Problem 81 - Algebra and Trigonometry

Question

Graph each polar equation. $$ r=\tan \theta, \quad-\frac{\pi}{2}<\theta<\frac{\pi}{2} \quad(\text {kappa curve}) $$

Step-by-Step Solution

Answer

Plot tangents of \( \theta \) in the polar plane within the range \( -\frac{\pi}{2} < \theta < \frac{\pi}{2} \).

Step 1 - Understand the polar equation

The given equation is in polar form: \[ r = \tan \theta \]where \[-\frac{\pi}{2} < \theta < \frac{\pi}{2} \].This means that for each angle \( \theta \), the radial distance \( r \) is \( \tan \theta \).

Step 2 - Determine key points

Evaluate \( r \) for specific values of \( \theta \) within the given range: 1. \( \theta = 0 \): \( r = \tan 0 = 0 \).2. \( \theta = \frac{\pi}{4} \): \( r = \tan \frac{\pi}{4} = 1 \).3. \( \theta = -\frac{\pi}{4} \): \( r = \tan -\frac{\pi}{4} = -1 \).4. As \( \theta \) approaches \( \frac{\pi}{2} \), \( r \) approaches \( \infty \). 5. As \( \theta \) approaches \( -\frac{\pi}{2} \), \( r \) approaches \( -\infty \).

Step 3 - Sketch the graph

Plot the key points determined in Step 2 on a polar coordinate system. These points include \( (0, 0) \), \( (\frac{\pi}{4}, 1) \), and \( ( -\frac{\pi}{4}, -1) \). Draw a curve through these points, noting that the curve stretches towards \( \infty \) as \( \theta \) approaches \( \frac{\pi}{2} \) and towards \( -\infty \) as \( \theta \) approaches \( -\frac{\pi}{2} \). This typical curve is known as a Tangent Curve or Kappa Curve.

Key Concepts

Polar Coordinates

Polar coordinates represent points in a plane using a distance and an angle. Unlike the Cartesian coordinate system, which uses \( x \) and \( y \) axes, the polar system uses \( r \) for the radial distance and \( \theta \) for the angle. This system is particularly useful for dealing with problems involving circular or rotational symmetry. For instance, the point \( (r, \theta) = (2, \frac{\pi}{4}) \) in polar coordinates means you are 2 units away from the origin at an angle of \( \frac{\pi}{4} \) radians from the positive x-axis.

Tangent Function

The tangent function, denoted as \( \tan(\theta) \), is one of the primary trigonometric functions. It relates the angle \( \theta \) to the ratio of the opposite side to the adjacent side in a right triangle. Mathematically, it's expressed as follows: \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). The function is periodic with a period of \( \pi \), and its values repeat every \( \pi \) radians. Importantly, tangent has vertical asymptotes—values where the function goes to infinity—at odd multiples of \( \frac{\pi}{2} \).

Graphing Polar Equations

Graphing polar equations involves plotting points in the polar coordinate system and connecting them smoothly. For the equation \( r = \tan(\theta) \), you start by evaluating \( r \) for key \( \theta \) values:

• \( \theta = 0 \) gives \( r = 0 \)
• \( \theta = \frac{\pi}{4} \) gives \( r = 1 \)
• \( \theta = -\frac{\pi}{4} \) gives \( r = -1 \)
• As \( \theta \) approaches \( \frac{\pi}{2} \), \( r \) approaches \( \infty \)
• As \( \theta \) approaches \( -\frac{\pi}{2} \), \( r \) approaches \( -\infty \)

Plot these points on a polar grid and draw the curve, known as the Kappa curve. This curve extends to positive infinity as \( \theta \) nears \( \frac{\pi}{2} \) and to negative infinity when \( \theta \) nears \( -\frac{\pi}{2} \).