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Precalculus · Question of the Day

Precalculus Question of the Day

A fresh daily question to build accuracy, reinforce recall, and turn practice into a steady habit.
Tuesday, September 8, 2026

In right triangle YZAYZA, the right angle is at ZZ. If the hypotenuse is YA=10YA = 10 and ∠Y=60∘\angle Y = 60^\circ, what is the exact length of the leg YZYZ (adjacent to ∠Y\angle Y)?

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Question of the Day

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In right triangle YZAYZA, the right angle is at ZZ. If the hypotenuse is YA=10YA = 10 and ∠Y=60∘\angle Y = 60^\circ, what is the exact length of the leg YZYZ (adjacent to ∠Y\angle Y)?

  1. 55 (correct answer)
  2. 535\sqrt{3}
  3. 10310\sqrt{3}
  4. 103\frac{10}{\sqrt{3}}

Explanation: This question tests the ability to solve right triangles using trigonometric ratios. Trigonometric ratios relate the angles of a right triangle to the ratios of its sides: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, and tan(θ) = opposite/adjacent. Since we know the hypotenuse of length 10 and need to find the adjacent side to angle Y of 60°, we use the cosine ratio: cos(60°) = adjacent/hypotenuse, so adjacent = 10 × cos(60°) = 10 × 0.5 = 5. Choice A is correct because it uses the cosine ratio with the given angle of 60° and hypotenuse of 10, correctly identifying the adjacent side. Choice B uses sine when the problem requires cosine, confusing which ratio relates the adjacent side. Remember the SOH-CAH-TOA mnemonic for choosing the correct trig ratio: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent, where opposite and adjacent are always relative to the angle in question. Key to right triangle problems: first identify the right angle and hypotenuse (longest side, opposite the right angle), then decide whether you have enough information for Pythagorean theorem (two sides known) or need trigonometry (one side and one angle known).