Problem 58 - 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$. $$ x=-3 $$

Step-by-Step Solution

Answer

θ is π, representing a line through the negative x-axis.

Step 1: Understanding Rectangular and Polar Coordinates

Rectangular coordinates are expressed as \((x, y)\), while polar coordinates are expressed as \((r, \theta)\), where \(r\) is the radius (distance from the origin) and \(\theta\) is the angle from the positive x-axis.

Step 2: Identifying Given Information

In this exercise, we are provided with \(x = -3\). The task is to convert this coordinate from rectangular to polar form. Here, we do not have \(y\) specified, which generally indicates a vertical line.

Step 3: Calculating r

Even though we don't need to solve for \(r\) in Exercises 60-63, it is useful for understanding. The general equation for \(r\) is \(r = \sqrt{x^2 + y^2}\). However, since the line is x=-3, it tells us directly if only considering a specific point with y.

Step 4: Solving for theta

We use \(\tan(\theta) = \frac{y}{x}\), but since y is not specified and x is a horizontal line, the value of \(y\) will affect \(\theta\). We focus here on the line \(x = -3\), then \(\theta = \pi\), representing a line through the negative x-axis direction.

Step 5: Conclusion of the Solution

Since \(x=-3\) represents a vertical line where y is not specified, in terms of angle \(\theta\), this is equivalent to the direction of angle \(\pi\) because it lies on the negative x-axis.

Key Concepts

Polar Coordinates

Polar coordinates are a way to represent points in a plane using a distance and an angle. Instead of using the traditional x and y coordinates like the rectangular system, polar coordinates use the radius \( r \) and the angle \( \theta \). This system can be particularly useful for dealing with problems involving circles or directions from a point.

To locate a point in polar coordinates:

• The radius \( r \) represents how far the point is from the origin (similar to how a circle's radius works).
• The angle \( \theta \) represents the direction of that point, measured in radians or degrees, starting from the positive x-axis and rotating counter-clockwise.

Using polar coordinates is like pinpointing a location by saying "how far" and "which way." This approach simplifies many mathematical problems, especially in trigonometry and complex numbers.

Rectangular Coordinates

Rectangular coordinates, or Cartesian coordinates, express a location in a plane using two values: \( x \) and \( y \). These refer to the horizontal and vertical displacements from a reference point known as the origin, where both \( x \) and \( y \) are zero.

There are some essential things to remember about rectangular coordinates:

• The \( x \)-coordinate shows the horizontal position, moving left if negative or right if positive.
• The \( y \)-coordinate shows the vertical position, moving down if negative or up if positive.
• This coordinate system is quite intuitive for graphing algebraic equations, as it divides the plane into four quadrants.

Rectangular coordinates are straightforward and are most commonly used with linear equations. They're like the basic grid system for plotting points in a plane.

Angle Theta

Theta \( (\theta) \) is the angle used in polar coordinates to indicate direction. This angle measures how much to rotate from the positive \( x \)-axis to reach a point.

Understanding and calculating \( \theta \) is crucial in coordinate conversion.

• When \( x \) is negative, such as when \( x = -3 \), \( \theta \) might be \( \pi \) radians, pointing directly across the x-axis to the left, this represents the direction from the origin along the negative x-axis.
• Calculating \( \theta \) can involve trigonometric functions like \( \tan \theta = \frac{y}{x} \), which helps find the angle when both \( x \) and \( y \) are known.

In the exercise mentioned, since \( x = -3 \) is a vertical line on the negative side, the angle \( \theta \) will equal \( \pi \), providing the direction of the line. Understanding \( \theta \) helps translate between the two systems of polar and rectangular coordinates.