Problem 48 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 45-78: Sketch the graph of the polar equation. $$ \theta=\pi / 4 $$

Step-by-Step Solution

Answer

The graph is a straight line at a 45-degree angle, passing through the origin.

Step 1: Understanding the Polar Equation

The given polar equation is \[ \theta = \frac{\pi}{4} \]In a polar coordinate system, \(\theta\) represents the angle from the positive x-axis. This equation indicates that the angle \(\theta\) is constant at \(\pi/4\) radians, which is equivalent to 45 degrees.

Step 2: Identify the Graph Type

Since the angle \(\theta\) is constant, the graph represents a straight line. In the polar coordinate system, this line passes through the origin and is at the angle of \(\pi/4\) radians with respect to the positive x-axis.

Step 3: Sketch the Line

To sketch the line, draw a line starting from the origin. This line should extend through all quadrants, making an angle of \(\pi/4\) (or 45 degrees) with the positive x-axis. Remember that in polar coordinates, the line extends infinitely in both directions from the origin.

Key Concepts

Polar Coordinate System

The polar coordinate system is an alternative to the Cartesian, or rectangular, coordinate system. Instead of using x and y coordinates, it uses a distance from a reference point and an angle. The reference point is commonly referred to as the pole, equivalent to the origin in Cartesian coordinates. The angle, measured in radians or degrees, typically starts from the positive x-axis (known as the polar axis) and sweeps counterclockwise.

• Each point in this system is represented as ( θ, r).
• θ is the angle made with the polar axis, and r is the radial distance from the pole.

Because the angles can result in overlapping paths, each point can have multiple representations. For example, a point at 90 degrees (or π/2 radians) could also be found by moving π/2 radians in the opposite direction and reversing direction by using a negative radius.

Graph Sketching

In the context of polar equations, graph sketching involves translating angles and radii into visual representations. This can be quite different from Cartesian graphing because you are dealing with rotations and arcs instead of direct lines and curves using x and y coordinates. When sketching a graph in polar coordinates:

• Always start by determining the nature of the equation. For example, θ = constant results in a line through the origin.
• Identify key points that are easy to plot; these often occur at strategic angles like π/2 or π.
• Use symmetry properties of the polar coordinate system to simplify your graph.

Utilize a polar grid that displays concentric circles and radial lines as guides when sketching graphs. This helps to better visualize how the radius and angle relate to each other in polar coordinates.

Radians

Radians are a way of measuring angles based on the radius of a circle. This measure is often preferred in mathematics because it relates directly to the properties of circles. One complete revolution around a circle is 2π radians, which is equivalent to 360 degrees.

• rac{ π}{4} radians is equivalent to 45 degrees, being one-eighth of a full circle rotation.
• The conversion between degrees and radians is given by the formula: degrees = (180/ π) * radians.

Using radians simplifies calculations of trigonometry, calculus, and many mathematical situations because of its direct relationship with the unit circle and the natural periodic properties of functions. Thinking in radians can make it easier to visualize angles in circular trigonometric terms.

Constant Angle

When discussing a polar equation such as θ = π/4, the angle θ is constant. This means all points on the graph are oriented at this fixed angle with respect to the polar axis. It’s a concept akin to what you might call a bearing in navigation.

• Regardless of the value of the radial variable r, the orientation remains π/4 radians, forming a straight, constant angle line.
• This implies any point plotted will maintain this directional angle from the polar center.

This constancy means every solution (point) corresponding to this equation will lie on the same line, but at different distances, showcasing inherent symmetry in polar coordinates.

Straight Line

In a polar coordinate system, an equation like θ = constant represents a straight line, despite the seemingly unconventional representation. This line passes through the origin because the angle θ is constant everywhere.

• The line effectively cuts through all quadrants because it indefinitely extends with every additional multiple of π since polar coordinates wrap around the circle.
• This graph does not 'stop' at a specific point unlike some Cartesian lines, due to its nature in polar coordinates.

This understanding reflects a fusion of angular geometry with linear paths, showing how diverging segments form when conceptualizing directions in polar coordinates.