2024 · Q18
From a point X on a bridge across a river, the angles of depression of two points P and Q on the banks on opposite side of the river are α and β respectively. If the point X is at a height h above the surface of the river, what is the width of the river if α and β are complementary?
Elementary MathematicsTrigonometry
- A2h(tanα + cotα)
- BHtanα.tanβ
- Ch cotα.cotβ
- Dhsecα.cosecα
Explanation
The width is h·cotα + h·cotβ (X's projection splits the river). Since α+β = 90°, cotβ = tanα, so width = h(cotα + tanα) = h(1+tan²α)/tanα = h·sec²α/tanα = h/(sinαcosα) = h·secα·cosecα. This is option D.