Active and passive voice: rules for every tense
How the verb changes in each tense, pronoun changes, modals and imperative sentences, with SSC examples.
Study notes
Heights and distances: angle setups, results and solved examples
Angles of elevation and depression, the 30°, 45° and 60° triangles, two-observer setups and ready results for SSC questions.
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The key terms
- Line of sight: the line from the eye to the object.
- Angle of elevation: the angle between the horizontal and the line of sight when you look up at an object.
- Angle of depression: the angle between the horizontal and the line of sight when you look down at an object.
The angle of elevation of B from A equals the angle of depression of A from B, because they are alternate angles between parallel horizontal lines.
The basic ratio
tan θ = Height/Horizontal distance
sin θ = Height/Line of sight · cos θ = Horizontal distance/Line of sight
sin θ = Height/Line of sight · cos θ = Horizontal distance/Line of sight
Almost every question uses tan θ, because height and ground distance are what you are given or asked for.
Standard triangles
| Angle of elevation | tan θ | Height : distance |
|---|---|---|
| 30° | 1/√3 | 1 : √3 |
| 45° | 1 | 1 : 1 |
| 60° | √3 | √3 : 1 |
In a 30°–60°–90° triangle the sides opposite 30°, 60° and 90° are in the ratio 1 : √3 : 2. In a 45°–45°–90° triangle they are 1 : 1 : √2.
- At 45°, height = distance.
- At 30°, distance = height × √3.
- At 60°, height = distance × √3.
Two observation points on the same side
A tower of height h is seen from two points on the same straight line, d apart, at angles α (nearer point) and β (farther point).
h = d × tan α × tan β/(tan α − tan β)
| Angles | Height h | Nearer point from the foot |
|---|---|---|
| 60° and 30° | d√3/2 | d/2 |
| 60° and 45° | d(3 + √3)/2 | d(√3 + 1)/2 |
| 45° and 30° | d(√3 + 1)/2 | d(√3 + 1)/2 |
Complementary angles rule
If the angles of elevation of a tower from two points at distances a and b from its foot (on the same line) are complementary (add up to 90°), then:
h = √(ab)
Example: points 4 m and 9 m from the foot give h = √36 = 6 m.
Shadows
When the sun’s elevation is θ, an object of height h casts a shadow of length h/tan θ. At 45° the shadow equals the height. At 60° it is h/√3, and at 30° it is h√3. As the sun rises, the shadow gets shorter.
Shortcut: draw a rough figure every time. Mark the right angle at the foot of the tower, then write the 1 : √3 : 2 or 1 : 1 : √2 sides straight onto it. Most answers appear without any equation.
Use √3 ≈ 1.732 and √2 ≈ 1.414 when options are in decimals.
Solved examples
- A tower is 50√3 m high. Find the angle of elevation of its top from a point 150 m from its foot.
tan θ = 50√3/150 = √3/3 = 1/√3, so θ = 30°. - From a point on the ground, the angle of elevation of a tower’s top is 30°. After walking 40 m towards the tower, it becomes 60°. Find the height.
Using the 60°/30° result, h = d√3/2 = 40√3/2 = 20√3 m ≈ 34.64 m. - From the top of a 75 m lighthouse, the angles of depression of two ships on the same side are 30° and 45°. Find the distance between the ships.
Ship at 45°: distance = 75 m. Ship at 30°: distance = 75√3 m. Gap = 75√3 − 75 = 75(√3 − 1) ≈ 75 × 0.732 = 54.9 m.
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