1. Understanding Height–Distance Problems
Most height and distance problems can be solved by creating a right-angled triangle using the given situation. The unknown height or distance becomes one of the sides of the triangle. The angle of elevation or depression gives the angle inside the triangle.
Commonly used ratio:
\( \tan \theta = \dfrac{\text{opposite}}{\text{adjacent}} \)
2. Steps to Solve a Height–Distance Problem
- Identify the observer and the object.
- Draw a horizontal line of sight from the observer.
- Mark the angle of elevation or depression.
- Form the right-angled triangle using height and distance.
- Choose the appropriate trigonometric ratio (usually tan).
- Substitute values and solve.
3. Example 1: Finding Height Using Angle of Elevation
Problem: A person standing 40 m away from a building sees its top at an angle of elevation of \(35^\circ\). Find the height of the building above the person’s eye level.
\( \tan 35^\circ = \dfrac{h}{40} \)
\( h = 40 \tan 35^\circ \)
This is the vertical height from eye level to the top.
4. Example 2: Finding Distance Using Angle of Depression
Problem: From the top of a lighthouse 60 m high, the angle of depression to a ship is \(25^\circ\). Find the horizontal distance of the ship from the lighthouse.
\( \tan 25^\circ = \dfrac{60}{d} \)
\( d = \dfrac{60}{\tan 25^\circ} \)
This gives the distance from the base of the lighthouse to the ship.
5. Example 3: Two Angles from Two Points
Problem: A person moves 20 m closer to a tower. The angle of elevation changes from \(30^\circ\) to \(45^\circ\). Find the height of the tower.
Let the initial distance from the tower be \(x\).
\( \tan 30^\circ = \dfrac{h}{x} = \dfrac{1}{\sqrt{3}} \Rightarrow h = \dfrac{x}{\sqrt{3}} \)
New distance = \(x - 20\)
\( \tan 45^\circ = \dfrac{h}{x - 20} = 1 \Rightarrow h = x - 20 \)
Equating heights:
\( x - 20 = \dfrac{x}{\sqrt{3}} \)
6. Example 4: Shadow Problems
Problem: A pole 12 m high casts a shadow 8 m long. Find the angle of elevation of the sun.
\( \tan \theta = \dfrac{12}{8} = \dfrac{3}{2} \)
\( \theta = \tan^{-1}(3/2) \)
7. Example 5: When Observer and Object Are at Different Heights
Problem: A person standing on a 10 m platform observes the top of a tree at an angle of elevation of \(40^\circ\). The horizontal distance to the tree is 30 m. Find the total height of the tree.
\( \tan 40^\circ = \dfrac{h - 10}{30} \)
\( h - 10 = 30 \tan 40^\circ \)
\( h = 10 + 30 \tan 40^\circ \)