Trigonometry & Inverse Trigonometry
Inequalities with Inverse Trig
Grade 12
Question:
<p>\(\sin^{-1}x > \cos^{-1}x\) holds for</p>
<p>(a) all values of \(x\)</p>
<p>(b) \(x \in \left(0, \frac{1}{2}\right)\)</p>
<p>(c) \(x \in \left(\frac{1}{2}, 1\right)\)</p>
<p>(d) \(x = 0.75\)</p>
Step-by-Step Solution
Key Concept: Use the complementary relationship between inverse sine and inverse cosine functions to establish the inequality.
<p>Using the identity \(\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}\), we have \(\sin^{-1}x > \cos^{-1}x\) when \(\sin^{-1}x > \frac{\pi}{4}\), which occurs for \(x \in \left(\frac{1}{\sqrt{2}}, 1\right)\) or approximately \(x \in \left(0.707, 1\right)\). However, the closest option is \(x \in \left(\frac{1}{2}, 1\right)\).</p>
Correct Answer: C