Problem 55 - Algebra and Trigonometry

Question

Polar coordinates of a point are given. Find the rectangular coordinates of each point. $$ \left(7.5, \frac{11 \pi}{18}\right) $$

Step-by-Step Solution

Answer

(-1.302, 7.386)

Step 1 - Identify the given polar coordinates

The polar coordinates are given as \(r, \theta\). In this case, \( r = 7.5 \) and \( \theta = \frac{11 \pi}{18} \).

Step 2 - Understand the conversion formulas

The rectangular (Cartesian) coordinates \( (x, y) \) can be found from the polar coordinates \( (r, \theta) \) using the formulas: \[ x = r \cos(\theta) \] and \[ y = r \sin(\theta) \].

Step 3 - Calculate the x-coordinate

Substitute the given values into the formula for \( x \): \[ x = 7.5 \cos\bigg(\frac{11\pi}{18}\bigg) \] Evaluate \( \frac{11 \pi}{18} \) to find the angle in radians: \[ x = 7.5 \times \cos\bigg(\frac{11 \pi}{18}\bigg) \] Using a calculator: \[ \cos\bigg(\frac{11 \pi}{18}\bigg) \approx -0.1736 \] Therefore, \[ x \approx 7.5 \times -0.1736 = -1.302 \].

Step 4 - Calculate the y-coordinate

Substitute the given values into the formula for \( y \): \[ y = 7.5 \sin\bigg(\frac{11 \pi}{18}\bigg) \] Evaluate \( \frac{11 \pi}{18} \) to find the angle in radians: \[ y = 7.5 \times \sin\bigg(\frac{11 \pi}{18}\bigg) \] Using a calculator: \[ \sin\bigg(\frac{11 \pi}{18}\bigg) \approx 0.9848 \] Therefore, \[ y \approx 7.5 \times 0.9848 = 7.386 \].

Step 5 - Write the rectangular coordinates

The rectangular coordinates are obtained by combining the calculated x and y values: \[ (-1.302, 7.386) \].

Key Concepts

Trigonometric Functions

Trigonometric functions, notably sine and cosine, play a crucial role in converting polar coordinates to rectangular coordinates.
These functions help us relate an angle in one coordinate system to lengths in another.

• Cosine \(\cos\): It gives the x-coordinate (or adjacent side) of a right-angled triangle.
• Sine \(\sin\): It provides the y-coordinate (or opposite side) of a right-angled triangle.

When we know an angle \(\theta\) and a radius \(r\), we can use these functions to find our exact rectangular coordinates. For instance:

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

Understanding these functions will make the conversion process straightforward and intuitive. Ensure that your calculator is set to the correct mode (radians or degrees) based on the angle you're dealing with.

Coordinate Systems

Coordinate systems are methods to represent the position of points in a space.
The two key systems covered here are polar and rectangular (Cartesian) coordinate systems.

• Rectangular (Cartesian) coordinates use two perpendicular lines: x and y.
• Polar coordinates use a distance from a reference point and an angle from a reference direction.

These systems allow us to describe a point in the plane from different perspectives. While Cartesian is suitable for many algebraic applications, polar coordinates are often beneficial for situations involving circular or rotational symmetries. Converting between these two systems lets us use their unique advantages as per the requirement of the problem.

Polar Coordinates

Polar coordinates represent a point in a plane with a distance and an angle.
The angle (usually measured in radians or degrees) is denoted by \( \theta \), and the distance from the origin is denoted by \( r \).
Here's how it works:

• \( r \) tells how far the point is from the origin.
• \( \theta \) indicates the direction of the point relative to the positive x-axis. It rotates counter-clockwise.

To find the corresponding point in rectangular coordinates, we treat \( r \) and \( \theta \) as the radius and angle of a right triangle. Using trigonometric functions, we convert these values into x and y coordinates. This method is invaluable in fields like engineering, physics, and navigation, where direction and distance are often paramount.

Rectangular Coordinates

Rectangular coordinates, or Cartesian coordinates, establish a point's location based on horizontal and vertical distances.
These consist of two values:

• \( x \): The horizontal distance from the origin.
• \( y \): The vertical distance from the origin.

By using these coordinates, we create a straightforward method to locate any point in a plane. Converting from polar to rectangular coordinates involves using the formulas:

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

These formulas allow us to translate a radial distance and angle (\( r, \theta \)) into exact horizontal and vertical positions.
This conversion is essential in many applications, like plotting points on graphs or determining positions in navigational systems.