Chapter in a nutshell: This is the application of trigonometry to find unknown heights and distances using the angle of elevation (looking up) or angle of depression (looking down). You draw a right triangle and use sin, cos, tan (mostly tan) with the standard angle values.
1. Key Ideas & Terms
- Line of sight: the line from the observer's eye to the object.
- Angle of elevation: angle above the horizontal (object higher than eye).
- Angle of depression: angle below the horizontal (object lower than eye).
- The angle of depression from A to B = the angle of elevation from B to A (alternate angles).
2. Method
- Draw the situation as a right-angled triangle, marking the angle and the known/unknown sides.
- Choose the ratio linking the known and unknown sides:
- Substitute the standard value and solve.
3. Formulas (quick reference)
- $\tan\theta=\dfrac{\text{height (opposite)}}{\text{distance (adjacent)}}$
- $\sin\theta=\dfrac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta=\dfrac{\text{adjacent}}{\text{hypotenuse}}$
- Angle of depression (from top) = angle of elevation (from bottom).
4. Prerequisite Formulas (earlier classes — in full)
Trig ratios: $\sin=\frac{\text{opp}}{\text{hyp}}$, $\cos=\frac{\text{adj}}{\text{hyp}}$, $\tan=\frac{\text{opp}}{\text{adj}}=\frac{\sin}{\cos}$. Standard angle values:| θ | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cos | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tan | 0 | 1/√3 | 1 | √3 | ∞ |
5. Worked Example (one, for the method)
Q. The angle of elevation of the top of a tower from a point 30 m away is 30°. Find the height. Solution: $\tan30^\circ=\dfrac{h}{30}\Rightarrow h=30\times\dfrac{1}{\sqrt3}=10\sqrt3\approx17.3$ m.6. Common Mistakes to Avoid
- Mixing up elevation (up) and depression (down).
- Measuring the depression from the vertical instead of the horizontal.
- Choosing the wrong ratio (use tan when height & base distance are involved).
- Forgetting to add the observer's height when the eye is above the ground.
- Using degrees/values incorrectly (memorise the standard table).
7. Likely Exam Questions (with crisp answers)
- Define angle of elevation. → The angle above the horizontal to an object higher than the eye.
- Define angle of depression. → The angle below the horizontal to an object lower than the eye.
- Relation between elevation and depression for two points? → They are equal (alternate angles).
- Which ratio links height and horizontal distance? → tan.
- $\tan60^\circ$? → √3.
- A 30° elevation at 30 m gives what height? → $10\sqrt3\approx17.3$ m.
- Which line makes the angle with the horizontal? → The line of sight.
- $\sin45^\circ$? → $1/\sqrt2$.
- Approx value of $\sqrt3$? → 1.732.
- When the object is below, which angle is used? → The angle of depression.
Extended & Prerequisite Formulas (full)
- tanθ = height/horizontal distance; sinθ=opp/hyp; cosθ=adj/hyp; depression∠ (top) = elevation∠ (bottom).
- Special triangles: 45-45-90 sides 1:1:√2; 30-60-90 sides 1:√3:2.
- Values: sin/cos/tan at 0,30,45,60,90 (see table); √3≈1.732, √2≈1.414. Pythagoras hyp²=opp²+adj².