Problem 27 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 24-35, at what points, if any, are the functions discontinuous? $$ r(\theta)=\tan \theta $$

Step-by-Step Solution

Answer

The function is discontinuous at \( \theta = \frac{\pi}{2} + n\pi \), where \( n \) is an integer.

Step 1: Identify Function

The given function is the tangent function, expressed as \( r(\theta) = \tan \theta \). This function is commonly known and we need to determine where it is undefined to find points of discontinuity.

Step 2: Determine Undefined Points

The tangent function \( \tan \theta \) is undefined at points where the cosine of \( \theta \) equals zero, as it is defined as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Thus, we solve for \( \cos \theta = 0 \).

Step 3: Solve for Discontinuities

To find the zeros of \( \cos \theta \), let's consider the unit circle where \( \cos \theta \) equals zero. This occurs at odd multiples of \( \frac{\pi}{2} \), thus the points of discontinuity are \( \theta = \frac{\pi}{2} + n\pi \), where \( n \) is an integer.

Step 4: Conclude Discontinuity Points

After determining the zeroes of the cosine function, conclude that the tangent function is discontinuous at \( \theta = \frac{\pi}{2} + n\pi \). This schedule of discontinuity holds for all integer values of \( n \).

Key Concepts

Tangent Function

The tangent function is a fundamental concept in trigonometry, often denoted as \( \tan \theta \). It is derived from the sine and cosine functions and is defined as:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

The tangent function can represent various phenomena, such as angles and slopes, in different mathematical contexts. However, it is crucial to recognize where the tangent function becomes undefined. Since it involves division by \( \cos \theta \), the function will be undefined wherever the cosine function equals zero. As a result, these points are known as discontinuities. Understanding these discontinuities is essential for plotting and analyzing trigonometric functions accurately.

Unit Circle

The unit circle is a powerful tool in trigonometry. It's a circle with a radius of one and is centered at the origin of the coordinate system. This simple construct helps visualize the behavior of trigonometric functions such as sine, cosine, and tangent.

• The unit circle allows you to see how angles (measured in radians) are related to the coordinates on the circle.
• At various angles around the circle, the cosine value is the x-coordinate, and the sine value is the y-coordinate.
• The tangent of an angle can be visualized as the slope of a line drawn through the point on the circle and the origin.

Particularly, the unit circle helps identify where certain trigonometric functions are undefined. For instance, the tangent function becomes undefined at angles where the cosine is zero, which can be visualized easily on the unit circle at odd multiples of \( \frac{\pi}{2} \). Each time you encounter these points, the value of tangent tends to infinity or changes direction abruptly, indicating a discontinuity.

Cosine Function

The cosine function is a critical component in understanding trigonometric concepts. Written as \( \cos \theta \), it represents the x-coordinate of a point on the unit circle that corresponds to a given angle \( \theta \).

• It oscillates between -1 and 1, creating a wavelike pattern as \( \theta \) varies over time or angle.
• The zeros of the cosine function are significant because they indicate points where the tangent function is undefined.

To determine these points:

• Remember that \( \cos \theta = 0 \) implies points where the tangent function will not exist.
• These occur at odd multiples of \( \frac{\pi}{2} \) or \( \theta = \frac{\pi}{2} + n\pi \), where \( n \) is an integer.

Keeping these attributes in mind helps you grasp the periodic nature of trigonometric functions and understand why discontinuities in the tangent function occur.