<p>(a) \(\tan 1 > \tan^{-1} 1\)</p>
<p>(b) \(\sin 1 > \cos 1\)</p>
<p>(c) \(\tan 1 < \sin 1\)</p>
<p>(d) \(\cos(\cos 1) > \frac{1}{2}\)</p>
Step-by-Step Solution
Key Concept: We need to compare trigonometric values with inverse trigonometric values, and understand that 1 radian ≈ 57.3° is in the first quadrant where specific inequalities hold. Using the value 1 (radian) requires careful numerical comparison with reference values.
<p><strong>Step 1: Analyze option (a): tan 1 > tan⁻¹ 1</strong></p><p>Here 1 is in radians. We know tan⁻¹ 1 = π/4 ≈ 0.7854.</p><p>For tan(1 radian): Since 1 radian ≈ 57.3°, and tan(57.3°) ≈ 1.557.</p><p>Therefore: tan 1 ≈ 1.557 > π/4 ≈ 0.7854. <strong>TRUE</strong></p><p><strong>Step 2: Analyze option (b): sin 1 > cos 1</strong></p><p>Since 1 radian ≈ 57.3°, which is greater than π/4 ≈ 45°, we are in the region where sine exceeds cosine.</p><p>More precisely: sin 1 ≈ 0.841 and cos 1 ≈ 0.540.</p><p>Therefore: sin 1 > cos 1. <strong>TRUE</strong></p><p><strong>Step 3: Analyze option (c): tan 1 < sin 1</strong></p><p>From Step 1: tan 1 ≈ 1.557. From Step 2: sin 1 ≈ 0.841.</p><p>Therefore: tan 1 ≈ 1.557 > 0.841 ≈ sin 1, so tan 1 > sin 1.</p><p>Option (c) is <strong>FALSE</strong></p><p><strong>Step 4: Analyze option (d): cos(cos 1) > 1/2</strong></p><p>First: cos 1 ≈ 0.540 (this is the value of cosine at 1 radian).</p><p>Next: We need cos(0.540). Since 0.540 radians ≈ 30.9°, and cos(30.9°) ≈ 0.857.</p><p>We need to verify: 0.857 > 0.5? Yes, clearly true.</p><p>Alternatively, since 0 < cos 1 < π/2, and cosine is decreasing on [0,π], we have cos(cos 1) > cos(π/2) = 0.</p><p>More precisely, cos 1 ≈ 0.540 < π/3 ≈ 1.047, so cos(cos 1) > cos(π/3) = 1/2.</p><p>Therefore: cos(cos 1) > 1/2. <strong>TRUE</strong></p><p><strong>∴ Answer: a,b,d</strong></p>
Correct Answer: a,b,d