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Chapter 10: Problem 27

Sketch the polar curve. $$r=\tan \theta.$$

Short Answer

Expert verified
The sketch of the polar curve \( r = \tan \theta \) manifests as a series of vertical lines for each period of the tangent function, tending towards positive infinity as \( \theta \) approaches \( \frac{\pi}{2} \) and to negative infinity as \( \theta \) approaches \( \frac{3\pi}{2} \)

Step by step solution

01

Understanding the polar curve

The polar curve is given by the equation \( r = \tan \theta \). Here, \( \theta \) is the angle made with the positive x-axis (in radians), and r is the distance from the origin. We can note that \(\tan \theta \) is undefined at \( \frac{\pi}{2} \) plus any multiple of \( \pi \), where \( \pi \) is 180 degrees in the unit circle. This might create a discontinuity in the graphing of the curve, so we need to keep this in mind when plotting the points.
02

Plotting points for the positive and negative values of \( r \)

It is convenient to start with the angles for which we know the tangent. As \( \theta=0 \), we have \( r=0 \). As \( \theta \) approaches \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \), the value of \( r \) tends to positive and negative infinity respectively. This process should be repeated until we have enough points to see the emerging pattern of the curve.
03

Sketching the curve

Now that we have plotted our points, we can sketch our curve. Draw a line connecting all the points and continuing towards the asymptotes. Repeat this for all periods of the tangent function to get the full sketch of the curve. Remember, the polar curve will look different from the typical Cartesian graph of the tangent function.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Tangent Function
The tangent function, denoted as \(\tan \theta\), is a trigonometric function that relates an angle \(\theta\) to the ratio of the opposite side to the adjacent side in a right-angled triangle. It measures how "steep" the angle is, much like the slope in algebra. This function is periodic, with a period of \(\pi\), meaning it repeats its values every \(\pi\) radians.

Key characteristics of the tangent function include:
  • It is undefined at certain points, specifically at \(\theta = \frac{\pi}{2} + k\pi\) where \(k\) is an integer.
  • It tends to \(+\infty\) or \(-\infty\) as it approaches these undefined points, resulting in vertical asymptotes.
  • The function is zero at multiples of \(\pi\), i.e., \(\theta = k\pi\).
  • Tangent is an odd function, so \(\tan(-\theta) = -\tan(\theta)\), exhibiting symmetry about the origin.

In the context of polar coordinates, \(r = \tan \theta\) expresses the radial distance \(r\) in terms of the angle \(\theta\). When graphing the curve \(r=\tan \theta\), one needs to handle the function's periodicity and asymptotes carefully to sketch it accurately.
Graphing Polar Curves
Graphing polar curves can initially seem daunting because it involves plotting based on angles and radii rather than \(x\) and \(y\) coordinates. A polar curve is defined by an equation \(r=f(\theta)\). To graph the curve \(r=\tan \theta\), follow these steps to gain clarity:

  • Convert angle measurements to radians, as polar coordinates often use radians rather than degrees.
  • For several values of \(\theta\), calculate \(r = \tan \theta\). Start with known values as landmarks, such as multiples of \(\frac{\pi}{4}\).
  • Note that as \(\theta\) nears \(\frac{\pi}{2}\) and \(\frac{3\pi}{2}\), \(r\) approaches infinity, mapping these to vertical asymptotes on the graph.
  • Plot \(\theta\) on the polar coordinate system鈥檚 angular grid and extend the radial lines of the points already calculated.
  • Join the plotted points smoothly, acknowledging the sharp transitions near asymptotes.

While devising the graph of \(r=\tan \theta\), remember to cover the entire range of \(\theta\) that spans its period. This ensures that the repeating pattern of the tangent function is accurately represented.
Discontinuities in Graphs
In mathematics, a discontinuity occurs where a function is not smoothly connected. Discontinuities can disrupt the graphing process, leading to gaps, jumps, or undefined areas. In the polar curve \(r = \tan \theta\), the tangent function's discontinuities pose challenges.

Here's what to look for:
  • Vertical asymptotes at \(\theta = \frac{\pi}{2} + k\pi\), where the function is undefined and \(r\) becomes infinitely large.
  • Sudden transitions in \(r\) as \(\theta\) approaches \(\frac{\pi}{2}\) or \(\frac{3\pi}{2}\), where \(r\) jumps from positive to negative infinity or vice versa.
  • Ensuring smoothly plotted sections between asymptotes by carefully choosing enough sample points to illustrate the behavior of \(\tan \theta\).

Addressing these discontinuities is crucial when graphing; it involves visualizing and drawing the sections of the curve that behave regularly, while accurately representing the jumps and infinite stretches with careful attention to the behavior of tangent as \(\theta\) changes.

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