Problem 60 - College Trigonometry

Question

60\( through \)\\# 63\(. In Exercises 60 - 63 , you need to solve for … # Convert the equation from rectangular coordinates into polar coordinates. Solve for \)r\( in all but \)\\# 60\( through \)\\# 63\(. In Exercises 60 - 63 , you need to solve for \)\theta$. $$ y=0 $$

Step-by-Step Solution

Answer

\(\theta = 0\) for positive \(x\), \(\theta = \pi\) for negative \(x\), when \(y = 0\).

Step 1: Understand Rectangular and Polar Coordinates

Rectangular coordinates represent points on a plane using \(x, y\) values, while polar coordinates use \(r, \theta\) values, where \( \) is the radius (distance from origin) and \( heta\) is the angle from the positive x-axis.

Step 2: Use Coordinate Conversion Formulas

The formula to convert a point from rectangular to polar coordinates is given by \(r = \sqrt{x^2 + y^2}\) for the radius, and \( heta = \tan^{-1}(y/x)\) for the angle. Since we are focusing on \( heta\), consider only the angle conversion equation.

Step 3: Substitute Given Equation into Conversion Formula

The given equation is \(y = 0\), which means the point lies on the x-axis. Substitute \(y = 0\) into \( heta = \tan^{-1}(y/x)\), resulting in \( heta = \tan^{-1}(0/x)\).

Step 4: Solve for \(\theta\)

For any non-zero \(x\), \(\tan^{-1}(0/x) = 0\). This indicates the angle \(\theta\), when \(y = 0\) and \(x > 0\), is 0 radians. If \(x < 0\), the angle would be \pi\ radians due to the negative direction along the x-axis.

Step 5: Conclude the General Solution

Thus, for the equation \(y = 0\), the polar coordinate \(\theta\) can be characterized as either 0 radians for positive x-values or \pi\ radians for negative x-values.

Key Concepts

Coordinate Conversion

Coordinate conversion is the process of changing from one type of coordinate system to another. In mathematics, the most common conversions occur between rectangular (Cartesian) and polar coordinates.

This process involves using specific formulas that relate the two systems:

• The polar radius is calculated using: \[ r = \sqrt{x^2 + y^2} \]The radius \(r\) is the distance from the origin to the point in the plane.
• The polar angle is calculated using the arctangent function: \[ \theta = \tan^{-1}(\frac{y}{x}) \]This angle \(\theta\) is measured from the positive x-axis to the line segment connecting the origin to the point.

These formulas help you shift between these two perspectives of the same point. Understanding both helps in visualizing and solving problems more effectively, especially when the problem suits one system better than the other.

Rectangular Coordinates

Rectangular coordinates, known as Cartesian coordinates, represent a point in a plane using two values: \(x\) and \(y\). These coordinates reference a point's horizontal and vertical distances from the origin, making them straightforward for interpreting many types of mathematical and real-world data.

In any given coordinate plane:

• \(x\): Describes the position along the horizontal axis
• \(y\): Describes the position along the vertical axis

This method is intuitive for plotting points on graph paper or digital displays, as each point has a unique \( (x, y) \) representation.

Angles in Polar Coordinates

Understanding angles in polar coordinates is crucial for fully grasping this coordinate system. In the polar system, the angle \(\theta\) describes the direction of the radius \(r\), from the positive x-axis.

• When \( y = 0 \), points lie directly on the x-axis. Here, \(\theta\) becomes critical for determining direction:
• If \(x > 0\), the angle \(\theta\) is 0 radians, meaning the point is directed to the right along the positive x-axis.
• If \(x < 0\), the angle \(\theta\) is \(\pi\) radians, pointing directly to the left along the negative x-axis.

Angles in polar coordinates provide a unique way to view and solve problems, often simplifying expressions by focusing on direction instead of specific x and y values. This understanding is particularly useful in various fields, including physics and engineering, where direction plays a vital role in determining outcomes.