evaluate the following expression. your answer must be in exact form: for example, type pi/6 for π/6 or dne…

Explanation:

Step1: Recall the property of inverse - sine

The function (y = \arcsin(x)) has a domain ([- 1,1]) and range (\left[-\frac{\pi}{2},\frac{\pi}{2}\right]). Also, (\arcsin(\sin(x))=x) when (x\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]).

Step2: Analyze the angle (-\frac{2\pi}{7})

The angle (x =-\frac{2\pi}{7}\approx - 0.8976) radians. Since (-\frac{\pi}{2}\approx - 1.57) and (\frac{\pi}{2}\approx1.57), and (-\frac{\pi}{2}<-\frac{2\pi}{7}<\frac{\pi}{2}).

Step3: Apply the property

Since (-\frac{2\pi}{7}) is in the range of (y = \arcsin(x)), then (\arcsin\left(\sin\left(-\frac{2\pi}{7}\right)\right)=-\frac{2\pi}{7}).

Answer:

(-\frac{2\pi}{7})