Problem 21 - Algebra and Trigonometry

Question

Use one or more of the six sum and difference identities to solve Exercises \(13-54\) Find the exact value of each expression. $$ \tan \left(\frac{\pi}{6}+\frac{\pi}{4}\right) $$

Step-by-Step Solution

Answer

The exact value for \( \tan \left( \frac{\pi}{6} + \frac{\pi}{4}\right) \) is \( \sqrt{3} + 3\).

Step 1: Substitute given angles into the identity

Substitute \( \alpha = \frac{\pi}{6} \) and \( \beta = \frac{\pi}{4} \) into the identity to get the equation: \( \tan \left( \frac{\pi}{6} + \frac{\pi}{4}\right) = \frac{\tan(\frac{\pi}{6}) + \tan(\frac{\pi}{4})}{1 - \tan(\frac{\pi}{6})\tan(\frac{\pi}{4})} \).

Step 2: Calculate the tan values

From trigonometric standard angles, we know that \( \tan(\frac{\pi}{6}) = \sqrt{3}/3 \) and \( \tan(\frac{\pi}{4}) = 1\). Sub in these values to get: \( \tan \left( \frac{\pi}{6} + \frac{\pi}{4}\right) = \frac{\sqrt{3}/3 + 1}{1 - \sqrt{3}/3}\).

Step 3: Simplify the expression

After carrying out the arithmetic operations in the numerator and denominator, we get: \( \tan \left( \frac{\pi}{6} + \frac{\pi}{4}\right) = \sqrt{3} + 3 = 0.\).

Key Concepts

Exact Value of Trigonometric Expressions

Understanding the exact value of trigonometric expressions is essential for solving problems in trigonometry with precision. In trigonometry, the 'exact value' refers to the solution of a trigonometric function that is expressed as a precise number, rather than a decimal approximation. These values are often found using the unit circle or trigonometric identities.

To calculate the exact value of a trigonometric function, especially that of more complex expressions like \( \tan \big(\frac{\fpi}{6}+\frac{\fpi}{4}\big) \), we can make use of known identities and properties of trigonometric functions. For instance, the sum and difference identities allow us to find the tangent of a sum of angles by relating it to the tangents of the individual angles.

Tan Identity

While solving for the tangent of an angle created by the addition of two standard angles, sum and difference identities are vital. The tan identity for the sum of two angles, \( \tan(\theta) \textrm{ and } \fphi) \), is given as \( \tan(\theta + \fphi) = \frac{\tan(\theta) + \tan(\fphi)}{1 - \tan(\theta)\tan(\fphi)} \). Utilizing this identity simplifies the process of finding exact trigonometric values.

The tan identity showcases how the sum of two angles within the tangent function can be decomposed into individual components. This is particularly useful when dealing with the addition of common angles for which the tangent values are known, such as the standard angles \( \frac{\fpi}{6} \textrm{ and } \fpi}{4}) \). By substituting these known tangent values into the identity, we can find the exact value of the expression without resort to calculators or approximate values.

Trigonometric Standard Angles

Trigonometric standard angles, like \( 0, \frac{\fpi}{6}, \frac{\fpi}{4}, \frac{\fpi}{3}, \frac{\fpi}{2}\), and their multiples, play a significant role in simplifying trigonometric expressions. The tangents of these fundamental angles are commonly memorized values, enabling the exact computation of more complex angles.

Known trigonometric ratios for these standard angles facilitate the application of various trigonometric identities, including those involving the tangent function. By recognizing that \( \tan(\frac{\fpi}{6}) = \frac{\fsqrt{3}}{3} \) and \( \tan(\frac{\fpi}{4}) = 1 \), one can easily embark on solving for an expression that incorporates these or their sum, making a clear path to the exact value without relying on approximation or numerical methods.