Problem 8 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 3-8: Change the polar coordinates to rectangular coordinates. $$ \left(10, \arccos \left(-\frac{1}{3}\right)\right) $$

Step-by-Step Solution

Answer

The rectangular coordinates are \( \left(-\frac{10}{3}, \frac{20\sqrt{2}}{3}\right) \).

Step 1: Identify the Components of Polar Coordinates

The polar coordinates are given as \( (r, \theta) \). For the problem \( (r, \theta) = \left(10, \arccos \left(-\frac{1}{3}\right)\right) \). This tells us that the radius \( r = 10 \) and the angle \( \theta = \arccos \left(-\frac{1}{3}\right) \).

Step 2: Use Conversion Formulas

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

Step 3: Calculate the Rectangular Coordinate x

Substitute the values into the formula for \( x\):\[ x = 10 \cdot \cos\left(\arccos\left(-\frac{1}{3}\right)\right) \] \( \cos(\arccos(x)) = x \), so \( \cos\left(\arccos\left(-\frac{1}{3}\right)\right) = -\frac{1}{3} \). Therefore, \[ x = 10 \cdot \left(-\frac{1}{3}\right) = -\frac{10}{3} \].

Step 4: Calculate the Rectangular Coordinate y

Since \( \sin^2(\theta) + \cos^2(\theta) = 1 \), we have:\[ \sin(\theta) = \sqrt{1 - (\cos(\theta))^2} = \sqrt{1 - \left(-\frac{1}{3}\right)^2} = \sqrt{1 - \frac{1}{9}} = \sqrt{\frac{8}{9}} = \frac{2\sqrt{2}}{3} \].\Then, calculate \( y \) using the formula: \[ y = 10 \cdot \frac{2\sqrt{2}}{3} = \frac{20\sqrt{2}}{3} \].

Step 5: Write the Rectangular Coordinates

With the calculations completed, the rectangular coordinates are \( \left(-\frac{10}{3}, \frac{20\sqrt{2}}{3}\right) \).

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing locations in a plane using a distance and an angle. They're particularly useful in situations where phenomena have circular or rotational symmetry, such as wave patterns or the paths of celestial bodies. In polar coordinates, each point on a plane is defined by two values:

• the radial distance, often denoted as \( r \), representing how far the point is from the origin
• the angular coordinate, \( \theta \), which is the angle measured from a reference direction (typically the positive x-axis)

For example, a polar coordinate \((r, \theta)\) could be \((10, \arccos(-\frac{1}{3}))\), where 10 is the distance from the origin, and \(\theta\) is the angle in radians. Expressing points this way is useful when dealing with cycles, rotations, and oscillations of points.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, describe a point in a plane using two perpendicular number lines intersecting at an origin. It forms the basis for most parts of traditional geometry. The coordinates are expressed with two values:

• \( x \) - the horizontal distance from the y-axis
• \( y \) - the vertical distance from the x-axis

The intersection of these two lines, known as the origin, has coordinates \((0, 0)\). This method is ideal for detailing positions in a space where linear relationships between points are significant. Converting from polar to rectangular coordinates allows for easier integration into algebraic calculations and graphical representations.

Trigonometric Functions

Trigonometric functions are essential tools in the conversion between polar and rectangular coordinates. These functions describe relationships between angles and sides of triangles, particularly right triangles. The primary trigonometric functions that come into play are:

• \( \cos(\theta) \) - the cosine function, which represents the ratio of the adjacent side to the hypotenuse in a right triangle
• \( \sin(\theta) \) - the sine function, which represents the ratio of the opposite side to the hypotenuse

In coordinate conversion, \( \cos(\theta) \) is used to transform the radial distance into the x-coordinate, whereas \( \sin(\theta) \) is used to find the y-coordinate. Understanding these functions is crucial for correctly deriving the rectangular coordinates from polar ones.

Coordinate Conversion Formulas

Converting between polar and rectangular coordinates involves specific mathematical formulas derived from trigonometry. The common formulas used in this transformation are:

• \( x = r \cdot \cos(\theta) \) - This formula helps calculate the rectangular x-coordinate by using the cosine of the angle.
• \( y = r \cdot \sin(\theta) \) - This one assists in finding the y-coordinate through the sine of the angle.

Let's delve into an example: Given a polar coordinate \( (10, \arccos(-\frac{1}{3})) \), we convert it to rectangular coordinates as follows:

• The x-coordinate is found by calculating: \( x = 10 \cdot \cos\left(\arccos\left(-\frac{1}{3}\right)\right) = -\frac{10}{3} \).
• The y-coordinate requires finding the sine: \( y = 10 \cdot \frac{2\sqrt{2}}{3} = \frac{20\sqrt{2}}{3} \).

This results in the rectangle coordinates \((x, y) = \left(-\frac{10}{3}, \frac{20\sqrt{2}}{3}\right)\), which can now be used in a variety of geometric applications.