Problem 18 - Precalculus Mathematics for Calculus

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 expression simplifies to \( \frac{\sqrt{3}}{3} \).

Step 1: Identify the Addition Formula

The given expression is \( \frac{\tan \frac{\pi}{18} + \tan \frac{\pi}{9}}{1 - \tan \frac{\pi}{18} \tan \frac{\pi}{9}} \). This has the form of the tangent addition formula: \( \tan(A + B) = \frac{\tan(A) + \tan(B)}{1 - \tan(A) \tan(B)} \). Recognize that the expression represents \( \tan(\frac{\pi}{18} + \frac{\pi}{9}) \).

Step 2: Simplify the Angle Sum

Calculate \( \frac{\pi}{18} + \frac{\pi}{9} \) by finding a common denominator. Convert \( \frac{\pi}{9} \) to \( \frac{2\pi}{18} \): \[ \frac{\pi}{18} + \frac{2\pi}{18} = \frac{3\pi}{18} \]. Simplify the fraction: \( \frac{3\pi}{18} = \frac{\pi}{6} \).

Step 3: Evaluate the Trigonometric Function

The expression simplifies to \( \tan(\frac{\pi}{6}) \). Use the known value of \( \tan(\frac{\pi}{6}) \), which is \( \frac{1}{\sqrt{3}} \).

Step 4: Simplification of Known Tangent Value

Further simplify \( \frac{1}{\sqrt{3}} \) to its exact form. The simplified form of \( \frac{1}{\sqrt{3}} \) is \( \frac{\sqrt{3}}{3} \), after rationalizing the denominator.

Key Concepts

Trigonometric Functions

Trigonometric functions are essential mathematical tools that relate the angles of a right triangle to the ratios of its sides. In trigonometry, the most commonly used functions are sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)). Each function offers a unique perspective under different contexts, but they primarily work together to describe geometric properties and periodic phenomena.

• Sine Function (\( \sin \)): Represents the ratio of the length of the opposite side to the hypotenuse of a right-angled triangle.
• Cosine Function (\( \cos \)): Denotes the ratio of the length of the adjacent side to the hypotenuse.
• Tangent Function (\( \tan \)): Is the quotient of the sine and cosine, representing the ratio of the opposite side to the adjacent side.

Trigonometric functions aren't just for triangles. They're pivotal in modeling waves, oscillations, and other periodic patterns in disciplines like physics, engineering, and even economics. By mastering these functions, you open the door to a deeper understanding of how they describe both simple and complex systems.

Angle Sum Identity

The angle sum identity is a critical formula in trigonometry, providing a method to find the values of trigonometric functions for the sums or differences of angles. Specifically, the tangent addition formula is a vital identity:For two angles, \( A \) and \( B \), the tangent addition formula is expressed as:\[\tan(A + B) = \frac{\tan(A) + \tan(B)}{1 - \tan(A) \tan(B)}\]This formula is particularly useful when you want to simplify expressions or solve problems where angles are combined. In our example, the expression \( \frac{\tan \frac{\pi}{18} + \tan \frac{\pi}{9}}{1 - \tan \frac{\pi}{18} \tan \frac{\pi}{9}} \) is reshaped using this formula. By recognizing this form, the expression can be simplified to \( \tan (\frac{\pi}{18} + \frac{\pi}{9}) \). This not only makes computations easier but also allows for determining exact trigonometric values for specific angles.

Exact Trigonometric Values

Exact trigonometric values are specific, well-known values of trigonometric functions for certain standard angles. These include angles like \( 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2} \), etc. These values are fundamental in trigonometry and help in solving various problems without the need for a calculator.

• \( \tan(\frac{\pi}{6}) \): This is one of the exact values, specifically \( \frac{1}{\sqrt{3}} \), which simplifies to \( \frac{\sqrt{3}}{3} \) after rationalizing the denominator.
• Memorization Tip: Remember the acronym "SOHCAHTOA" to recall sine, cosine, and tangent relationships, and use practice to familiarize yourself with these exact values.

Such exact values are crucial because they frequently appear in mathematical proofs and exercises, allowing for simplification and precise calculation without computational tools. Understanding and memorizing these standard angle values enriches your trigonometric knowledge and problem-solving skills.