Trigonometry: Heights & Distances

Height and distance problems are primarily word problems based on right-angled triangles.

• Line of Sight: The imaginary straight line drawn from the observer's eye to the object.
• Angle of Elevation: The angle formed above the horizontal line when looking up at an object (e.g., looking at the top of a tower).
• Angle of Depression: The angle formed below the horizontal line when looking down at an object.

Instead of manually calculating $\tan \theta$ every time, memorize the standard ratio sides of common right triangles to solve visually.

• The 30°-60°-90° Triangle:
  The sides opposite to the angles 30°, 60°, and 90° are always in the ratio $1 : \sqrt{3} : 2$.
  Application: If the height (opposite 30°) is 10m, the shadow length (opposite 60°) is $10\sqrt{3}$m.
• The 45°-45°-90° Triangle:
  The sides opposite to the angles are in the ratio $1 : 1 : \sqrt{2}$.
  Application: If the angle of elevation is 45°, the height of the object and the length of its shadow are exactly equal.

When an observer moves towards an object, the angle of elevation increases.

• Moving 30° to 45°: Distance moved = $h(\sqrt{3} - 1)$
• Moving 30° to 60°: Distance moved = $h(2/\sqrt{3})$
• Moving 45° to 60°: Distance moved = $h(1 - 1/\sqrt{3})$