Question:What is the value of \(\sin\left(\frac{-401\pi}{4}\right)\)?

1. SIMPLIFY using periodicity

• The key insight: Sine repeats every \(2\pi\), so we can reduce large angles
• Since \(2\pi = \frac{8\pi}{4}\), we need to find how many complete periods fit in \(\frac{401\pi}{4}\)
• Calculate: \(401 \div 8 = 50.125\) (use calculator)
• This means 50 complete periods: \(50 \times \frac{8\pi}{4} = \frac{400\pi}{4}\)
• Reduce the angle: \(-\frac{401\pi}{4} + \frac{400\pi}{4} = -\frac{\pi}{4}\)

2. INFER the strategy for negative angles

• Now we have \(\sin(-\frac{\pi}{4})\) instead of the original massive angle
• Since sine is an odd function, we can use: \(\sin(-x) = -\sin(x)\)
• This gives us: \(\sin(-\frac{\pi}{4}) = -\sin(\frac{\pi}{4})\)

3. APPLY special angle knowledge

• From the unit circle: \(\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}\)
• Therefore: \(\sin(-\frac{401\pi}{4}) = -\sin(\frac{\pi}{4}) = -\frac{\sqrt{2}}{2}\)

Answer: A. \(-\frac{\sqrt{2}}{2}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skills: Students see the large angle \(-\frac{401\pi}{4}\) and either panic or attempt to work with it directly without using periodicity.

They might try to convert \(-\frac{401\pi}{4}\) to degrees (getting \(-18,045°\)) and then get lost in the massive number. Or they might attempt to use their calculator directly on \(\sin(-\frac{401\pi}{4})\) and get a decimal approximation without recognizing the exact answer. This leads to confusion and random guessing among the answer choices.

Second Most Common Error:

Missing conceptual knowledge: Students successfully reduce to \(-\frac{\pi}{4}\) but forget that sine is an odd function.

Instead of applying \(\sin(-\frac{\pi}{4}) = -\sin(\frac{\pi}{4})\), they might incorrectly think \(\sin(-\frac{\pi}{4}) = \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}\). This leads them to select Choice D (\(\frac{\sqrt{2}}{2}\)) instead of the correct negative value.

The Bottom Line:

This problem tests whether students can systematically break down intimidating large angles using periodicity. The key insight is that no matter how large the angle, sine's periodic nature always allows reduction to a manageable reference angle.