Question

Sketch the graph of the given cylindrical or spherical equation. $$ \theta=\pi / 6 $$

Short answer

The graph is a straight line at a 30° angle from the x-axis.

Step-by-step solution

Understanding the Equation

The equation given is in cylindrical coordinates. In cylindrical coordinates, the variable \( \theta \) represents the azimuthal angle, which is the angle measured from the positive x-axis in the xy-plane.

Identifying the Angle

The given equation \( \theta = \pi / 6 \) specifies that the angle between the positive x-axis and any point on the graph is consistently \( \pi / 6 \).

Interpreting in the Cartesian Plane

In the Cartesian plane, \( \theta = \pi / 6 \) corresponds to a line that passes through the origin and makes a \( 30^\circ \) angle with the positive x-axis. This is because \( \pi / 6 \) radians is equivalent to \( 30^\circ \).

Drawing the Graph

To sketch the graph, draw a straight line starting from the origin that creates a \( 30^\circ \) angle with the positive x-axis. This line extends infinitely in both directions through the origin.

Key concepts

Graph Sketching

Graph sketching in the context of cylindrical coordinates involves visualizing the equation provided and representing it geometrically on a plane or space. In the exercise, the equation \( \theta = \pi / 6 \) needs to be graphed. This type of equation generally represents a direction rather than a shape, like a line or curve, such as those common in Cartesian equations.

When sketching this graph, you should think of how this angle interacts with the coordinate system. Here, the angle \( \pi / 6 \) is taken from the positive x-axis in the cylindrical coordinate system. This means that every point on this line shares this direction. All points lie along a ray or line, ensuring that each is distanced radially from the line that consists of points making an angle of \( \pi / 6 \) with the x-axis.

Sketching the graph means drawing a straight line that represents this continuous angle, extending infinitely across the plane in both directions from the origin.

Azimuthal Angle

The azimuthal angle, often denoted as \( \theta \), is a crucial component of cylindrical and spherical coordinate systems. It quantifies the angle between a projection of a point onto the xy-plane and the positive x-axis. In this particular exercise, the azimuthal angle \( \theta = \pi / 6 \) describes a specific linear path from the origin.

Azimuthal angle relies heavily on the concept of direction, this time described in radian measure.

• It's measured in the xy-plane.
• Its value is determined against the positive x-axis.
• The measure \( \pi / 6 \) equates to 30 degrees.

The ability to interpret azimuthal angles is vital in transforming cylindrical equations into familiar Cartesian forms, allowing for the derivation of corresponding lines or curves on a two-dimensional plot.

Cartesian Plane

The Cartesian plane is the fundamental framework for graph sketching, identified by x and y axes intersecting perpendicularly at the origin. In terms of cylindrical coordinates, translations often occur wherein aspects of the cylindrical descriptor are transformed into the Cartesian framework.

In the context of \( \theta = \pi / 6 \):

• This condition describes a line from the origin at a 30-degree angle (since \( \pi / 6 \) equals 30 degrees).
• The Cartesian representation simplifies understanding by designating the path as a straight line moving radially in the plane starting from the origin.

Sketching in the Cartesian plane allows one to draw straight lines representing angular directions as given by the cylindrical system, demonstrating the stark distinctions yet interconnectedness between these coordinate representations.

Coordinate Transformation

Coordinate transformation is the pivotal process of translating information from one coordinate system to another. In this scenario, cylindrical coordinates specified by \( \theta = \pi / 6 \) must be expressed within the Cartesian plane to effectively illustrate their geometric interpretation.

Transformations involve mathematical operations to convert angles into lines on a plane.

• The process involves understanding the radial distance from the origin and the angle in relation to the x-axis.
• The angle \( \theta = \pi / 6 \) indicates a perpetual direction rather than a fixed position, leading to a line in the x-y plane.

These transformations ensure that the same data can be interpreted in multiple forms, enriching analysis and enabling continual interpretation across different systems. Thus, understanding how to transform this information fosters a deeper grasp of the broader mathematical landscape as coordinated actions between these systems.