Problem 28 - Algebra and Trigonometry

Question

Find the exact value of the trigonometric function. $$ \tan \frac{5 \pi}{6} $$

Step-by-Step Solution

Answer

\( \tan \frac{5 \pi}{6} = -\frac{\sqrt{3}}{3} \).

Step 1: Understand the Angle

We need to find the value of \( \tan \frac{5 \pi}{6} \). The angle \( \frac{5 \pi}{6} \) is in radians and lies in the second quadrant because \( \frac{\pi}{2} < \frac{5\pi}{6} < \pi \).

Step 2: Know the Reference Angle

Since \( \frac{5 \pi}{6} \) is in the second quadrant, its reference angle is \( \pi - \frac{5 \pi}{6} \), which equals \( \frac{\pi}{6} \).

Step 3: Use the Tangent Identity

In the second quadrant, the tangent function is negative. Therefore, \( \tan \frac{5 \pi}{6} = -\tan \frac{\pi}{6} \).

Step 4: Calculate \( \tan \frac{\pi}{6} \)

The tangent of \( \frac{\pi}{6} \) is \( \frac{1}{\sqrt{3}} \) or its rationalized form \( \frac{\sqrt{3}}{3} \).

Step 5: Determine the Exact Value

Since \( \tan \frac{5 \pi}{6} = -\tan \frac{\pi}{6} \), we have \( \tan \frac{5 \pi}{6} = -\frac{\sqrt{3}}{3} \).

Key Concepts

Tangent Function

The tangent function, often abbreviated as "tan," is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the opposite side over the adjacent side. In the unit circle, the tangent of an angle \(\theta\) is given by the formula:\[tan(\theta) = \frac{\text{sin}(\theta)}{\text{cos}(\theta)}\]This formula means that tangent is the quotient of the y-coordinate (sine) and the x-coordinate (cosine) at a particular angle on the unit circle.

The tangent function has some unique properties:

• It is periodic with a period of \(\pi\), meaning it repeats values every \(\pi\) radians or 180 degrees.
• It is undefined for angles where cosine equals zero, such as \(\pi/2\), 3\(\pi/2\), etc.
• The function's range is all real numbers \((-\infty, \infty)\) as it can take any real value.

Understanding these properties is crucial when solving problems involving the tangent of an angle, especially when determining exact values based on known values of sine and cosine.

Reference Angle

Reference angle is a key concept in trigonometry that helps simplify the solution of trigonometric functions by taking into consideration only the acute angle. The reference angle is the smallest angle that the terminal side of the given angle makes with the x-axis. This angle is always positive and is between 0 and \(\pi/2 \) radians.

• To find a reference angle for an angle \(\theta\) in different quadrants:
  • In the first quadrant, the reference angle is simply \(\theta\).
  • In the second quadrant, it is \(\pi - \theta\).
  • In the third quadrant, use \(\theta - \pi\).
  • Finally, in the fourth quadrant, it's \(2\pi - \theta\).

The main idea behind using a reference angle is that trigonometric functions of any angle are based on the trigonometric functions of its reference angle, though they may have a positive or negative sign based on the quadrant. For instance, in our exercise, the reference angle for \(\frac{5\pi}{6}\) is \(\frac{\pi}{6}\). This tells us that the behavior of trigonometric functions is similar at \(\frac{5\pi}{6}\) and \(\frac{\pi}{6}\) angles, but their signs may differ due to the different quadrants.

Quadrants in Trigonometry

Quadrants help us determine the sign of trigonometric functions based on the position of the angle's terminal side in the coordinate plane. A circle is divided into four quadrants:

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

In our exercise, \(\frac{5\pi}{6}\) is located in the second quadrant. Thus, by these rules, \(\tan(\frac{5\pi}{6})\) turns out to be negative because tangent is negative in the second quadrant. This understanding aids significantly in quickly determining the sign of the trigonometric values without complex calculations.

Exact Values in Trigonometry

Exact values in trigonometry denote the known values of trigonometric functions for specific angles, typically reference angles like \(\pi/6, \pi/4,\) and \(\pi/3\). Memorizing these exact values is extremely helpful in solving trigonometric problems quickly and accurately without using a calculator.

For example:

• \(\tan(\pi/6) = \frac{1}{\sqrt{3}}\), and rationalized it is \(\frac{\sqrt{3}}{3}\).
• \(\tan(\pi/4) = 1\).
• \(\tan(\pi/3) = \sqrt{3}\).

By relating an unfamiliar angle to a familiar reference angle, one can use these known values to solve more complex problems, like in our exercise. Understanding that \(\tan(\frac{5\pi}{6}) = -\tan(\pi/6)\) relies on both knowing the exact value \(\tan(\pi/6) = \frac{1}{\sqrt{3}}\) and recognizing the quadrant's impact on the function's sign. Therefore, we conclude that \(\tan(\frac{5\pi}{6}) = -\frac{\sqrt{3}}{3}\).