Problem 16 - Algebra and Trigonometry

Question

\(13-18\) m 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 expression simplifies to \(\frac{\sqrt{3}}{3}\).

Step 1: Identify the formula to use

The given expression is of the form \( \frac{\tan A + \tan B}{1 - \tan A \tan B} \), which corresponds to the tangent addition formula \( \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \).

Step 2: Assign angles to A and B

Identify \(A\) and \(B\) in our problem: \(A = \frac{\pi}{18}\) and \(B = \frac{\pi}{9}\).

Step 3: Simplify the expression

Using the tangent addition formula, the expression simplifies to \( \tan(A + B) = \tan(\frac{\pi}{18} + \frac{\pi}{9}) \).

Step 4: Find the sum of angles

Calculate \( \frac{\pi}{18} + \frac{\pi}{9} \). Convert \(\frac{\pi}{9}\) into a common denominator: \(\frac{2\pi}{18}\). Thus, the sum is \(\frac{\pi}{18} + \frac{2\pi}{18} = \frac{3\pi}{18} = \frac{\pi}{6}\).

Step 5: Calculate the exact value

Now find \(\tan(\frac{\pi}{6})\). The tangent of \(\frac{\pi}{6}\) is \(\frac{1}{\sqrt{3}}\) or \(\frac{\sqrt{3}}{3}\).

Key Concepts

Tangent Addition Formula

The Tangent Addition Formula is a powerful identity in trigonometry that helps us evaluate the tangent of the sum of two angles. It states that if you have two angles, say \( A \) and \( B \), then:

• \[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]

This formula allows us to express the tangent of a sum of angles using the tangents of the individual angles, which can often be more straightforward to calculate.
This formula is particularly useful in problems where you need to simplify expressions involving tangents, as seen in the original exercise. By using this identity, we can turn a potentially complex fraction into a single argument for the tangent function. This reduces computational complexity and can sometimes bring us to a standard angle where the value is known.

Angle Sum Identities

Angle Sum Identities are invaluable tools in trigonometry that help us find the sine, cosine, and tangent of sums (or differences) of angles. In addition to the tangent identity, they include:

• Sine addition: \( \sin(A + B) = \sin A \cos B + \cos A \sin B \)
• Cosine addition: \( \cos(A + B) = \cos A \cos B - \sin A \sin B \)

The tangent addition formula is derived from these identities and gives a direct method to solve for \( \tan(A + B) \).
Using these identities requires practice and a solid understanding of basic trigonometric relationships. They simplify complex expressions and help reduce them to simpler forms. This is particularly handy when dealing with known values that apply to frequently encountered angles, like those in the unit circle such as \( \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3} \), etc.

Exact Trigonometric Values

Exact Trigonometric Values are the key numbers that come from trigonometric functions like sine, cosine, and tangent at specific angles. These values are crucial for simplifying expressions and solving trigonometric equations without needing a calculator.

• For example, the tangent of \( \frac{\pi}{6} \) is \( \frac{1}{\sqrt{3}} \), often simplified to \( \frac{\sqrt{3}}{3} \).

Similarly, knowing the sine and cosine of typical angles like \( \frac{\pi}{4} \) or \( \frac{\pi}{3} \) helps in swift calculations. Memorizing these values can make solving trigonometric problems on tests or homework quicker and easier.
In the original exercise, knowing the exact value of \( \tan(\frac{\pi}{6}) \) allowed us to conclude the problem efficiently. Hence, becoming familiar with these key values can aid significantly in understanding and applying trigonometric principles.