Problem 72 - Algebra and Trigonometry

Question

use reference angles to find the exact value of each expression. Do not use a calculator. $$ \tan \frac{9 \pi}{2} $$

Step-by-Step Solution

Answer

The exact value of \(\tan(\frac{9 \pi}{2})\) is 'undefined'

Step 1: Find the Associated Acute Reference Angle

First, convert the given angle \(\frac{9 \pi}{2}\) into a form that is between \(0\) and \(2\pi\), in order to find an equivalent acute reference angle. To do this, use the periodic nature of the tangent function. The period of the tangent function is \(\pi\), which means every \(\pi\), the function repeats itself. So, subtract multiple of \(\pi\) from \(\frac{9 \pi}{2}\) until it is less than or equal to \(\pi\). You will find that \(\frac{9 \pi}{2} - 4\pi = \frac{1 \pi}{2}\). So, the acute reference angle is \(\frac{\pi}{2}\).

Step 2: Apply the Properties of the Tangent Function

Next, apply the reference angle to the tangent function. The tangent of an angle in the second quadrant is negative. Therefore, \(\tan(\frac{9 \pi}{2}) = \tan(\frac{\pi}{2}) = undefined\). Note that \(\tan(\frac{\pi}{2}) = undefined\) because the cosine of \(\frac{\pi}{2}\) is zero and tangent equals sine divided by cosine.

Key Concepts

Tangent Function Periodicity

Understanding the periodic nature of the tangent function is crucial in solving trigonometric problems without a calculator. The key feature of the tangent function is that it repeats its values after a specific interval called the period, which for tangent, is \(\pi\) radians. This essentially means that for any angle \(\theta\), the value of \(\tan(\theta)\) will be the same as \(\tan(\theta + n\pi)\) for every integer \(n\).

In the example of finding \(\tan\frac{9\pi}{2}\), it's important to subtract multiples of \(\pi\) to find an angle that is co-terminal with \(\frac{9\pi}{2}\) and lies within the fundamental period from \(0\) to \(\pi\). This concept applies not just to homework exercises, but also provides a foundational understanding for further study in trigonometry, physics, and engineering where waveform periodicity is a recurring theme.

Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent are fundamental in mathematics, especially when dealing with right triangles and circular motion. Each function provides a relationship between the angles of a triangle and its side lengths.

Sine is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse, while cosine is the ratio of the length of the adjacent side to the hypotenuse. Tangent, which we focus on in the discussed exercise, is the ratio of the sine and cosine of the same angle. It's written as \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\).

When working with these functions, it's essential to remember that they are defined based on the unit circle, and each has distinct properties and graphs. For instance, while sine and cosine are defined for all real numbers, tangent has points of discontinuity (like at \(\frac{\pi}{2}\) and \(\frac{3\pi}{2}\)) where the function is undefined, which corresponds to vertical asymptotes on its graph.

Exact Trigonometric Values

Exact trigonometric values are preferred over decimal approximations in many mathematical contexts because they maintain precision. Some angles, such as 0, \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{3}\), and \(\frac{\pi}{2}\), have exact values that can be derived geometrically or through the unit circle. For example, \(\sin(\frac{\pi}{6})\) is exactly \(\frac{1}{2}\), and \(\cos(\frac{\pi}{3})\) is exactly \(\frac{1}{2}\).

These exact values are building blocks for solving more complex trigonometric equations and for understanding the behavior of trigonometric functions. In practice, this means you can evaluate the trigonometric functions of many angles without a calculator, which is essential in standardized tests and when working with theoretical problems. For our case, the reference angle \(\frac{\pi}{2}\) has an exact value for sine, but leads to an undefined value for tangent due to a zero in the denominator, showing the importance of knowing these exact values and their implications.