Let's break down how to evaluate the given expression:
Recall the definitions of inverse trigonometric functions:
cos−1(x) gives the angle (between 0 and
Ï€ ) whose cosine is
x.
sin‌−1(x) gives the angle (between
−π∕2 and
π∕2 ) whose sine is
x.
Now, let's consider the angles involved:
35π∕18 is greater than
2Ï€ (a full circle). To work with angles within a single circle, we can subtract multiples of
2Ï€ :
‌−2π=‌−‌=−‌−π∕18 lies within the interval
[−π,0], where the cosine function is positive and the sine function is negative. This is important for the inverse functions.
Let's apply these ideas to our expression:
Since
cos(−x)=cos(x) and
sin‌(−x)=−sin‌(x) :
Now, we can apply the definitions of the inverse functions. Because
π∕18 is within the range of
cos−1, the first term simplifies directly:
=‌−sin‌−1(−sin‌‌)For the second term, we need to find the angle between
−π∕2 and
π∕2 whose sine is
−sin‌(π∕18). Since the sine function is odd, this angle is simply
−π∕18 :
=‌−(−‌)=‌Therefore, the correct answer is Option B.