Problem 18 - Algebra and Trigonometry

Question

Use an Addition or Subtraction Formula to write the expression as a trigonometric function of one number, and then find its exact value. $$ \frac{\tan \frac{\pi}{18}+\tan \frac{\pi}{9}}{1-\tan \frac{\pi}{18} \tan \frac{\pi}{9}} $$

Step-by-Step Solution

Answer

The exact value is \( \frac{\sqrt{3}}{3} \).

Step 1: Identify the Addition Formula for Tangent

The given expression resembles the tangent addition formula: \( \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \). Our goal is to recognize \( a \) and \( b \) such that the expression can be simplified using this formula.

Step 2: Match the Expression to the Formula

Here, \( a = \frac{\pi}{18} \) and \( b = \frac{\pi}{9} \). Therefore, using the formula we have: \( \frac{\tan \frac{\pi}{18} + \tan \frac{\pi}{9}}{1 - \tan \frac{\pi}{18} \tan \frac{\pi}{9}} = \tan\left(\frac{\pi}{18} + \frac{\pi}{9}\right) \).

Step 3: Simplify the Angle

Calculate \( \frac{\pi}{18} + \frac{\pi}{9} \) as \( \frac{\pi}{18} + \frac{2\pi}{18} = \frac{3\pi}{18} \). Simplifying gives \( \frac{\pi}{6} \).

Step 4: Determine the Exact Value

Now evaluate \( \tan\left(\frac{\pi}{6}\right) \). We know from trigonometric identities that \( \tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} \).

Key Concepts

Addition Formula

The addition formula for the tangent function is a marvelous tool for simplifying expressions involving tangents of multiple angles. It is expressed as: \[ \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \].This formula allows us to take two angles, add them together, and find the tangent of the resulting angle.
This is particularly useful in scenarios where you want to transform an expression into a simpler form to find an exact trigonometric value.
In our original exercise, identifying that the expression matches the tangent addition formula helps us simplify it into the tangent of a single angle, making it easier to find the exact value.

Subtraction Formula

While our problem focuses on addition, the subtraction formula for tangent is equally important. It is given by: \[ \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} \]. This formula helps decode expressions where the tangent of the difference between two angles is required.
Understanding both the addition and subtraction formulas is vital as they often appear together in more complex problems.
Both formulas help in transforming complex trigonometric problems into simpler ones. They reveal a wide range of interconnected trigonometric identities, demonstrating how powerful these identities can be in mathematical simplification.

Tangent Function

The tangent function, denoted as \( \tan(\theta) \), represents the ratio of the opposite side to the adjacent side in a right-angled triangle.
It is one of the basic trigonometric functions and is pivotal in solving problems involving angles and side lengths.
In our task, identifying the tangent function in the provided expression enabled us to apply the addition formula, linking the angles \( \frac{\pi}{18} \) and \( \frac{\pi}{9} \).

• Positive for angles in the first and third quadrants.
• Negative for angles in the second and fourth quadrants.
• Periodic with a period of \( \pi \).

Understanding these properties aids in predicting the behavior of the tangent function across different angles.

Exact Trigonometric Values

Exact trigonometric values are special because they provide precise measurements without approximation.
These are typically derived from special angles like \( \frac{\pi}{6} \), \( \frac{\pi}{4} \), and \( \frac{\pi}{3} \), among others.
For instance, the value of \( \tan\left(\frac{\pi}{6}\right) \) is \( \frac{\sqrt{3}}{3} \).

• Memorizing these values can save time.
• Crucial for solving problems in pure mathematics where precision is key.
• They also provide insights into the behavior of trigonometric functions.

In our exercise, knowing the exact value of \( \tan\left(\frac{\pi}{6}\right) \) helped us conclude the problem with an exact, precise result.

Angle Simplification

Angle simplification is a method used to reduce complex angle expressions into simpler forms.
This involves using basic algebraic identities and formulas to combine and simplify angle measures.
In our example, we combined \( \frac{\pi}{18} \) and \( \frac{\pi}{9} \) into \( \frac{\pi}{6} \) using simple arithmetic.

• This process not only simplifies calculations but also helps in identifying standard angles for which exact values are known.
• It's a crucial technique in trigonometry, particularly when dealing with complex equations.

Understanding angle simplification aids in recognizing patterns, making it easier to solve trigonometric equations.