)) - this describes the direction of the point from the origin, measured in radians or degrees relative to the positive x-axis.

A point in polar coordinates is written as \((r, \theta)\). This system differs from the more familiar rectangular or Cartesian coordinate system, where a point is defined by how far it is along two perpendicular axes, x and y. Understanding how this circular system works is crucial for various fields such as physics, engineering, and navigation.
For example, the point \((\sqrt{2}, \frac{11\pi}{6})\) means that the point is \(\sqrt{2}\) units from the origin and positioned at an angle of \(\frac{11\pi}{6}\) from the positive x-axis. Polar coordinates provide an alternative approach to position points in contexts where circular or rotational motion is more intuitive.

Coordinate Conversion

The process of coordinate conversion is essential when it comes to translating points from polar coordinates to rectangular coordinates and vice versa. This conversion is based on simple trigonometric relationships due to the inherent geometric connections between the two systems.
To convert from polar coordinates \((r, \theta)\) to rectangular coordinates \((x, y)\), we use:

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

These formulas work because:

1. \(\cos\theta\) determines the horizontal projection of the radius.
2. \(\sin\theta\) calculates the vertical projection of the radius.

Thus, these formulas break the radius into x and y components based on the angle given.As an example, converting the polar point \((\sqrt{2}, \frac{11\pi}{6})\) involves calculating:

• \(x = \sqrt{2} \cos \frac{11\pi}{6}\), resulting in \(\frac{\sqrt{6}}{2}\).
• \(y = \sqrt{2} \sin \frac{11\pi}{6}\), which gives \(-\frac{\sqrt{2}}{2}\).

The ability to effectively transition between these coordinate pairs aids in visualizing and solving problems across various scientific domains.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, play a pivotal role in the conversion process between polar and rectangular coordinates. These functions relate angles to side ratios in right triangles and are foundational in the study of circles.

• Sine (\(\sin\theta\)) measures the ratio of the opposite side to the hypotenuse of a right triangle.
• Cosine (\(\cos\theta\)) measures the ratio of the adjacent side to the hypotenuse.

When using these functions to convert coordinates:

• The cosine function helps determine the x-coordinate, which represents the horizontal distance from the origin.
• The sine function helps determine the y-coordinate, which indicates the vertical distance.

For instance, consider \(\theta = \frac{11\pi}{6}\), both \(\sin\frac{11\pi}{6} = -\frac{1}{2}\) and \(\cos\frac{11\pi}{6} = \frac{\sqrt{3}}{2}\) are needed to translate the point from polar to rectangular coordinates. Understanding these trigonometric functions allows you to harness their power for calculations involving angles and distances, reinforcing the connection between different mathematical systems.