Some Applications of Trigonometry — Notes
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9.1 Line of Sight
The line of sight is the line drawn from the observer’s eye to the point being viewed.
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9.2 Angle of Elevation
When the observed point is above the observer’s horizontal level, the angle between the line of sight and horizontal is the angle of elevation.
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9.3 Angle of Depression
When the observed point is below the observer’s horizontal level, the angle between the line of sight and horizontal is the angle of depression. With parallel horizontal lines, the relevant alternate interior angle can be used in the ground triangle.
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9.4 Solving Heights and Distances
Draw a right-triangle model first. Mark the known height, horizontal distance and angle. Then choose sin, cos or tan according to the sides involved.
Typical pattern
If height h and horizontal distance d are involved with angle θ, tan θ = h/d. If a ladder or rope is the hypotenuse, sin or cos may be more appropriate.
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9.5 Practical Problems
NCERT applications include towers, buildings, ladders, shadows, bridges, rivers, poles, lighthouses and moving objects.
⚡ Quick Revision
- Draw the figure before calculating.
- Elevation → object above horizontal; depression → object below horizontal.
- Use parallel horizontal lines to relate depression to an elevation angle.
- Choose the ratio that contains the required side.
🎯 Board Exam Focus
- Include observer height when the question gives eye level.
- Do not confuse the horizontal distance with the line-of-sight distance.
- Keep units consistent.
🚀 Last-Minute Revision
✔ Read every formula with its conditions and symbols.
✔ Write given data first and keep units consistent.
✔ For proofs, state the theorem and show logical steps.
✔ Check signs, substitutions and final units.
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