Problem 71 - Algebra and Trigonometry

Question

use reference angles to find the exact value of each expression. Do not use a calculator. $$ \tan \frac{9 \pi}{4} $$

Step-by-Step Solution

Answer

The exact value of \(tan(\frac{9\pi}{4})\) is 1.

Step 1: Express the Given Angle in Terms of a Reference Angle

Because the period of the tangent function is \(\pi\), any angle \(θ\) and \(θ + \pi\) have the same tangent value. To find an angle equivalent to \(\frac{9\pi}{4}\) that lies in the interval \([0, \pi]\), express \(\frac{9\pi}{4}\) as \(\frac{8\pi}{4} + \frac{\pi}{4}\) . This is equal to \(2\pi + \frac{\pi}{4}\) , which means the angle is more than one full rotation of the unit circle plus an additional \(\frac{\pi}{4}\).

Step 2: Determine the Tangent of the Reference Angle

The additional \(\frac{\pi}{4}\) rotation puts the terminal side of the angle in the first quadrant, where \(x\) and \(y\) values are both positive. Thus the tangent value, being \(\frac{y}{x}\), is also positive. The tangent of an angle of \(\frac{\pi}{4}\) radian in the unit circle is 1, so \(tan(\frac{\pi}{4}) = 1\).

Step 3: Formulate the Final Answer

Since the reference angle is within the first rotation of the unit circle and maintains the same tangent value as the original angle, the exact value of \(tan(\frac{9\pi}{4})\) is the same as \(tan(\frac{\pi}{4})\), which is 1.

Key Concepts

Tangent Function

The tangent function is one of the basic functions in trigonometry, often symbolized as \( \tan(\theta) \), where \( \theta \) is an angle. It is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle. However, within the context of the unit circle, which is a circle of radius 1 centered at the origin of a coordinate system, \( \tan(\theta) \) can also be expressed as the ratio of the y-coordinate to the x-coordinate of a point on the circle's circumference.

To understand the behavior of the tangent function, it's important to know that it is periodic, with a period of \( \pi \). This means that the value of the tangent function repeats every \( \pi \) radians. Additionally, the tangent function has asymptotes, which are lines that the graph of the function approaches but never quite touches, occurring at \( \theta = \frac{(2n+1)\pi}{2} \), where \( n \) is any integer. These are the points where the cosine function, or the x-coordinate on the unit circle, equals zero, and thus the tangent function is undefined as the ratio would involve division by zero.

When using reference angles to find the value of the tangent function, it's essential to note which quadrant the terminal side of the angle is in because the sign of the tangent value will depend on the signs of the x and y coordinates in that quadrant. For instance, in the first and third quadrants, the tangent values are positive, while in the second and fourth quadrants, they are negative.

Unit Circle

The unit circle is an incredibly useful concept in trigonometry. It's a circle with a radius of precisely one unit that's centered at the origin (0, 0) on the coordinate plane. The significance of the unit circle comes from its ability to relate angles to coordinates effortlessly. By drawing a line from the origin to a point on the circumference, one creates an angle with the positive x-axis. The coordinates of the point (x, y) on the circumference are linked to the trigonometric functions of the angle \( \theta \).

The x-coordinate corresponds to the cosine of the angle, \( \cos(\theta) \), and the y-coordinate to the sine of the angle, \( \sin(\theta) \). Because of this relationship, the unit circle makes it straightforward to find the values of sine and cosine for angles, which in turn, aids in determining the value of the tangent function as \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \), provided that \( \cos(\theta) \) is not zero.

When dealing with trigonometric calculations, especially with those involving reference angles, the unit circle emerges as a powerful tool. It not only helps in visualizing the angles and their corresponding trigonometric values but also simplifies finding the values of trigonometric functions for angles greater than \( 2\pi \) or less than \( 0 \), by essentially 'wrapping' the angle around the circle as many times as necessary.

Exact Trigonometric Values

In trigonometry, when we refer to exact values, we mean the precise trigonometric values that can be derived without the approximation typically associated with calculator computations. These values are particularly useful for special angles such as \( 0 \), \( \frac{\pi}{6} \), \( \frac{\pi}{4} \), \( \frac{\pi}{3} \), and \( \frac{\pi}{2} \) radians, correspondingly 0, 30, 45, 60, and 90 degrees. Knowing the exact values of the sine, cosine, and tangent functions for these angles is crucial for solving trigonometric problems efficiently.

These exact values are often memorized using various mnemonics or the unit circle. For instance, the tangent of the commonly referred angle \( \frac{\pi}{4} \) is one of these exact values. Since at this angle, the x and y coordinates are equal (because \( \sin(\frac{\pi}{4}) = \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \)), the tangent, which is the ratio \( \frac{y}{x} \), is exactly 1. Recognizing these key exact values is vital not just for academic purposes but also for their applications in fields such as physics, engineering, and computer graphics, where precise calculations are necessary.