Problem 50 - Algebra 2

Question

Which equation does NOT represent a vertical asymptote of the graph of \(y=\tan \theta ?\) F. \(\theta=-\frac{\pi}{2} \quad\) G. \(\theta=0\) H. \(\theta=\frac{\pi}{2} \quad\) J. \(\theta=\frac{3 \pi}{2}\)

Step-by-Step Solution

Answer

The equation that does NOT represent a vertical asymptote of the graph of \(y=\tan \theta\) is \(\theta = 0\)

Step 1: Identify where the tangent function is undefined

The tangent function \(y=\tan \theta\) is undefined at\(-\frac{\pi}{2}\), \(\frac{\pi}{2}\), and in general \(\frac{(2n+1)\pi}{2}\), where 'n' is an integer, because at these points, the function takes on the indeterminate form of either \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). These points are the vertical asymptotes of the function.

Step 2: Evaluate each choice

We need to look at each of the given choices and apply our knowledge of the tangent function. Choice F \(\theta = -\frac{\pi}{2}\), choice H \(\theta = \frac{\pi}{2}\), and choice J \(\theta = \frac{3\pi}{2}\) satisfy the condition \(\frac{(2n+1)\pi}{2}\), where 'n' is an integer. Hence, these are possible vertical asymptotes of the function. But choice G \(\theta = 0\) does not satisfy this condition.

Step 3: Provide the answer

Based on the above steps, we can conclude that the only choice that does not represent a vertical asymptote of the graph of the function \(y = \tan \theta\) is \(\theta = 0\)

Key Concepts

Tangent Function

The tangent function, denoted as \(y = \tan \theta\), is one of the fundamental trigonometric functions. It relates the angle \(\theta\) in a right triangle to the ratio of the opposite side to the adjacent side. This function is periodic and has a period of \(\pi\). This means its graph repeats every \(\pi\) units.

• **Expression**: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
• **Domain**: All real numbers, except where \(\cos \theta = 0\)
• **Range**: All real numbers

Due to its definition through sine and cosine, the tangent function will be undefined at angles where the cosine is zero, leading to vertical asymptotes.

Vertical Asymptotes

Vertical asymptotes occur where a function approaches infinity, indicating discontinuity at those points. For the tangent function \(y = \tan \theta\), vertical asymptotes arise because it becomes undefined when the cosine of the angle is zero.

These occur at:

• \(\theta = -\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2}, \ldots\)
• Generally expressed as \(\theta = \frac{(2n+1)\pi}{2}\), where 'n' is any integer.

At these points, the graph shoots upwards or downwards without touching, creating the asymptotic behavior seen in trigonometric graphs. Avoiding these points is key to understanding and sketching the tangent function correctly.

Trigonometric Graphs

Trigonometric graphs, such as that of the tangent function, display unique wave-like patterns which are crucial for visualizing trigonometric behavior. The graph of \(y = \tan \theta\) exhibits distinctive characteristics:

• **Periodic Nature**: Repeats every \(\pi\) units.
• **Vertical Asymptotes**: Indicated by dashed lines, the graph approaches but never crosses these asymptotes at \(\theta = \frac{(2n+1)\pi}{2}\).
• **Wave Form**: Crosses the x-axis at multiples of \(\pi\), forming a wave that extends infinitely.

Understanding these features allows students to predict the behavior of the graph and solve related problems effectively, like identifying points without asymptotes, such as \(\theta = 0\).