Problem 47 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 45-78: Sketch the graph of the polar equation. $$ \theta=-\pi / 6 $$

Step-by-Step Solution

Answer

The graph is a straight line at an angle of \( -30^{\circ} \) from the x-axis.

Step 1: Identify Equation Type

The given equation is a polar equation in the form of \[ \theta = -\frac{\pi}{6} \] This means it represents a line where the angle \( \theta \) is constant at \( -\frac{\pi}{6} \).

Step 2: Understand Polar Coordinates

In polar coordinates, \( \theta \) is the angle from the positive x-axis, and all points with the same \( \theta \) lie on a line passing through the origin forming an angle \( -\frac{\pi}{6} \) with the horizontal axis.

Step 3: Determine Line Orientation

The angle \( -\frac{\pi}{6} \) corresponds to 30 degrees in the negative direction (clockwise) from the positive x-axis. This means the line will slope downward to the right away from the vertical axis.

Step 4: Sketch the Polar Graph

Draw a horizontal line (x-axis) and vertical line (y-axis) to represent the coordinate system. From the origin, draw a line at an angle of \( -\frac{\pi}{6} \) from the positive x-axis. This is equivalent to drawing a line 30 degrees in the clockwise direction.

Key Concepts

Polar Coordinates

Polar coordinates provide a different approach to locating points in a plane compared to rectangular coordinates. Instead of using an ordered pair of horizontal (x) and vertical (y) distances from the origin, polar coordinates describe points through a distance and an angle. In this system:

• The distance from the origin to the point is called the radius, often denoted as \( r \).
• The angle is measured from the positive x-axis, known as \( \theta \).

In polar coordinates, a point is expressed as \( (r, \theta) \). This method is especially useful for dealing with points that are more naturally located by their direction and distance, like those moving in circular paths. The value of \( \theta \) can be expressed in radians or degrees, and it defines how far to turn from the x-axis to reach the point. This gives a more flexible and sometimes simpler way to describe certain graphs, especially those involving circles and spirals.

Angle in Polar Graphing

When working with angles in polar graphing, it is important to understand how angles are measured and represented. In polar equations, the angle \( \theta \) is crucial as it determines the orientation of lines or curves with respect to the x-axis. The angle is typically measured:

• From the positive x-axis as a starting point.
• In a counterclockwise direction if \( \theta \) is positive.
• In a clockwise direction if \( \theta \) is negative.

For the given exercise, \( \theta = -\frac{\pi}{6} \), which means the line representing this angle lies -30 degrees, or \(-\frac{\pi}{6}\) radians, clockwise from the positive x-axis. Understanding this angle helps in determining how to position the line on a polar graph. The concept of negative and positive angle measurement allows us to fully capture all directions and orientations of lines originating from the center of the polar graph.

Graphing Polar Equations

Graphing polar equations involves plotting points or lines based on their polar coordinates. The process starts by determining the nature of the equation to understand its geometry. In the case of a constant angle like \( \theta = -\frac{\pi}{6} \), the graph is a straight line that:

• Passes through the origin.
• Maintains a constant angle with respect to the x-axis.

To graph polar equations, it's effective to:

• Start by drawing the axes, which are the horizontal x-axis and vertical y-axis.
• Draw the line or curve following the angle \( \theta \), ensuring that it extends in both directions from the origin.

In our example, the line at \(-\frac{\pi}{6}\) will slope downwards, moving right from the origin, since the angle is measured clockwise. This direct approach simplifies understanding and sketching the geometric transformations that polar equations represent, making it easier to visualize mathematical concepts like this in a concrete, spatial form.