Problem 25 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

\( \cos \frac{\pi}{12} = \frac{\sqrt{2 + \sqrt{3}}}{2} \).

Step 1: Recognize the Half-Angle Identity

To solve this problem, we use the half-angle identity for cosine. The identity states that for any angle \( \theta \), the cosine of half of \( \theta \) is \( \cos \left( \frac{\theta}{2} \right) = \pm \sqrt{\frac{1 + \cos \theta}{2}} \). This identity helps us find \( \cos \frac{\pi}{12} \) by first determining \( \cos \frac{\pi}{6} \) and using it to find \( \cos \frac{\pi}{12} \).

Step 2: Identify Angles and Use Known Values

Figure out which known angle equals twice the given angle of \( \frac{\pi}{12} \). Notice that \( \frac{\pi}{6} = 2 \times \frac{\pi}{12} \). We need to find \( \cos \frac{\pi}{6} \), a standard known value, which is \( \frac{\sqrt{3}}{2} \).

Step 3: Substitute Known Values into the Formula

Now, substitute \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \) into the half-angle formula: \( \cos \frac{\pi}{12} = \sqrt{\frac{1 + \cos \frac{\pi}{6}}{2}} \). This becomes \( \cos \frac{\pi}{12} = \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}} \).

Step 4: Simplify the Expression

Simplify the expression \( \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}} \). First, add \( 1 + \frac{\sqrt{3}}{2} \), resulting in \( \frac{2}{2} + \frac{\sqrt{3}}{2} = \frac{2 + \sqrt{3}}{2} \). Then, divide by 2 to get \( \frac{2 + \sqrt{3}}{4} \). The expression simplifies to \( \cos \frac{\pi}{12} = \sqrt{\frac{2 + \sqrt{3}}{4}} \).

Step 5: Final Simplification

Further simplify \( \sqrt{\frac{2 + \sqrt{3}}{4}} \) by factoring out the square root. This is \( \frac{\sqrt{2 + \sqrt{3}}}{2} \). Therefore, \( \cos \frac{\pi}{12} = \frac{\sqrt{2 + \sqrt{3}}}{2} \).

Key Concepts

Understanding the Cosine Identity

The cosine identity is a fundamental tool in trigonometry. It helps simplify complex trigonometric expressions and solve trigonometric equations. One of the essential identities used for our purpose here is the half-angle identity for cosine.

This identity is expressed as:

• \( \cos \left( \frac{\theta}{2} \right) = \pm \sqrt{\frac{1 + \cos \theta}{2}} \)

This equation shows how to find the cosine of half an angle knowing the cosine of the original angle.

The sign (positive or negative) depends on the quadrant in which the angle \( \frac{\theta}{2} \) is located.
For angles in the first quadrant, where all trigonometric functions are positive, we choose the positive sign.

Using the cosine identity allows us to break down complex angles into more manageable parts, aiding in the precise calculation of their values.

The Role of Trigonometric Identities

Trigonometric identities are the backbone of solving trigonometric equations. They provide relationships between various trigonometric functions, allowing us to simplify expressions and calculations.

Important identities include the Pythagorean identities, angle sum and difference identities, and the double and half-angle identities.

• Pythagorean identities: Represent the relationship between sine, cosine, and tangent.
• Double angle identities: Used to express functions of double angles in terms of single angles.
• Half-angle identities: Useful to determine the value of trigonometric functions at half-angles, crucial for solving problems like the one here.

These identities enable mathematicians to transform and simplify angles for solving equations or calculating specific angle measures.

Understanding and remembering these identities are crucial to mastering trigonometry and working through such exercises with ease.

Angle Simplification Strategies

Angle simplification is a vital technique in trigonometry to manage complex expressions. By using identities, we can convert an unwieldy angle into simpler, known angles, facilitating easier computation.

In the case of finding \( \cos \frac{\pi}{12} \), we achieve simplification through the following steps:

• First, recognize that \( \frac{\pi}{12} \) is half of \( \frac{\pi}{6} \), which is a more familiar angle.
• Utilize the half-angle identity for cosine to transform the expression into terms involving known values like \( \cos \frac{\pi}{6} \), which equals \( \frac{\sqrt{3}}{2} \).
• Substitute \( \cos \frac{\pi}{6} \) into the identity to simplify further.
• Conduct the algebraic operations necessary, such as addition and division, to achieve the simplified result \( \frac{\sqrt{2 + \sqrt{3}}}{2} \).

These techniques not only help in calculations but also enhance understanding of trigonometric principles. By decomposing angles into more digestible parts, solving problems becomes less daunting.