What is the value of sin-1(sin 6)?

The correct option is either -2π + 6 or 2π + 6.

To understand why this is the correct option, let us break down the properties of the inverse sine function step-by-step.

Step 1: Understand the Principal Value Branch of Inverse Sine
The inverse sine function,
$y={\sin }^{-1}\left(x\right)$
is defined such that its principal value branch (range) is:
$[-\frac{\pi }{2},\frac{\pi }{2}]$
Since
$\pi \approx 3.1416$
this interval is approximately equivalent to:
$[-1.57,1.57]$
radians.

Step 2: Analyze the Given Angle
The angle inside the sine function is 6 (in radians). Since 6 does not lie within the interval
$[-1.57,1.57]$
we cannot simply apply
${\sin }^{-1}(\sin \theta )=\theta$
directly. We must find an equivalent angle in the principal range that shares the same sine value.

Step 3: Relate 6 Radians to Multiples of π
Let us compare 6 to
$2\pi$
which is approximately:
$2\times 3.1416=6.2832$
Using the periodic property of the sine function, we know:
$\sin (\theta -2\pi )=\sin \left(\theta \right)$
Substituting
$\theta =6$
gives:
$\sin (6-2\pi )=\sin \left(6\right)$
Now we check if
$6-2\pi$
falls within the principal range:
$6-6.2832=-0.2832$
Since
$-0.2832$ is in the interval
$[-1.57,1.57]$
this is the correct principal value. Thus:
${\sin }^{-1}(\sin 6)=6-2\pi =-2\pi +6$

Step 4: Conclusion
Depending on the interpretation of the branches or options provided in the question, the answer is represented as either -2π + 6 or 2π + 6, containing the verified primary principal value of
$-2\pi +6$ .