Which of the following is the value of $ ext{sin}^{-1}( ext{sin} a)$ given that $ rac{ ext{π}}{2} < a < rac{3 ext{π}}{2}$?
A - HSC - SSCE Mathematics Extension 1 - Question 5 - 2023 - Paper 1
Question 5
Which of the following is the value of $ ext{sin}^{-1}( ext{sin} a)$ given that $ rac{ ext{π}}{2} < a < rac{3 ext{π}}{2}$?
A. $a - ext{π}$
B. $ ext{π} - a$
C. $a$... show full transcript
Worked Solution & Example Answer:Which of the following is the value of $ ext{sin}^{-1}( ext{sin} a)$ given that $ rac{ ext{π}}{2} < a < rac{3 ext{π}}{2}$?
A - HSC - SSCE Mathematics Extension 1 - Question 5 - 2023 - Paper 1
Step 1
Determine the range of $a$
96%
114 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
We begin with the given condition: rac{ ext{π}}{2} < a < rac{3 ext{π}}{2}. This tells us that a is in the second or third quadrant.
Step 2
Understand the behavior of $ ext{sin}^{-1}( ext{sin} a)$
99%
104 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
The function extsin−1(x) retrieves values in the range [- rac{ ext{π}}{2}, rac{ ext{π}}{2}]. Therefore, knowing that a is in the second or third quadrant, we need to adjust a to fit within this range.
Step 3
Evaluate the possible values
96%
101 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
For values of a in the second quadrant ( rac{ ext{π}}{2} < a < ext{π}), we have:
extsin−1(extsina)=extπ−a.
For values in the third quadrant ( ext{π} < a < rac{3 ext{π}}{2}), we have:
extsin−1(extsina)=a−extπ.
Since both expressions bring us to the same conclusion within the specified a ranges.
Step 4
Select the correct option
98%
120 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
Given the conditions and evaluations, the correct option is B: a−extπ.
