The Solution:
Given the value below:
[tex]\cos \lbrack\sin ^{-1}(-0.845)\rbrack[/tex]
We are asked to find the value of the above expression rounded to the nearest thousandth.
So, applying the property rule:
[tex]\sin ^{-1}(-x)=-\sin ^{-1}(x)[/tex]
We have that:
[tex]\sin ^{-1}(-0.845)=-\sin ^{-1}(0.845)=-57.671936[/tex]
So, it follows that:
[tex]\cos \lbrack\sin ^{-1}(-0.845)\rbrack=\cos (-57.671936)[/tex]
Applying the property rule:
[tex]\cos (-x)=\cos (x_{})[/tex]
We have that:
[tex]\cos \lbrack\sin ^{-1}(-0.845)\rbrack=\cos (-57.671936)=\cos (57.671936)=0.534766[/tex]
Rounding to the nearest thousandth, we get
[tex]\cos \lbrack\sin ^{-1}(-0.845)\rbrack=\cos (-57.671936)=0.534766\approx0.535[/tex]
Therefore, the correct answer is 0.535
