Question

Let $f\left(n\right)=\left(\sin 1\right)\left(\sin 2\right)\left(\sin 3\right)\cdots \left(\sin \right(n\left)\right)\forall n\in N$ where $n$ is in radians. Then the number of elements in the set $A=\left\{f\right(1),f(2),\dots ,f(6\left)\right\}$ that are positive, is

• A. $2$
• B. $3$
• C. $4$
• D. $5$

Answer 1

The correct option is C $4$

Given that $f\left(n\right)=\left(\sin 1\right)\left(\sin 2\right)\left(\sin 3\right)\cdots \left(\sin \right(n\left)\right)$
Clearly $1,2,3\in (0,\pi )$ and $4,5,6\in (\pi ,2\pi )$
So, $\sin 1,\sin 2,\sin 3$ are positive
and $\sin 4,\sin 5,\sin 6$ are negative.
So, $f\left(1\right),f\left(2\right),f\left(3\right),f\left(5\right)$ are positive and $f\left(4\right),f\left(6\right)$ are negative.