Which of the following is the value of sin^{-1}(sin a) given that \( \pi < a < \frac{3\pi}{2} \)?
A - HSC - SSCE Mathematics Extension 1 - Question 5 - 2023 - Paper 1
Question 5
Which of the following is the value of sin^{-1}(sin a) given that \( \pi < a < \frac{3\pi}{2} \)?
A. a - \pi
B. \pi - a
C. a
D. -a
Worked Solution & Example Answer:Which of the following is the value of sin^{-1}(sin a) given that \( \pi < a < \frac{3\pi}{2} \)?
A - HSC - SSCE Mathematics Extension 1 - Question 5 - 2023 - Paper 1
Step 1
Determine the range of sin^{-1}
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Answer
The function ( sin^{-1}(x) ), also known as arcsine, has a range of ( -\frac{\pi}{2} ) to ( \frac{\pi}{2} ). Since ( a ) is in the interval ( (\pi, \frac{3\pi}{2}) ), we need to determine how ( sin a ) behaves in this range.
Step 2
Calculate sin a
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Answer
For ( a ) in the range ( \pi < a < \frac{3\pi}{2} ), sin a is negative. Therefore, we find that ( sin a = -sin(a - \pi) ).
Step 3
Apply the inverse sine function
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Answer
Since ( sin a ) is negative and lies in the range of arcsine, we can write:
[
sin^{-1}(sin a) = a - \pi
]
Step 4
Conclude with the correct option
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Answer
Thus, the value of ( sin^{-1}(sin a) ) when ( \pi < a < \frac{3\pi}{2} ) is ( a - \pi ). The correct answer is option B: ( \pi - a ).
