Class 10 Maths Notes Chapter 9 — Some Applications of Trigonometry#
Key terms#
- Line of sight — the line from the observer’s eye to the point on the object being viewed.
- Horizontal — the line through the observer’s eye, parallel to the ground.
- Angle of elevation — the angle between the line of sight and the horizontal when the object is above eye level (you raise your head).
- Angle of depression — the angle between the line of sight and the horizontal when the object is below eye level (you lower your head).
The angle of depression from A to B equals the angle of elevation from B to A. They are alternate angles, because both horizontals are parallel.
Solving heights and distances#
- Draw a neat figure and mark the right angle, the known lengths and the angles.
- Pick the right triangle that contains the unknown and one known side.
- Choose the ratio that links them. $\tan\theta$ is used most, because it links height and horizontal distance: $$\tan\theta = \frac{\text{height}}{\text{horizontal distance}}$$ Use $\sin\theta$ or $\cos\theta$ when the slant length (rope, ladder, string, line of sight) is involved.
- Substitute the standard value and solve.
| $30°$ | $45°$ | $60°$ | |
|---|---|---|---|
| $\sin$ | $\frac{1}{2}$ | $\frac{1}{\sqrt{2}}$ | $\frac{\sqrt{3}}{2}$ |
| $\cos$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{\sqrt{2}}$ | $\frac{1}{2}$ |
| $\tan$ | $\frac{1}{\sqrt{3}}$ | $1$ | $\sqrt{3}$ |
Common set-ups#
- Two angles from two points on the same line (object moves, or observer walks): write $\tan$ for each triangle, form two equations, and subtract or substitute.
- Observer’s height: subtract the eye height from the object’s height before using the ratio. A 1.5 m boy looking at a 30 m building sees a height of 28.5 m above his eyes.
- Object on top of another (a statue on a pedestal, a tower on a building): use the bottom and the top as two separate angles from the same point.
- Angle of 45°: height = horizontal distance, because $\tan 45° = 1$.
- Speed and time: find the distance covered, then use time = distance ÷ speed. Often the time comes out as a simple ratio, as in “6 seconds then 3 seconds”.
Common mistakes#
- Measuring the angle of depression from the vertical instead of the horizontal.
- Using the full height of the object when the observer’s eye is above the ground.
- Choosing $\sin$ when the slant length is not given or needed. Use $\tan$ for height and distance.
- Rationalising wrongly: $\frac{30}{\sqrt{3}} = 10\sqrt{3}$, not $30\sqrt{3}$.