Mathematics

Mathematics

Problem

Evaluate sin⁡30∘⋅cos⁡60∘tan⁡45∘\dfrac{\sin 30^\circ \cdot \cos 60^\circ}{\tan 45^\circ}. a) 34\dfrac{\sqrt3}{4} b) 11 c) 14\dfrac14 d) 12\dfrac12
Intuitive explanation

This problem tests whether you know the exact values of sine, cosine, and tangent at the most common special angles. The key insight is that 30°30°, 45°45°, and 60°60° are the "golden angles" of trigonometry—their trig values are simple fractions and radicals that you can either memorize or derive from a 45-45-90 or 30-60-90 triangle. Once you substitute those values, the arithmetic simplifies immediately.

Intuitive explanation

This problem tests whether you know the exact values of sine, cosine, and tangent at the most common special angles. The key insight is that 30°30°, 45°45°, and 60°60° are the "golden angles" of trigonometry—their trig values are simple fractions and radicals that you can either memorize or derive from a 45-45-90 or 30-60-90 triangle. Once you substitute those values, the arithmetic simplifies immediately.

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