Trigonometry & Inverse Trigonometry
Inverse Sine and Cosecant
Grade 12
Question:
<p>If \(\sin^{-1}\left(\frac{x}{5}\right) + \text{cosec}^{-1}\left(\frac{5}{4}\right) = \frac{\pi}{2}\), then \(x\) is</p>
<p>(a) 4</p>
<p>(b) 5</p>
<p>(c) 1</p>
<p>(d) 3</p>
Step-by-Step Solution
Key Concept: Apply the complementary angle relationship for inverse sine and cosecant.
<p><strong>Solution:</strong> Use the identity \(\sin^{-1}u + \text{cosec}^{-1}v = \frac{\pi}{2}\) when applicable.</p><p>Given: \(\sin^{-1}\left(\frac{x}{5}\right) + \text{cosec}^{-1}\left(\frac{5}{4}\right) = \frac{\pi}{2}\)</p><p>This implies: \(\sin^{-1}\left(\frac{x}{5}\right) = \frac{\pi}{2} - \text{cosec}^{-1}\left(\frac{5}{4}\right) = \sin^{-1}\left(\frac{4}{5}\right)\)</p><p>Therefore: \(\frac{x}{5} = \frac{4}{5}\), so \(x = 4\)</p>
Correct Answer: D