Sketch the polar curve.
$$\theta=\frac{2}{3} \pi.$$
Short Answer
Expert verified
To sketch the polar curve \(\theta=\frac{2}{3}\pi\), draw the angle \(\frac{2}{3}\pi\) in the coordinate plane, which is approximately 120 degrees. Since the radius r can be any non-negative value, the curve will be a line of points with the same angle \(\frac{2}{3}\pi\) and varying radii. Connect these points with a straight line passing through the origin. Label the angle \(\theta\) as \(\frac{2}{3}\pi\) to complete the sketch.
Step by step solution
01
Understand polar coordinates
In the polar coordinate system, each point is described using a radius (r) and an angle \(\theta\). The radius is the distance from the origin (also known as the pole) to the point, and \(\theta\) is the angle formed between the positive x-axis and the line connecting the origin to the point.
02
Analyze the given equation
The given polar equation is \(\theta=\frac{2}{3}\pi\). This equation specifies the angle \(\theta\), but not the radius r. This means that the points on this curve will have a fixed angle \(\theta=\frac{2}{3}\pi\) with respect to the positive x-axis, but the radius r can be any non-negative value.
03
Plot the points
To plot the polar curve, start by drawing the angle \(\theta=\frac{2}{3}\pi\) in the coordinate plane (which is approximately equal to 120 degrees, since \(\pi\) radians is equal to 180 degrees). Since the radius r can be any non-negative value, the curve will be a line of points with the same angle \(\frac{2}{3}\pi\) and varying radii. You can draw these points by sweeping a line from the origin to any point with the angle \(\frac{2}{3}\pi\) with respect to the positive x-axis.
04
Finalize the sketch
Connect the points with a straight line passing through the origin. This line represents the polar curve \(\theta=\frac{2}{3}\pi\). Make sure to label the angle \(\theta\) to show that it is fixed at \(\frac{2}{3}\pi\). You have successfully sketched the polar curve.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polar Curve
A polar curve represents the collection of points in the polar coordinate system. Each point on a polar curve has a specific angle and radius, denoted as \(r, \theta\). In our case, the polar curve is defined by a constant angle of \(\theta=\frac{2}{3}\pi\). This means:
The angle is fixed at \(\frac{2}{3}\pi\), or approximately 120 degrees.
The radius can vary, so the curve consists of all points with this angle and any possible non-negative radius.
To visualize this curve, imagine tracing a line starting from the origin. This line emanates out at an angle of \(\frac{2}{3}\pi\), extending infinitely outwards as the radius grows. This forms a polar curve shaped like a straight line on a traditional graph, intersecting the origin and extending indefinitely in one direction.
Angle Measurement
In polar coordinates, the angle measurement is critical in defining the location of points. Angles are typically measured in radians for polar graphing, helping to determine how point locations relate to the center or origin.To define the angle in radians, recall:
One full circle around the origin is equivalent to \(2\pi\) radians.
Our given angle is \(\frac{2}{3}\pi\), which is less than \(\pi\), meaning it forms a sharp angle as opposed to being flat or looping around the origin.
Thinking of \(\pi\), remember it is half a full circle (Visualizing \(\theta=\frac{2}{3}\pi\)Understanding the angle measurement allows us to comprehend the direction in which our line radiates, essentially determining the overall path of our polar curve.
Coordinate Systems
Coordinate systems provide a framework to describe locations in space. Two often-discussed systems are the Cartesian and polar coordinates.
Cartesian Coordinate System: Describes points using (x, y) pairs, representing the horizontal and vertical distances from the origin.
Polar Coordinate System: Specifies each point using a radius \(r\) and an angle \(\theta\). Here, \(r\) is the distance from the origin, and \(\theta\) is the direction measured from the positive x-axis.
The polar system is especially useful for curves and designs involving circles or symmetries about a point. It redefines positions based on distances and directional angles, offering a different perspective for plotting and understanding geometrical figures. This can be seen in our task where the angle \(\theta=\frac{2}{3}\pi\) specifies direction while leaving the distance variable.
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