Problem 15 - Algebra and Trigonometry

Question

Find the exact value of the trigonometric function at the given real number. $$ \begin{array}{llll}{\text { (a) } \tan \frac{5 \pi}{6}} & {\text { (b) } \tan \frac{7 \pi}{6}} & {\text { (c) } \tan \frac{11 \pi}{6}}\end{array} $$

Step-by-Step Solution

Answer

(a) \(-\frac{\sqrt{3}}{3}\), (b) \(\frac{\sqrt{3}}{3}\), (c) \(-\frac{\sqrt{3}}{3}\).

Step 1: Determine the Reference Angle for (a)

For \( \tan \frac{5\pi}{6} \), we start by finding the reference angle. The angle \( \frac{5\pi}{6} \) is in the second quadrant. The reference angle is \( \pi - \frac{5\pi}{6} = \frac{\pi}{6} \).

Step 2: Identify the Sign of Tangent for (a)

In the second quadrant, the tangent function is negative. Thus, \( \tan \frac{5\pi}{6} = - \tan \frac{\pi}{6} \). Since \( \tan \frac{\pi}{6} = \frac{1}{\sqrt{3}} \) or \( \frac{\sqrt{3}}{3} \), \( \tan \frac{5\pi}{6} = -\frac{\sqrt{3}}{3} \).

Step 3: Determine the Reference Angle for (b)

For \( \tan \frac{7\pi}{6} \), the angle is in the third quadrant. The reference angle here is \( \frac{7\pi}{6} - \pi = \frac{\pi}{6} \).

Step 4: Identify the Sign of Tangent for (b)

In the third quadrant, tangent is positive. So, \( \tan \frac{7\pi}{6} = \tan \frac{\pi}{6} = \frac{\sqrt{3}}{3} \).

Step 5: Determine the Reference Angle for (c)

For \( \tan \frac{11\pi}{6} \), we find that the angle is in the fourth quadrant. The reference angle is \( 2\pi - \frac{11\pi}{6} = \frac{\pi}{6} \).

Step 6: Identify the Sign of Tangent for (c)

In the fourth quadrant, the tangent is negative. Therefore, \( \tan \frac{11\pi}{6} = -\tan \frac{\pi}{6} = -\frac{\sqrt{3}}{3} \).

Key Concepts

Tangent Function

The tangent function, often denoted as \( \tan \theta \), is one of the six primary trigonometric functions. It relates the angle in a right triangle to the ratio of the opposite side over the adjacent side. In the unit circle, it is represented as the ratio of the sine and cosine of an angle, given by \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

• It shows periodic behavior, with a period of \( \pi \), meaning that \( \tan(\theta + \pi) = \tan \theta \).
• Unlike sine and cosine, the tangent function can take any real value, and it is undefined where \( \cos \theta = 0 \).

Understanding the tangent function helps in solving trigonometric equations and analyzing periodic phenomena in both mathematics and physical sciences.

Reference Angle

A reference angle is the smallest angle a given angle makes with the x-axis. It is always acute, meaning that it is between 0 and \( \frac{\pi}{2} \) radians (or 0 and 90 degrees).

• It is essential because it allows the calculation of trigonometric functions of any angle using their known values for acute angles.
• The reference angle can be found by subtracting the given angle from \( \pi \) or \( 2\pi \), depending on which quadrant the angle lies in. For instance, for an angle in the second quadrant \( \text{(like } \frac{5\pi}{6}\text{)} \), the reference angle is \( \pi - \frac{5\pi}{6} = \frac{\pi}{6} \).

Knowing the reference angle simplifies complex angle problems in trigonometry, making any angle manageable by reducing it to a more familiar acute case.

Quadrant Analysis

Quadrant analysis is the study of how the signs of trigonometric functions vary according to the quadrant in which their angles lie. The unit circle is divided into four quadrants:

• First Quadrant (0 to \( \frac{\pi}{2} \)): All trigonometric functions are positive.
• Second Quadrant (\( \frac{\pi}{2} \) to \( \pi \)): Sine is positive, but cosine and tangent are negative.
• Third Quadrant (\( \pi \) to \( \frac{3\pi}{2} \)): Tangent is positive, while sine and cosine are negative.
• Fourth Quadrant (\( \frac{3\pi}{2} \) to \( 2\pi \)): Cosine is positive, and sine and tangent are negative.

Through quadrant analysis, understanding the sign of each trigonometric function becomes straightforward, aiding in determining values such as \( \tan \frac{11\pi}{6} = -\frac{\sqrt{3}}{3} \) when working with angles in different quadrants.

Exact Values

Exact values in trigonometry refer to the precise, accurate values of trigonometric functions at specific angles. These angles usually include \( \frac{\pi}{6} \), \( \frac{\pi}{4} \), \( \frac{\pi}{3} \), etc., which correspond to 30°, 45°, and 60° in degrees.

• These values are derived from geometric relations of special triangles like the 30-60-90 and 45-45-90 triangles.
• Using these exact values, it becomes possible to compute the trigonometric functions of any angle by recognizing its reference angle and corresponding quadrant.

For example, knowing \( \tan \frac{\pi}{6} = \frac{\sqrt{3}}{3} \) allows you to determine other values like \( \tan \frac{7\pi}{6} = \frac{\sqrt{3}}{3} \) or \( \tan \frac{11\pi}{6} = -\frac{\sqrt{3}}{3} \) by considering their position on the unit circle. Exact values enable students to work with trigonometric functions more effectively, without relying on a calculator.