Assignment -1
Grade Class 12
Question:
<p>The value of sin<sup>-1</sup><span class="math-tex">\(\left(\cos \frac{3 \pi}{5}\right)\)</span> is</p>
<p style="display:inline"><span class="math-tex">\(\frac{-3 \pi}{5}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{-\pi}{10}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{\pi}{10}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{3 \pi}{5}\)</span></p>
Step-by-Step Solution
Key Concept: Convert cosine to sine using the complementary angle identity cos(θ) = sin(π/2 - θ), then apply the principal range restriction of sin⁻¹ which is [-π/2, π/2]. Since sin⁻¹(sin(x)) = x only when x ∈ [-π/2, π/2], angles outside this range must be adjusted using sin(π - x) = sin(x).
<p><span class="math-tex">$ \sin ^{-1}\left(\cos \frac{3 \pi}{5}\right) $</span><br />
<span class="math-tex">$= \sin ^{-1}\left(\sin \left(\frac{\pi}{2}-\frac{3 \pi}{5}\right)\right) $</span><br />
<span class="math-tex">$= \sin ^{-1} \sin \left(\frac{5 \pi-6 \pi}{10}\right) $</span><br />
<span class="math-tex">$= \sin ^{-1} \sin \left(-\frac{\pi}{10}\right) $</span><br />
<span class="math-tex">$= -\frac{\pi}{10} \epsilon\left[-\frac{\pi}{2} , \frac{\pi}{2}\right]$</span></p>
Correct Answer: B