Question:If theta is a real number such that cos theta = 1, what is the value of \(\sin(10\theta)\)?-10(sqrt(2))/21

1. INFER what cos θ = 1 tells us about θ

• Given: \(\cos \theta = 1\)
• Key insight: We need to identify which angles have cosine equal to 1
• From the unit circle, \(\cos \theta = 1\) occurs at \(\theta = 0, 2\pi, 4\pi\), etc.
• General form: \(\theta = 2\pi k\) for any integer k

2. SIMPLIFY by substituting into the expression

• We need to find \(\sin(10\theta)\)
• Substitute \(\theta = 2\pi k\): \(\sin(10\theta) = \sin(10 \cdot 2\pi k) = \sin(20\pi k)\)

3. INFER the final value

• Since \(20\pi k = 2\pi(10k)\), this represents an integer multiple of \(2\pi\)
• The sine of any integer multiple of \(2\pi\) equals 0
• Therefore: \(\sin(20\pi k) = 0\)

Answer: (B) 0

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual gap about unit circle values: Students may not remember or recognize that \(\cos \theta = 1\) only occurs at specific angles (multiples of \(2\pi\)). They might think cosine can equal 1 at other angles like \(\pi/4\) or \(\pi/2\), leading to incorrect values for \(\theta\).

This leads to confusion and incorrect substitution, causing them to guess among the answer choices.

Second Most Common Error:

Weak INFER skill: Students might recognize that \(\cos \theta = 1\) means \(\theta = 0\) (correctly), but then substitute \(\theta = 0\) directly without considering the general case \(\theta = 2\pi k\). This still gives \(\sin(10 \cdot 0) = \sin(0) = 0\), which happens to be correct, but shows incomplete understanding.

Alternatively, they might get confused about what \(\sin(20\pi k)\) equals, particularly if they don't remember that sine of multiples of \(2\pi\) equals zero.

The Bottom Line:

This problem tests whether students can connect unit circle knowledge to algebraic manipulation. The key insight is recognizing that \(\cos \theta = 1\) constrains \(\theta\) to very specific values, which then determines the final answer regardless of the specific integer k.