Problem 8 - Algebra 2

Question

Use the graph of \(y=\tan \theta\) to find each value. If the tangent is undefined at that point, write undefined. $$ \tan \frac{3 \pi}{2} $$

Step-by-Step Solution

Answer

The value of \( \tan \frac{3\pi}{2} \) is undefined.

Step 1: Identification of the necessary value

First of all, recognize the required value of \( \theta \) which is \( \frac{3\pi}{2} \).

Step 2: Recalling the behavior of the tangent function

Remember that the function \(y=\tan \theta\) is not defined at \( \frac{\pi}{2} + k\pi \) where \(k\) is an integer.

Step 3: Compare \(\theta\) with the range of undefined values

Since \( \frac{3\pi}{2} = \frac{\pi}{2} + \pi \), this falls into the category where the tangent function is undefined.

Key Concepts

Tangent Function

The tangent function, often written as \( y = \tan \theta \), is one of the primary trigonometric functions. It represents the ratio of the sine function to the cosine function. In other words, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Understanding this function is crucial for solving many trigonometric problems.

• **Periodic Nature:** The tangent function is periodic with a period of \( \pi \). This means it repeats its pattern every \( \pi \) units.
• **Graph Characteristics:** The graph of \( y = \tan \theta \) has asymptotes, which are vertical lines where the function is undefined. It appears as a series of rising lines that shoot up to infinity and then continue from negative infinity.
• **Usage:** It's often used in right triangles to find unknown side lengths and angles, especially in the context of slope and angle measures.

Although the tangent function can seem tricky because of its undefined values, understanding its properties can greatly help you in tackling related math problems.

Radians

Radians are an alternative to degrees for measuring angles. While degrees divide a circle into 360 parts, radians use the radius of the circle to define angle measures. This method can be more natural in many mathematical contexts.

• **Definition:** One full revolution around a circle is \( 2\pi \) radians, which equals 360 degrees. Hence, \( \pi \) radians equals 180 degrees.
• **Conversion:** To convert degrees to radians, use the formula \( \, \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \, \). Conversely, to convert radians to degrees, multiply by \( \frac{180}{\pi} \).

This scale is essential for expressing angles in calculus, trigonometry, and other advanced mathematics fields. Understanding radians can simplify computations, especially involving trigonometric functions.

Undefined Values

In trigonometry, certain functions, like the tangent function, can be undefined for some angle measures. These so-called "undefined values" occur due to the division by zero problem in trigonometric equations.

• **Tangent Function:** The tangent function \( \tan \theta \) becomes undefined whenever \( \cos \theta = 0 \), as this translates to dividing by zero.
• **Identifying Undefined Points:** For \( \tan \theta \), values are undefined at \( \theta = \frac{\pi}{2} + k\pi \), where \( k \) is any integer. This pattern repeats every \( \pi \) radians due to the function's periodic nature.
• **Graph Implications:** On the graph of \( \tan \theta \), these undefined values correspond to vertical asymptotes, which are breaks where the line approaches but never actually meets.

Remembering where trigonometric functions become undefined helps in accurately sketching graphs and interpreting the behavior of these functions across different intervals.