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Geometry · Question of the Day

Geometry Question of the Day

A fresh daily question to build accuracy, reinforce recall, and turn practice into a steady habit.
Friday, November 6, 2026

A right triangle △DEF\triangle DEF is shown with ∠E\angle E marked as a right angle. The acute angles are labeled θ=∠D\theta=\angle D and φ=∠F\varphi=\angle F. Which expression represents sin⁡(θ)\sin(\theta) in terms of cos⁡(φ)\cos(\varphi)?

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Question of the Day

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A right triangle △DEF\triangle DEF is shown with ∠E\angle E marked as a right angle. The acute angles are labeled θ=∠D\theta=\angle D and φ=∠F\varphi=\angle F. Which expression represents sin⁡(θ)\sin(\theta) in terms of cos⁡(φ)\cos(\varphi)?

  1. sin⁡(θ)=cos⁡(φ)\sin(\theta)=\cos(\varphi) (correct answer)
  2. sin⁡(θ)=sin⁡(φ)\sin(\theta)=\sin(\varphi)
  3. sin⁡(θ)=cos⁡(θ)\sin(\theta)=\cos(\theta)
  4. sin⁡(θ)=cos⁡(90∘)\sin(\theta)=\cos(90^\circ)

Explanation: This problem involves expressing sin(θ) in terms of cos(φ) for complementary angles. In right triangle DEF with right angle at E, angles θ = ∠D and φ = ∠F satisfy θ + φ = 90°. For angle θ, the opposite side is EF and the adjacent side is DE, giving sin(θ) = EF/DF. For angle φ, the opposite side is DE and the adjacent side is EF, yielding cos(φ) = EF/DF. Both expressions equal EF/DF, so sin(θ) = cos(φ). The incorrect choice sin(θ) = sin(φ) assumes the angles are equal rather than complementary. To master this concept, practice identifying which sides play opposite and adjacent roles for each acute angle in the triangle.