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Which statement is always true for real numbers a and b where −1 ≤ a < b ≤ 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 ≤ a < b ≤ 1? A. sec a < sec b B. sin⁻¹ a < sin⁻¹ b C. arccos a < arccos b D. cos⁻¹ a + sin⁻¹ a ... show full transcript
Worked Solution & Example Answer:Which statement is always true for real numbers a and b where −1 ≤ a < b ≤ 1? A - HSC - SSCE Mathematics Extension 1 - Question 7 - 2023 - Paper 1
Step 1
B. sin⁻¹ a < sin⁻¹ b
Answer
To evaluate the statement, we can use the properties of the inverse sine function, sin⁻¹ x, which is defined for values in the range [-1, 1] and outputs angles in [−π/2, π/2].
Given the condition −1 ≤ a < b ≤ 1, since both a and b are within the domain of the function, and because the inverse sine function is strictly increasing, we have:
If a < b, then sin⁻¹ a < sin⁻¹ b.
This means that option B is always true when −1 ≤ a < b ≤ 1.