Problem 5 - Precalculus with Calculus Previews

Question

Find the indicated value without the use of a calculator. $$ \tan \frac{9 \pi}{2} $$

Step-by-Step Solution

Answer

The value \( \tan \frac{9\pi}{2} \) is undefined.

Step 1: Understand the Tangent Function

The tangent function, \( \tan(x) \), is periodic with a period of \( \pi \). This means that \( \tan(x + n\pi) = \tan(x) \) for any integer \( n \). Our goal is to simplify the given expression by finding an equivalent angle within the most common period \([0, \pi)\) or \([-\frac{\pi}{2}, \frac{\pi}{2})\).

Step 2: Simplify the Angle

The given angle is \( \frac{9\pi}{2} \). To find an equivalent angle within the range \([-\frac{\pi}{2}, \frac{\pi}{2})\), we subtract multiples of \( \pi \) (since \( \tan(x) \) has a periodicity of \( \pi \)). First, divide \( \frac{9\pi}{2} \) by \( \pi \) which results in \( \frac{9}{2} = 4.5 \), telling us that \( \frac{9\pi}{2} \) is equivalent to \( 4\pi + \frac{\pi}{2} \).

Step 3: Determine the Equivalent Angle

From Step 2, we know \( \frac{9\pi}{2} = 4\pi + \frac{\pi}{2} \). Now, we simplify the angle to \( \frac{\pi}{2} \), as the multiples of \( \pi \) do not affect the tangent function. Thus, the task is reduced to finding \( \tan \frac{\pi}{2} \).

Step 4: Evaluate \( \tan \frac{\pi}{2} \)

The tangent function \( \tan(x) \) approaches infinity as \( x \) approaches \( \frac{\pi}{2} \) from the left, and negative infinity from the right. Therefore, \( \tan \frac{\pi}{2} \) is undefined due to the vertical asymptote at that point.

Key Concepts

Understanding the Tangent Function

The tangent function, often denoted as \( \tan(x) \), is one of the principal trigonometric functions. It relates angles in a right triangle to the ratio of the sides. Specifically, it is the ratio of the opposite side to the adjacent side. This function is very useful in solving problems across geometry and calculus.

A key characteristic of the tangent function is its periodicity. This means the function repeats its values at regular intervals. For \( \tan(x) \), this interval or period is \( \pi \). For any angle \( x \), \( \tan(x + n\pi) = \tan(x) \) holds true for any integer \( n \). This property is particularly useful in finding equivalent angles within certain ranges, helping simplify trigonometric calculations.

Angle Simplification Techniques

Simplifying angles, especially when dealing with trigonometric functions, is crucial for easier computation. In this context, we take the angle \( \frac{9\pi}{2} \) and convert it into an equivalent angle within a more manageable range for the tangent function.

To do this, divide the original angle by the tangent’s period \( \pi \). So, \( \frac{9\pi}{2} \div \pi = 4.5 \). This tells us how many full periods fit into the angle. Thus, \( \frac{9\pi}{2} \) can be rewritten as \( 4\pi + \frac{\pi}{2} \), breaking it down to a simpler angle, \( \frac{\pi}{2} \).

• Divide by the function's period to find full cycles.
• Express the angle as a sum of a multiple of periods plus a remainder.

These steps help to reduce the angle into a standard form where evaluating the tangent function is straightforward.

Exploring the Periodicity of Tangent

The periodic behavior of \( \tan(x) \) is a defining feature, making certain calculations much easier. As mentioned, the periodicity of the tangent function is \( \pi \), which means that every \( \pi \) units along the x-axis, the function values repeat.

Because of this periodicity, once we determine \( \tan(\frac{\pi}{2}) \), we know that \( \tan(\frac{9\pi}{2}) \) will behave similarly due to its simplification to \( \frac{\pi}{2} \). This predictability is very advantageous when solving trigonometric equations and estimating values without using complex calculations or technology.

• Periodic functions repeat their values after a specific interval.
• This concept allows simplification of angles, making computation simpler.

Understanding Undefined Values in Trigonometry

In trigonometry, some angles lead to undefined values in certain functions due to division by zero or vertical asymptotes. For the tangent function, \( \tan(x) \) is undefined at \( \frac{\pi}{2} \) and every odd multiple of \( \frac{\pi}{2} \).

The reason \( \tan(\frac{\pi}{2}) \) is undefined is because the cosine of \( \frac{\pi}{2} \) (the denominator in \( \tan(x) = \frac{\sin(x)}{\cos(x)} \)) is zero. This results in division by zero, causing the tangent function to approach infinity, thus making it undefined.

• Understand division by zero leads to undefined outcomes.
• Vertical asymptotes occur at angles where the tangent function has undefined values.

Recognizing these conditions helps avoid potential errors when solving trigonometric equations.