Problem 16 - Precalculus Essentials

Question

In Exercises \(9-16\), evaluate the trigonometric function at the quadrantal angle, or state that the expression is undefined. $$\tan \frac{\pi}{2}$$

Step-by-Step Solution

Answer

The expression \(\tan\left(\frac{\pi}{2}\right)\) is undefined.

Step 1: Analyze the Given Function

We are to evaluate the function \(\tan\left(\frac{\pi}{2}\right)\). This corresponds to evaluating the tangent of 90 degrees.

Step 2: Understand the Meaning of the Function

The tangent function, \(\tan(x)\), can be understood as the ratio \(\frac{\sin(x)}{\cos(x)}\). We know that \(\cos(\frac{\pi}{2})\) equals zero while \(\sin(\frac{\pi}{2})\) equals one.

Step 3: Evaluate the Function

Substituting the angle into the ratio, we have \(\frac{\sin(\frac{\pi}{2})}{\cos(\frac{\pi}{2})}\) = \(\frac{1}{0}\). The division by zero is undefined in mathematical operations.

Key Concepts

Quadrantal Angles

Understanding quadrantal angles is crucial in evaluating trigonometric functions. A quadrantal angle is an angle that is a multiple of 90 degrees (or \(\frac{\pi}{2}\) radians) and coincides with the axes on the unit circle. These angles are 0 degrees, 90 degrees (or \(\frac{\pi}{2}\) radians), 180 degrees (or \(\pi\) radians), and 270 degrees (or \(\frac{3\pi}{2}\) radians).

When dealing with trigonometric functions like sine, cosine, or tangent at these specific angles, certain rules apply. For instance, the sine of a quadrantal angle will be either 1, 0, or -1, and the cosine will follow a similar pattern. This makes calculations easier, but one must also remember that some functions like tangent can lead to undefined mathematical expressions at certain quadrantal angles, such as when the cosine part of the ratio is zero.

Tangent Function

The tangent function often causes confusion, yet it's an essential trigonometric ratio. Represented as \(\tan(x)\), it can be defined as the ratio of the opposite side to the adjacent side in a right-angled triangle, or equivalently, as the ratio of sine to cosine: \(\tan(x) = \frac{\sin(x)}{\cos(x)}\).

Due to its definition as a ratio, it's particularly important to look at both the numerator and the denominator. When evaluating \(\tan(x)\) at quadrantal angles, if the cosine is zero, the ratio becomes \(\frac{\sin(x)}{0}\), which leads to an undefined value because division by zero does not exist in standard arithmetic. This is one of the key factors to bear in mind to avoid misunderstandings with the tangent function.

Undefined Mathematical Expressions

In mathematics, undefined mathematical expressions are expressions that do not have meaning or cannot be evaluated according to conventional arithmetic rules. The most common example of this in trigonometry is division by zero. In the context of trigonometric functions, this typically occurs when evaluating the tangent function, \(\tan(x)\), at certain angles where the cosine function is equal to zero.

For example, \(\tan(\frac{\pi}{2})\) or \(\tan(90^\circ)\) is undefined because it corresponds to \(\frac{1}{0}\). When you encounter this, it's important to simply state that the expression is undefined rather than attempting to compute a value. Recognizing undefined expressions helps to ensure mathematical arguments remain valid and avoid the creation of false paradoxes or faulty conclusions.