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A polar equation describes the relationship between the radius and the angle in a two-dimensional plane. Unlike Cartesian coordinates, which use ordered pairs \(x, y\), polar coordinates express points using distances from a central point, called the "origin" or "pole," and an angle from a reference direction. The radius is often denoted by \(r\), and the angle by \(θ\). A polar equation can be written in the form \(r = f(θ)\), where \(r\) is a function of the angle \(θ\).

In our given problem, the polar equation is \(r = 3\). This tells us that no matter the angle \(θ\), the radius remains constant at 3. This is the defining characteristic of a circle in polar coordinates.

• The radius \(r\) is constant (here, \(r=3\)).
• The angle \(θ\) can take any value between \(0\) and \(2Ï€\) radians.

This equation therefore describes a complete circle with a fixed radius centered at the origin.

Sketching a circle in polar coordinates from a polar equation involves understanding how the radius changes with varying angles. In our example, because \(r=3\) is constant, every point at a distance of 3 units from the origin lies on the circle.

Start by placing a point 3 units from the origin. As you adjust the angle \(θ\), maintain that point at a radius of 3 units from the center. Each location of \(θ\) gives a point on the circle, and as \(θ\) covers its complete range, these points connect to form a full circle.

• Draw the horizontal x-axis and the vertical y-axis, with the origin at the center. This is your reference framework.
• Visualize or sketch a point at a distance \(r=3\) from the center, constantly moving as you rotate the angle.
• Trace the circumference by connecting all these points as \(θ\) spans from \(0\) to \(2Ï€\).

This method effectively allows you to sketch the circle defined by the equation \(r=3\), illustrating that it is a complete circle centered at the origin.

The angle \(θ\) in polar coordinates is crucial for defining the position of a point in the polar system. It indicates the direction in which you measure the radius, providing a complete 360-degree view of the plane as \(θ\) varies from \(0\) to \(2π\) radians.

When sketching a polar plot like our circle equation \(r=3\), \(θ\) plays the role of guiding the tracing of the circle. The line formed by the angle \(θ\) rotates around the origin and aligns along the positive x-axis initially. As \(θ\) increases, this line effectively rotates, marking a new position on the circular path.

• Begin with \(θ=0\), aligning the line with the positive x-axis.
• Increase \(θ\), and mark the point at radius \(r=3\).
• Continue to rotate this line until \(θ\) returns to \(2Ï€\) to complete the circle.

Angle \(θ\) gives directionality and motion to the polar sketch, with \(r=3\) maintaining the consistency of the circle's radius.