Problem 44 - Thomas Calculus

Question

Evaluate \(\cos \frac{11 \pi}{12}\) as \(\cos \left(\frac{\pi}{4}+\frac{2 \pi}{3}\right)\)

Step-by-Step Solution

Answer

\( \cos \frac{11\pi}{12} = -\frac{\sqrt{2} + \sqrt{6}}{4} \).

Step 1: Identify the formula

We will use the angle sum identity for cosine: \( \cos(a + b) = \cos a \cdot \cos b - \sin a \cdot \sin b \). Identify \( a = \frac{\pi}{4} \) and \( b = \frac{2\pi}{3} \).

Step 2: Calculate \( \cos a \) and \( \sin a \)

Since \( a = \frac{\pi}{4} \), we have \( \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \) and \( \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \).

Step 3: Calculate \( \cos b \) and \( \sin b \)

Since \( b = \frac{2\pi}{3} \), we have \( \cos \frac{2\pi}{3} = -\frac{1}{2} \) and \( \sin \frac{2\pi}{3} = \frac{\sqrt{3}}{2} \).

Step 4: Apply the angle sum identity

Substitute the values into the identity: \( \cos(a + b) = \frac{\sqrt{2}}{2} \cdot \left(-\frac{1}{2}\right) - \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} \).

Step 5: Simplify the expression

Simplify each term: \(-\frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4} = -\frac{\sqrt{2} + \sqrt{6}}{4} \).

Step 6: Present the final result

The evaluated expression is \( \cos \frac{11\pi}{12} = -\frac{\sqrt{2} + \sqrt{6}}{4} \).

Key Concepts

Angle Sum Identity

The angle sum identity for cosine is a valuable tool in trigonometry that assists in evaluating the cosine of sum angles. This formula states: \( \cos(a + b) = \cos a \cdot \cos b - \sin a \cdot \sin b \). It helps us find the cosine of an angle that is expressed as the sum of two known angles.

By breaking down the angle into components you know, complex trigonometric evaluations become more manageable. In this exercise, the angle \( \frac{11\pi}{12} \) was expressed as \( \frac{\pi}{4} + \frac{2\pi}{3} \), where both \( \frac{\pi}{4} \) and \( \frac{2\pi}{3} \) are angles with known cosine and sine values.

This identity not only simplifies computations but also aids in understanding how observations for smaller, simpler angles can combine to give answers for more complex angles. Remembering formulas like these is crucial for confidently tackling trigonometric problems.

Trigonometric Evaluation

Trigonometric evaluation involves finding the values of trigonometric functions for specific angles. In this case, you need to evaluate \( \cos \frac{11\pi}{12} \).

Instead of approaching it directly, it's more efficient to use angle identities to break it into simpler parts. Firstly, consider known angles such as \( a = \frac{\pi}{4} \) and \( b = \frac{2\pi}{3} \) where we can easily determine:

• \( \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \)
• \( \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \)
• \( \cos \frac{2\pi}{3} = -\frac{1}{2} \)
• \( \sin \frac{2\pi}{3} = \frac{\sqrt{3}}{2} \)

Substituting these values back into the angle sum formula provides us with a simpler path to the solution.

Evaluating trigonometric values through such methods aids in swiftly unraveling the complexity.

Angle Simplification

Angle simplification is key in solving trigonometry problems effectively. It involves expressing a complex angle as the sum or difference of simpler angles. In this exercise, simplifying \( \cos \frac{11\pi}{12} \) into \( \cos \left(\frac{\pi}{4} + \frac{2\pi}{3}\right) \) is the crucial first step.

Breaking down a difficult angle helps us leverage known values for standard angles.

• These known angles are often found on the unit circle or
• can be calculated using fundamental trigonometric ratios.
• Such procedures not only make calculations faster and more accurate, but they also deepen your understanding of angle relationships.
• Grasping this skill means fewer errors and an increased ability to evaluate trigonometric expressions with confidence.

Keep practicing angle decompositions to enhance your problem-solving toolkit in trigonometry.