What is the value of sin 42pi?

1. TRANSLATE the problem information

• We need to find: \(\sin 42\pi\)
• Given answer choices suggest the result will be a familiar trigonometric value

2. INFER the key insight

• The sine function repeats every \(2\pi\) units (this is its period)
• Since \(42\pi\) is much larger than \(2\pi\), we can use periodicity to simplify
• Key strategy: Find how many complete periods of \(2\pi\) fit into \(42\pi\)

3. SIMPLIFY using the period

• Divide the angle by the period: \(42\pi \div 2\pi = 21\)
• This means: \(42\pi = 21 \times 2\pi\)
• So we have exactly 21 complete rotations around the unit circle

4. INFER the final position

• After any whole number of complete rotations, we return to the starting point
• The starting point on the unit circle is \((1, 0)\)
• Since sine gives the y-coordinate: \(\sin 42\pi = \) y-coordinate of \((1, 0) = 0\)

Answer: A. 0

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may not recognize that they can use the periodic property of sine to simplify large angles. Instead, they might try to evaluate \(\sin 42\pi\) directly or get overwhelmed by the large number, leading to random guessing among the answer choices.

Second Most Common Error:

Conceptual confusion about unit circle positions: Students might correctly identify that \(42\pi\) represents multiple rotations but incorrectly determine the final position. They may confuse the starting point \((1, 0)\) with other key points like \((0, 1)\) or \((-1, 0)\), leading them to select Choice D (1) thinking of the x-coordinate instead of the y-coordinate.

The Bottom Line:

Success on this problem requires recognizing that trigonometric functions are periodic, allowing you to reduce any angle to an equivalent angle within one period. The key insight is that "large" angles aren't actually more complicated - they just require using periodicity first.