First, let's analyze each part of the expression separately:
1. We need to evaluate
cos−1[cos(−680∘)].
The cosine function is periodic with a period of
360∘. Therefore, we can reduce the angle
−680∘ as follows:
−680∘+2×360∘=−680∘+720∘=40∘ Thus, we have:
cos−1[cos(−680∘)]=cos−1[cos(40∘)] Since
40∘ is within the range that
cos−1 can take, we get:
cos−1[cos(40∘)]=40∘2. We need to evaluate
sin‌−1[sin‌(−600∘)].
The sine function is periodic with a period of
360∘. Therefore, we can reduce the angle
−600∘ as follows:
−600∘+2×360∘=−600∘+720∘=120∘ Thus, we have:
sin‌−1[sin‌(−600∘)]=sin‌−1[sin‌(120∘)]Since
120∘ is not within the range of the inverse sine function (which is from
−90∘ to
90∘ ), we need to convert
120∘ to a value within this range. Note that:
sin‌(120∘)=sin‌(180∘−120∘)=sin‌(60∘)And
60∘ is within the range, so we have:
sin‌−1[sin‌(120∘)]=60∘ 3. We need to evaluate
cos−1(sin‌270∘).
We know that:
sin‌270∘=−1Therefore:
cos−1(sin‌270∘)=cos−1(−1)We know that the value of
cos−1(−1) is:
cos−1(−1)=180∘ Now, let's sum up all the evaluated parts:
40∘+60∘−180∘=−80∘Since we need the answer in radians, let's convert:
−80∘×‌=−‌Therefore, the answer is:
Option C:
−‌