Some Applications of Trigonometry
Elevation and depression
Use angles of elevation and depression to find heights.
Most of this chapter is one right triangle: you know one angle and one side, and you want another side. Only four words are new.
| Word | Meaning |
|---|---|
| line of sight | the straight line from the observer's eye to the thing they look at |
| horizontal | a line from the observer, parallel to the ground |
| angle of elevation | you look up: the angle between the line of sight and the horizontal |
| angle of depression | you look down: the angle between the line of sight and the horizontal |
Two common slips: both angles are measured from the horizontal, never from the vertical. And the horizontal is drawn at the observer's eye, not at the object.
Plan before you calculate. To find the height of the Qutub Minar, what must you know? The distance from its foot, the angle of elevation, and (if you are careful) your own eye height.
Pick the ratio by asking: which two sides do I have, and which do I want?
| Sides involved | Use |
|---|---|
| height and ground distance | tan (or cot) |
| height and a sloping length (ladder, rope, kite string) | sin (or cosec) |
| ground distance and a sloping length | cos (or sec) |
Example: a tower is seen at 60° from 15 m away. tan 60° = h/15, so h = 15√3 m.
Eye height: if the observer's eyes are 1.5 m above the ground, draw the horizontal at eye level. Find the height above the eyes, then add 1.5 m back.
Always: read slowly and draw as you read. Leave √3 in the answer unless the question gives a value for it. End with a sentence: 'So the height of the tower is … m.'
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