Problem 6 - A Graphical Approach to Precalculus with Limits

Question

Use identities to find the exact value of each expression. Do not use a calculator. $$\tan \frac{\pi}{12}$$

Step-by-Step Solution

Answer

The exact value of \( \tan \frac{\pi}{12} \) is \( 2 - \sqrt{3} \).

Step 1: Recognize the Identity Needed

To find the exact value of \( \tan \frac{\pi}{12} \), we can use the tangent subtraction identity: \( \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} \). Since \( \frac{\pi}{12} = \frac{\pi}{4} - \frac{\pi}{6} \), we'll apply this identity using these angles.

Step 2: Apply Known Values of Tangents

We know that \( \tan \frac{\pi}{4} = 1 \) and \( \tan \frac{\pi}{6} = \frac{1}{\sqrt{3}} \). Substitute these into the tangent subtraction identity: \( \tan \left( \frac{\pi}{4} - \frac{\pi}{6} \right) = \frac{1 - \frac{1}{\sqrt{3}}}{1 + 1 \cdot \frac{1}{\sqrt{3}}} \).

Step 3: Simplify the Expression

Simplify the expression: \( \frac{1 - \frac{1}{\sqrt{3}}}{1 + \frac{1}{\sqrt{3}}} = \frac{\sqrt{3} - 1}{\sqrt{3} + 1} \).

Step 4: Rationalize the Denominator

To simplify further, multiply the numerator and the denominator by the conjugate of the denominator \( \sqrt{3} - 1 \): \[ \frac{(\sqrt{3} - 1)(\sqrt{3} - 1)}{(\sqrt{3} + 1)(\sqrt{3} - 1)} = \frac{(\sqrt{3} - 1)^2}{2} \].

Step 5: Expand and Simplify Again

Expand the numerator: \((\sqrt{3} - 1)^2 = 3 - 2\sqrt{3} + 1 = 4 - 2\sqrt{3} \). So, the expression becomes: \[ \frac{4 - 2\sqrt{3}}{2} = 2 - \sqrt{3} \].

Step 6: Conclude with the Exact Value

The exact value of \( \tan \frac{\pi}{12} \) is \( 2 - \sqrt{3} \). This concludes our solution using trigonometric identities without a calculator.

Key Concepts

Tangent Subtraction Identity

The tangent subtraction identity is a powerful tool in trigonometry that allows you to find the tangent of an angle that is the difference between two known angles. It is particularly used when calculating the exact value of angles using standard trigonometric measures. The identity states as follows:

• \( \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} \)

Here, \( a \) and \( b \) are angles for which you already know their tangent values. For example, when you want to find \( \tan \frac{\pi}{12} \), you can break it down into \( \frac{\pi}{4} - \frac{\pi}{6} \), for which the tangent values are known. This approach simplifies the calculations and helps in finding the exact value of complex angles without using a calculator.

Exact Values in Trigonometry

Understanding exact values in trigonometry is a valuable skill. This enables you to solve trigonometric problems without needing a calculator, which is crucial for exams and theoretical studies. The key is to remember the tangent, sine, and cosine values of common angles, such as \( \frac{\pi}{4} \), \( \frac{\pi}{6} \), and \( \frac{\pi}{3} \). These angles are often used in conjunction with identities to deduce the tangent, sine, or cosine of new angles.For instance:

• \( \tan \frac{\pi}{4} = 1 \)
• \( \tan \frac{\pi}{6} = \frac{1}{\sqrt{3}} \)

These exact values are substituted in identities to derive the values of more complex angles. For \( \frac{\pi}{12} \), the known values are placed in the tangent subtraction identity, which helps in reducing the problem to algebraic simplification.

Rationalizing Denominators

Rationalizing the denominator is a process used to remove square roots or irrational numbers from the denominator of fractions. This makes expressions easier to interpret and calculate, especially within standard mathematical forms. The key step involves multiplying the numerator and the denominator by the conjugate of the denominator.For example, consider an expression like \( \frac{\sqrt{3} - 1}{\sqrt{3} + 1} \). To rationalize:

• Multiply both the numerator and denominator by the conjugate: \( \sqrt{3} - 1 \)
• Hence: \( \frac{(\sqrt{3} - 1)(\sqrt{3} - 1)}{(\sqrt{3} + 1)(\sqrt{3} - 1)} = \frac{(\sqrt{3} - 1)^2}{2} \)

Upon expanding the numerator and simplifying, we get a final expression \( 2 - \sqrt{3} \). This form is much neater and usually preferred when expressing final trigonometric exact values.