0580

Right Angled Triangles

Pythagoras and Trigonometry

Elevation and depression

  • An angle of elevation is measured up from the horizontal to the line of sight, and an angle of depression is measured down from it
  • Both are measured from the horizontal, never from the vertical, which is the usual misreading
An observer's eye with a dashed horizontal line drawn from it: the line of sight up to a bird makes the angle of elevation above that horizontal, and the line of sight down to a boat makes the angle of depression below it
Source: Elevation & depression by Save My Exams
  • Sketch the situation with the horizontal drawn in and the right angle marked, then apply SOHCAHTOA
The elevation problem turned into a right-angled triangle BFT: the dashed horizontal at the top of the tower makes the 35 degree angle of depression, which equals the 35 degree angle of elevation at B, with the 24 m height marked O and the sloping line of sight marked H
Source: Elevation & depression by Save My Exams
  • Tangent appears most often, because these questions usually pair a height with a horizontal distance
A vertical tower of height 24 m seen from a point B on the ground: the angle of elevation to the top T is 35 degrees measured down from the horizontal at T, and the angle to a point M part way up is 18 degrees at B, with the height x m of M still to be found
Source: Elevation & depression by Save My Exams
Worked example

An angle of elevation

From a point 40 m from the base of a tower, the angle of elevation of the top is 32°. Find the height of the tower, correct to 1 decimal place.

A right-angled triangle standing on horizontal ground. The ground from the observation point to the foot of the tower is labelled 40 m, the tower rises vertically with a right angle marked at its foot, and the angle of elevation at the observation point is marked 32 degrees. The height of the tower is unlabelled.

Solution:

  • The horizontal distance is adjacent to the 32° angle and the height is opposite it
  • O and A means tangent
  • tan 32° = height ÷ 40
  • height = 40 × tan 32°
  • height = 24.9947…
  • The tower is 25.0 m tall to 1 decimal place

Shortest distance to a line

  • The shortest distance from a point to a line is the perpendicular distance, measured at a right angle to the line
  • The syllabus names this directly, and these papers ask for a "shortest distance" in 6 of the 43
  • Drawing that perpendicular creates a right-angled triangle, which Pythagoras or SOHCAHTOA then solves
  • Bearings questions often combine with this, so a north line and a bearing may supply the angle