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Which statement is always true for real numbers a and b where \(-1 \leq a < b \leq 1\)? A - HSC - SSCE Mathematics Extension 1 - Question 7 - 2023 - Paper 1
Question 7
Which statement is always true for real numbers a and b where \(-1 \leq a < b \leq 1\)? A. sec a < sec b B. sin^{-1} a < sin^{-1} b C. arccos a < arccos b D. cos... show full transcript
Worked Solution & Example Answer:Which statement is always true for real numbers a and b where \(-1 \leq a < b \leq 1\)? A - HSC - SSCE Mathematics Extension 1 - Question 7 - 2023 - Paper 1
Step 1
B. sin^{-1} a < sin^{-1} b
Answer
Given the constraints (-1 \leq a < b \leq 1), we can analyze the behavior of the function (f(x) = sin^{-1}(x)).
The arcsine function (sin^{-1}(x)) is defined within the range of ([-\frac{\pi}{2}, \frac{\pi}{2}]) for inputs in the interval ([-1, 1]). It is also a strictly increasing function. Consequently, if (a < b), then it follows that:
This means statement B is always true given the stated conditions.