Height and distance: two angles of elevation
You cannot reach the foot of a temple tower, so you sight its top from two points on a straight line towards it. Change the two angles and the distance walked, and find the tower's height and how far away it is.
Core concept
If the top of a tower is seen at angles α and β from two points d apart on a line towards it, then h = d tan α tan β ÷ (tan β − tan α), and the near point is h ÷ tan β from the tower
Topic: Trigonometry
The angle: The amount of turn between two lines that meet, measured in degrees (°). All key words of elevation is the angle between the level ground and your line of sight up to the top of something. Looking down from a roof to a point on the ground, the angle with the level is the angle of depression, and it equals the angle of elevation from that point back up. Often you cannot reach the foot of a tower, such as a temple tower or Dharahara with a fence round it. Then sight its top from a far point A at angle α, walk d metres straight towards it to B and sight it again at a bigger angle β. With x the distance from B to the tower, tan β = h ÷ x and tan α = h ÷ (x + d). Solving the two gives h = d tan α tan β ÷ (tan β − tan α), and then x = h ÷ tan β.
What you will learn
- Explain angles of elevation and depression.
- Use tan to link a height and a distance in a right-angled triangle.
- Find a height from two angles of elevation and the distance between the points.
Before you start
- Know tan 30° = 1/√3, tan 45° = 1 and tan 60° = √3.
Materials
- A clinometer made from a protractor, a straw and a thread with a weight
Equipment
- A measuring tape
Procedure
- Record 30° and 60° with 20 m walked.
- Walk 50 m and record.
- Change the angles to 45° and 60° and record.
- Change the angles to 30° and 45° and record.
How this simulation is simplified
- The ground is level, both points are on a straight line to the foot of the tower, and the eye is at ground level.