What is the value of cos 41pi?

1. INFER the key strategy

• When dealing with trigonometric functions of large angles, use periodicity
• Cosine repeats every \(2\pi\) radians, so \(\cos(x) = \cos(x + 2\pi k)\) for any integer k
• Goal: Reduce \(41\pi\) to an equivalent angle we can easily evaluate

2. SIMPLIFY the angle using periodicity

• Write \(41\pi\) in terms of complete periods of \(2\pi\):

\(41\pi = 40\pi + \pi\)

• Factor out the period: \(40\pi = 20 \times 2\pi\)
• Therefore: \(41\pi = 20(2\pi) + \pi\)

3. INFER the final evaluation

• Apply periodicity: \(\cos(41\pi) = \cos(20(2\pi) + \pi) = \cos(\pi)\)
• From the unit circle: \(\cos(\pi) = -1\)

Answer: (A) -1

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that periodicity is the key to handling large angles like \(41\pi\).

Instead, they might attempt to convert \(41\pi\) to degrees (\(7380°\)) or try to approximate it numerically, leading to confusion about how to proceed. Without the periodicity insight, they get overwhelmed by the large number and resort to guessing.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students understand they need to use periodicity but struggle with the arithmetic of expressing \(41\pi\) in terms of \(2\pi\).

They might incorrectly calculate how many periods fit into \(41\pi\), perhaps writing something like \(41\pi = 20\pi + \pi\) instead of recognizing that \(40\pi = 20(2\pi)\). This leads to applying periodicity incorrectly and potentially selecting Choice (E) 1 if they mistakenly reduce to \(\cos(0)\).

The Bottom Line:

Large-angle trigonometry problems test whether students can connect the abstract concept of periodicity to practical problem-solving. The key insight is that no matter how large the angle, trigonometric functions always cycle back to familiar values.