Problem 26 - Algebra and Trigonometry

Question

\(17-28\) Use an appropriate Half-Angle Formula to find the exact value of the expression. $$ \tan \frac{5 \pi}{12} $$

Step-by-Step Solution

Answer

The exact value is \(-2 - \sqrt{3}\).

Step 1: Identify Reference Angles

Recognize that \( \frac{5\pi}{12} \) is not a standard angle, so we need to express it as a sum or difference of standard angles.\( \frac{5\pi}{12} = \frac{\pi}{3} + \frac{\pi}{4} \) because \( \frac{5}{12} = \frac{4}{12} + \frac{3}{12} \).

Step 2: Use Sum Formula for Tangent

Use the formula for the tangent of the sum of two angles: \( \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \cdot \tan B} \). Substitute \( A = \frac{\pi}{3} \) and \( B = \frac{\pi}{4} \).

Step 3: Evaluate Individual Tangents

Calculate \( \tan \frac{\pi}{3} = \sqrt{3} \) and \( \tan \frac{\pi}{4} = 1 \).

Step 4: Substitute and Simplify

Substitute these values into the sum formula: \( \tan \left( \frac{\pi}{3} + \frac{\pi}{4} \right) = \frac{\sqrt{3} + 1}{1 - (\sqrt{3} \times 1)} \).

Step 5: Simplify the Expression

Simplify the expression: \( \frac{\sqrt{3} + 1}{1 - \sqrt{3}} \). Multiply numerator and denominator by the conjugate of the denominator \( 1 + \sqrt{3} \) to remove the square root from the denominator.

Step 6: Calculate the Fraction

The multiplication gives \( (\sqrt{3} + 1)(1 + \sqrt{3}) = 3 + \sqrt{3} + \sqrt{3} + 1 = 4 + 2\sqrt{3} \) in the numerator and \((1-\sqrt{3})(1+\sqrt{3}) = 1 - 3 = -2 \) in the denominator.

Step 7: Final Simplification

Simplify \( \frac{4 + 2\sqrt{3}}{-2} = -2 - \sqrt{3} \).

Key Concepts

Tangent Sum Formula

The Tangent Sum Formula is a key tool in trigonometry, especially when dealing with angles that are not standard, such as \( \frac{5\pi}{12} \). To express tangent for the sum of two angles, we use the formula:

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

This formula allows us to break down a complex angle into simpler, standard angles. For instance, in our problem, \( \frac{5\pi}{12} \) is not a typical angle we easily recognize. However, we can split it into \( \frac{\pi}{3} \) and \( \frac{\pi}{4} \), both of which have known tangent values. Thus, applying the sum formula helps us calculate the tangent of a composite angle using more straightforward computations.

Reference Angles

Reference Angles are an essential concept when dealing with non-standard angles. They help simplify calculations by relating these angles to known ones within the first quadrant. Every angle can be associated with a reference angle, which is the acute angle that any given angle makes with the x-axis.
In the exercise, \( \frac{5\pi}{12} \) is not an angle we usually deal with directly. By expressing it as the sum of \( \frac{\pi}{3} \) and \( \frac{\pi}{4} \), we essentially find a way to reference it back to standard angles that are easy to work with,
allowing the use of known trigonometric values and properties. This makes solving trigonometric equations more manageable and efficient.

Trigonometric Identities

Trigonometric Identities are fundamental in simplifying and solving trigonometric equations. They are formulas expressing a relationship between the trigonometric functions. In this exercise, the identity used is the Tangent Sum Formula. However, there are numerous others, such as Pythagorean identities, which include:

• \( \sin^2\theta + \cos^2\theta = 1 \)
• \( 1 + \tan^2\theta = \sec^2\theta \)
• \( 1 + \cot^2\theta = \csc^2\theta \)

These identities allow us to connect various trigonometric functions and simplify expressions or solve equations that appear complex at first glance. By understanding and applying these identities, we can transform complicated trigonometric problems into more accessible parts.

Exact Trigonometric Values

Exact Trigonometric Values are indispensable for calculations in trigonometry. They give precise results rather than approximations, which can be important for ensuring accuracy in more complex calculations. For standard angles like \( \frac{\pi}{3} \), \( \frac{\pi}{4} \), and others, these values are readily available:

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

Having these values memorized or easily accessible allows quick insertion into formulas like the Tangent Sum Formula. Utilizing exact values is not just a short-cut, but a necessity to ensure that the final result is as precise as possible, especially important in high-stakes mathematical computations or applications.