A point starts at \((1, 0)\) on the unit circle and is rotated counterclockwise by 2023pi/4 radians about the origin....

1. TRANSLATE the problem information

• Given information:
  • Starting point: \((1, 0)\) on unit circle
  • Rotation: \(\frac{2023\pi}{4}\) radians counterclockwise
  • Need: y-coordinate of final position

2. INFER the coordinate approach

• On the unit circle, coordinates after rotation by \(\theta\) are \((\cos \theta, \sin \theta)\)
• Since we want the y-coordinate, we need \(\sin\left(\frac{2023\pi}{4}\right)\)
• The key insight: sine has period \(2\pi\), so we can reduce this large angle

3. SIMPLIFY the angle using modular arithmetic

• Since sine repeats every \(2\pi\), convert \(2\pi\) to fourths: \(2\pi = \frac{8\pi}{4}\)
• Find \(2023 \bmod 8\): \(2023 \div 8 = 252\) remainder \(7\)
• Therefore: \(\frac{2023\pi}{4} \equiv \frac{7\pi}{4} \pmod{2\pi}\)

4. INFER the quadrant and reference angle

• \(\frac{7\pi}{4} = 315°\), which places us in Quadrant IV
• In Quadrant IV, sine is negative
• Reference angle: \(2\pi - \frac{7\pi}{4} = \frac{\pi}{4}\)

5. SIMPLIFY to find the final value

• \(\sin\left(\frac{7\pi}{4}\right) = -\sin\left(\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2}\)

Answer: C \(\left(-\frac{\sqrt{2}}{2}\right)\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students attempt to work with the full angle \(\frac{2023\pi}{4}\) without reducing modulo \(2\pi\).

They might try to convert \(\frac{2023\pi}{4}\) directly to degrees (getting something like \(91,035°\)) and then struggle with such a large angle. This leads to confusion about which quadrant they're in and often results in guessing or selecting a positive value like Choice E \(\left(\frac{\sqrt{2}}{2}\right)\) by forgetting about sign considerations.

Second Most Common Error:

Conceptual confusion about modular arithmetic: Students correctly identify they need to reduce the angle but make errors in the division.

For example, they might incorrectly calculate \(2023 \div 8\) or make mistakes with the remainder, leading to the wrong reduced angle. This could lead them to evaluate sine at the wrong angle and select an incorrect answer like Choice B \(\left(-\frac{\sqrt{3}}{2}\right)\) or Choice D \((0)\).

The Bottom Line:

This problem tests whether students can handle large angle rotations efficiently. The key insight is that you don't need to track 252+ full rotations—just find where you end up after reducing modulo \(2\pi\).