Problem 82 - College Trigonometry

Question

Convert the equation from polar coordinates into rectangular coordinates. $$ \theta=\pi $$

Step-by-Step Solution

Answer

The rectangular equation is \( y = 0 \).

Step 1: Understand the Polar Equation

The given polar equation is \( \theta = \pi \). In polar coordinates, the angle \( \theta \) is the angle from the positive x-axis in the counter-clockwise direction.

Step 2: Recall Polar to Rectangular Transformation

To convert polar coordinates \((r, \theta)\) into rectangular coordinates \((x, y)\), we use the transformations: \( x = r \cos \theta \) and \( y = r \sin \theta \). We now need to analyze the given equation \( \theta = \pi \).

Step 3: Analyze the Angle \(\theta=\pi\)

When \( \theta = \pi \), the angle points directly opposite to the positive x-axis, which lies on the negative x-axis. In polar coordinates, all points corresponding to \( \theta = \pi \) lie on this line.

Step 4: Set Up Rectangular Coordinates Equation

Since \( \theta = \pi \) implies a line along the negative x-axis, it can be expressed in rectangular form by setting \( y=0 \). Thus, any point on this line has \( x=r \cos(\pi) = -r \).

Step 5: Finalize the Rectangular Equation

The rectangular representation of \( \theta = \pi \) is the line \( y = 0 \) and \( x \leq 0 \) since \( r \) could be any non-negative value indicating a point on the negative x-axis. So, the equation in rectangular coordinates is simply \( y = 0 \).

Step 6: Conclusion

The rectangular coordinate form represents a horizontal line along the x-axis, negative direction, so the equation simplifies to \( y = 0 \) with \( x \leq 0 \).

Key Concepts

Polar Coordinates

Polar coordinates represent a point in a plane based on its distance from a reference point and the angle from a reference direction. In this system, each point is determined by two values: the radial coordinate, often denoted as \( r \), and the angular coordinate, \( \theta \).

• The radial distance \( r \) is the distance from the point to the origin of the coordinate system.
• The angle \( \theta \) is measured from the positive x-axis, moving counter-clockwise.

This format is distinctly different from Cartesian or rectangular coordinates. It is especially useful in scenarios where symmetry and angles play a significant role, such as circular and rotational motion.

Rectangular Coordinates

Rectangular coordinates, often referred to as Cartesian coordinates, locate a point in a plane through two perpendicular axes, usually labeled as the x-axis and y-axis. Each point in this system is represented by an ordered pair \((x, y)\).

• Here, \( x \) is the horizontal distance from the origin, measured along the x-axis.
• \( y \) represents the vertical distance from the origin, measured along the y-axis.

This framework is highly intuitive and commonly used for plotting graphs and performing algebraic calculations, as it directly correlates with the object's horizontal and vertical position.

Angle Transformation

Angle transformation involves converting or utilizing angles from one coordinate system into another. In polar coordinates, the angle \( \theta \) plays a crucial role in defining the position. In the exercise, we consider the angle \( \theta = \pi \), which corresponds to the line that lies directly opposite the positive x-axis, effectively on the negative x-axis. This means that the entire line described by \( \theta = \pi \) in polar coordinates aligns horizontally along the x-axis's negative side, where the vertical component \( y \) remains zero. Understanding angle transformation is vital in accurately switching between coordinate systems. It helps in tracing the path or trajectory of an object relative to a fixed origin.

Coordinate Transformation

Coordinate transformation is the process of converting coordinates from one system to another to analyze and understand the geometric representation interchangeably.

• For transforming from polar to rectangular coordinates, we use the transformations \( x = r \cos \theta \) and \( y = r \sin \theta \).
• In the given problem, the expression \( \theta = \pi \) is analyzed by these transformations to conclude that \( y = 0 \) and \( x \leq 0 \).

These transformations allow us to express complex polar descriptions into simpler rectangular forms that can be easily interpreted and visualized. By understanding how \( r \) and \( \theta \) translate into \( x \) and \( y \), you can solve problems that require switching perspectives or formats in mathematical equations.