Problem

Evaluate \(\sin(30°)\) and \(\sin(45°)\) from their special-right-triangle ratios. Compare the exact values \(1/2\) and \(\sqrt{2}/2\) and state the resulting inequality with justification.

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What this problem is really about

The comparison must use evaluated trig ratios rather than the angle measures themselves. We’ll recall each exact sine value from its special triangle, place them over a common positive denominator, compare the positive numerators, and translate that fraction comparison back into a trig inequality.

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sin 30 degrees and sin 45 degrees are equal because both are sine values from special right triangles, so they represent the same ratio.

sin 30 degrees is greater than sin 45 degrees because 1/2 is greater than sqrt(2)/2 when the denominators are the same.

sin 30 degrees is greater than sin 45 degrees because sin 30 degrees = sqrt(3)/2 and sin 45 degrees = sqrt(2)/2, and sqrt(3)/2 > sqrt(2)/2.

sin 45 degrees is greater than sin 30 degrees because sin 45 degrees = sqrt(2)/2 and sin 30 degrees = 1/2, and sqrt(2)/2 > 1/2.

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Evaluate \(\sin(30°)\) and \(\sin(45°)\) from their special-right-triangle ratios. Compare the exact values \(1/2\) and \(\sqrt{2}/2\) and state the resulting inequality with justification.

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