Trigonometry & Inverse Trigonometry
Principal values of inverse trigonometric functions
Grade 12
Question:
<p>The principal value of <span class="math">\cos^{-1}\left(\cos\left(2\cot^{-1}(\sqrt{2}-1)\right)\right)</span> is equal to</p>
<p>(a) <span class="math">\frac{2\pi}{3}</span></p>
<p>(b) <span class="math">\frac{\pi}{4}</span></p>
<p>(c) <span class="math">\frac{3\pi}{4}</span></p>
<p>(d) <span class="math">\frac{5\pi}{6}</span></p>
Step-by-Step Solution
Key Concept: Apply the inverse cotangent to find the angle, then use the property that cos⁻¹(cos θ) = θ when θ is in the range [0, π].
<p><strong>Solution:</strong></p><p>We have, <span class="math">\cos^{-1}\left(\cos\left(2\cot^{-1}(\sqrt{2}-1)\right)\right)</span></p><p><span class="math">= \cos^{-1}\left(\cos\left(2 \cdot \frac{3\pi}{8}\right)\right)</span></p><p>Note: <span class="math">\cot\left(\frac{3\pi}{8}\right) = \sqrt{2}-1</span></p><p><span class="math">= \cos^{-1}\left(\cos\left(\frac{3\pi}{4}\right)\right) = \frac{3\pi}{4}</span></p><p>∴ Answer is (c).</p>
Correct Answer: C