Problem 4 - Algebra 2

Question

Use the graph of \(y=\tan \theta\) to find each value. If the tangent is undefined at that point, write undefined. $$ \tan \frac{\pi}{2} $$

Step-by-Step Solution

Answer

The value of \( \tan \frac{\pi}{2} \) is undefined

Step 1: Understanding the question

The exercise is asking to find the value of \( \tan \frac{\pi}{2} \). That can be done by using the trigonometric function of tangent.

Step 2: Find the value of the tangent function

As mentioned in the analysis, the tangent function is undefined at points where the cosine function is zero. Those points include \(\frac{\pi}{2}\), \(-\frac{\pi}{2}\), and their multiples. Since the point in question here, \(\frac{\pi}{2}\), is one of these points, the value of \( \tan \frac{\pi}{2} \) is undefined.

Key Concepts

Tangent Function

The tangent function, denoted as \( \tan \theta \), is a fundamental trigonometric function. It relates an angle in a right triangle to the ratio of the length of the opposite side to the adjacent side. This can be expressed as \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \). In the context of the unit circle, which is a circle with a radius of 1, the tangent of an angle \( \theta \) is the slope of the line that passes through the origin and the point on the circle corresponding to \( \theta \). Some key properties of the tangent function include:

• It is periodic with a period of \( \pi \), meaning \( \tan(\theta + \pi) = \tan \theta \).
• It has vertical asymptotes where the function is undefined.
• It is an odd function, satisfying \( \tan(-\theta) = -\tan(\theta) \).

When working with the tangent function, especially on the unit circle, understanding these properties is crucial for problem solving.

Graph Interpretation

The graph of the tangent function \( y = \tan \theta \) exhibits a distinct pattern. It's a series of repeating sections or cycles due to its periodic nature.Here's a closer look at key features of this graph:

• The graph has vertical asymptotes at points where the function is undefined. You'll notice these at the angles \( \frac{\pi}{2} + n\pi \), where \( n \) is an integer.
• The function increases from negative to positive infinity as it approaches each asymptote from the left, creating a pattern of upward curves.
• There are no maximum or minimum values because the function can take any real value.
• Each cycle between the asymptotes looks similar: starting from negative infinity, increasing continuously, and approaching positive infinity.

Understanding this graphical behavior is important for interpreting values or identifying undefined points, such as \( \tan \frac{\pi}{2} \). By visualizing or sketching the graph, you get a clearer picture of how the function behaves over its domain.

Undefined Values in Trigonometry

Trigonometric functions can be undefined at certain points, typically where their ratios involve division by zero. For the tangent function \( \tan \theta \), it is particularly undefined wherever the cosine of the angle is zero. Let's explore why:

• The tangent function is defined as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This quotient becomes undefined when \( \cos \theta = 0 \) because you cannot divide by zero.
• Key angles where \( \cos \theta = 0 \) include \( \theta = \frac{\pi}{2} + n\pi \), where \( n \) is a whole number.
• At these points, the graph of the tangent function will show vertical asymptotes, indicating places the function is undefined.

Being aware of these undefined points provides a deeper understanding of trigonometric functions. It helps in recognizing and avoiding potential pitfalls in calculations that involve these angles.