Step-by-Step Solution
Key Concept: The inverse cosine function cos⁻¹ has a range of [0, π], so we must express the argument in a form where the angle lies within this range. Since 13 radians exceeds π, we need to find a coterminal angle within [0, π].
<p><strong>Step 1:</strong> Identify the range of cos⁻¹. The function cos⁻¹(x) has range [0, π], meaning cos⁻¹(cos θ) = θ only if θ ∈ [0, π].</p><p><strong>Step 2:</strong> Check if 13 radians is in [0, π]. Since π ≈ 3.14159 and 13 > π, we need to reduce 13 to an equivalent angle in [0, π].</p><p><strong>Step 3:</strong> Reduce 13 using periodicity. We need to find how many complete periods of 2π fit into 13:<br/>13 ÷ (2π) ≈ 13 ÷ 6.283 ≈ 2.07<br/>So 13 = 2(2π) + r, where r is the remainder.<br/>r = 13 - 4π ≈ 13 - 12.566 ≈ 0.434<br/>Thus cos(13) = cos(13 - 4π)</p><p><strong>Step 4:</strong> Check if (13 - 4π) is in [0, π]. Since 13 - 4π ≈ 0.434 and 0 < 0.434 < π, this angle is already in the valid range.</p><p><strong>Step 5:</strong> Apply the definition. Since 13 - 4π ∈ [0, π], we have:<br/>cos⁻¹(cos 13) = cos⁻¹(cos(13 - 4π)) = 13 - 4π</p><p><strong>∴ Answer: 13 - 4π</strong></p>
Correct Answer: 13