What is the value of \(\cos\left(\frac{565\pi}{6}\right)\)?

1. INFER the approach needed

• Given: \(\cos(\frac{565\pi}{6})\)
• Key insight: This angle is much larger than \(2\pi\), so we need to use periodicity to reduce it
• Strategy: Find an equivalent angle between 0 and \(2\pi\) (or \(-\pi\) and \(\pi\))

2. SIMPLIFY using periodicity

• Since \(\cos(\theta) = \cos(\theta + 2\pi k)\) for any integer k, we can subtract multiples of \(2\pi\)
• Express \(2\pi\) in the same denominator: \(2\pi = \frac{12\pi}{6}\)
• Find how many complete periods fit: \(565 \div 12 = 47\) remainder \(1\)
• This means: \(\frac{565\pi}{6} = 47 \times (\frac{12\pi}{6}) + \frac{1\pi}{6} = 47 \times 2\pi + \frac{\pi}{6}\)

3. APPLY the periodicity property

• \(\cos(\frac{565\pi}{6}) = \cos(47 \times 2\pi + \frac{\pi}{6}) = \cos(\frac{\pi}{6})\)
• We've reduced the problem to finding \(\cos(\frac{\pi}{6})\)

4. RECALL the unit circle value

• From the unit circle: \(\cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}\)

Answer: C. \(\frac{\sqrt{3}}{2}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that large angles need to be reduced using periodicity. They might try to work directly with \(\frac{565\pi}{6}\), getting overwhelmed by the large numbers, or attempt to convert to degrees unnecessarily. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students understand they need to use periodicity but make arithmetic errors. For example, they might incorrectly calculate \(565 \div 12\), getting the wrong remainder, which leads them to an incorrect equivalent angle like \(\frac{5\pi}{6}\) instead of \(\frac{\pi}{6}\). Since \(\cos(\frac{5\pi}{6}) = -\frac{\sqrt{3}}{2}\), this could lead them to select a negative value if it were among the choices, or cause confusion when all choices are positive.

The Bottom Line:

The key insight is recognizing that trigonometric functions are periodic - no matter how large the angle, it can always be reduced to an equivalent angle in the standard interval. The arithmetic must be done carefully to ensure the correct equivalent angle is found.