<p>The principal value of \(\cos^{-1}\left(\cos\dfrac{7\pi}{4}\right)\) is ______.</p>
Step-by-Step Solution
Key Concept: The principal value of cos⁻¹ is restricted to [0, π], so we must reduce the angle to this range using the periodicity and symmetry of cosine before applying the inverse function.
<p><strong>Step 1:</strong> Identify the range of principal values for cos⁻¹: [0, π]</p><p><strong>Step 2:</strong> Check if 7π/4 is in [0, π]. Since 7π/4 ≈ 5.498 > π, it is not in the range.</p><p><strong>Step 3:</strong> Use the identity cos(2π - α) = cos(α). Here, 7π/4 = 2π - π/4, so:<br/>cos(7π/4) = cos(2π - π/4) = cos(π/4)</p><p><strong>Step 4:</strong> Apply cos⁻¹ to both sides:<br/>cos⁻¹(cos(7π/4)) = cos⁻¹(cos(π/4))</p><p><strong>Step 5:</strong> Since π/4 ∈ [0, π], the principal value is directly π/4.<br/>∴ Answer: <strong>π/4</strong></p>
Correct Answer: π/4