<p>Which of the following is the least?</p><p>(a) \(\sin 3\)</p><p>(b) \(\sin 2\)</p><p>(c) \(\sin 1\)</p><p>(d) \(\sin 7\)</p>
Step-by-Step Solution
Key Concept: Convert all angles to their reference angles in the first quadrant where sine is increasing, then compare the magnitudes.
<p><strong>Step 1:</strong> Express each angle in terms of known reference angles in the first quadrant.</p><p>\(\sin 3 = \sin[\pi - (\pi - 3)] = \sin(\pi - 3) = \sin(0.14)\)</p><p>\(\sin 2 = \sin[\pi - (\pi - 2)] = \sin(\pi - 2) = \sin(1.14)\)</p><p>\(\sin 7 = \sin[2\pi + (7 - 2\pi)] = \sin(7 - 2\pi) = \sin(0.72)\)</p><p><strong>Step 2:</strong> Compare the reference angles: \(1.14 > 1 > 0.72 > 0.14\)</p><p><strong>Step 3:</strong> Since sine is increasing in the first quadrant, we have:</p><p>\(\sin(1.14) > \sin 1 > \sin(0.72) > \sin(0.14)\)</p><p>Therefore, \(\sin 2 > \sin 1 > \sin 7 > \sin 3\)</p><p>∴ The answer is (a) \(\sin 3\) is the least.</p>
Correct Answer: A