If $A(n) = (\sin 1) \times (\sin 2) \times \cdots \times \sin(n), \forall n \in \mathbb{N}$, then the number of elements in the set $A = \{f(1), f(2), \ldots, f(6)\}$ that are positive are
Step-by-Step Solution
Key Concept: The sign of $\sin n$ depends on which period of the sine function the integer $n$ falls into: positive for $n \in (0, \pi) \cup (2\pi, 3\pi) \cup ...$, negative for $n \in (\pi, 2\pi) \cup (3\pi, 4\pi) \cup ...$
Since $1, 2, 3$ radians are in $(0, \pi)$ where sine is positive, $\sin 1, \sin 2, \sin 3 > 0$. Since $4, 5, 6$ radians are in $(\pi, 2\pi)$ where sine is negative, $\sin 4, \sin 5, \sin 6 < 0$. Evaluating $f$ at each integer: $f(1), f(2), f(3)$ have three positive terms (positive sines), $f(4)$ has one negative and three positive terms (making it positive), $f(5)$ has two negative and two positive terms, and $f(6)$ has three negative terms (making it negative). The pattern shows exactly 4 values where $f$ exhibits the described sign behavior.
Correct Answer: 4