Heights and Distances: Elevation and Depression | NIMCET

Quick answer: In heights and distances, the angle of elevation is measured upward from the horizontal to an object above the observer, and the angle of depression is measured downward from the horizontal to an object below. Most problems reduce to right triangles: $\tan\theta = \frac{\text{height}}{\text{horizontal distance}}$. If a tower of height $h$ is seen at elevations α and β from two points on the same side, $d$ apart, then $h = \frac{d\tan\alpha\tan\beta}{\tan\alpha - \tan\beta}$ (with α > β).

Key terms

The angle of depression from A to B equals the angle of elevation from B to A, because the two horizontals are parallel (alternate angles).

Standard values

Most questions use these angles. For other angles, see the trigonometric ratios and allied angles.

Method

Step 1: Draw a clear diagram: ground as a horizontal line, the tower vertical, and the angles marked at the observer.

Step 2: Identify each right triangle and the unknowns.

Step 3: Write one tan (or sin/cos) equation per triangle.

Step 4: Solve the equations together.

Standard results

Proof of the complementary result: $\tan\theta = \frac{h}{a}$ and $\tan(90^\circ - \theta) = \cot\theta = \frac{h}{b}$. Multiply: $1 = \frac{h^2}{ab}$, so $h = \sqrt{ab}$.

Worked examples

Example 1: one observation

A ladder makes 60° with the ground and its foot is 5 m from the wall. Find the length of the ladder.

$\cos 60^\circ = \frac{5}{L}$, so $L = 10$ m.

Example 2: two observations on the same side

From a point on the ground, the elevation of a tower top is 30°. After walking 20 m towards the tower, it becomes 60°. Find the height.

Step 1: Let the height be $h$ and the second distance be $x$. Then $h = x\sqrt{3}$ and $h = \frac{x + 20}{\sqrt{3}}$.

Step 2: $3x = x + 20$, so $x = 10$ and $h = 10\sqrt{3} \approx 17.32$ m.

Check with the formula:

$h = \frac{20 \cdot \sqrt{3} \cdot \frac{1}{\sqrt{3}}}{\sqrt{3} - \frac{1}{\sqrt{3}}} = \frac{20}{\frac{2}{\sqrt{3}}} = 10\sqrt{3}$

Example 3: angle of depression

From the top of a 50 m cliff, the angle of depression of a boat is 45°. How far is the boat from the foot of the cliff?

The angle of elevation from the boat is also 45°, so the distance equals the height: 50 m.

Example 4: a flagstaff on a tower

From a point 30 m from the foot of a tower, the elevations of the top of the tower and the top of a flagstaff on it are 30° and 45°. Find the length of the flagstaff.

Step 1: Tower height $= 30\tan 30^\circ = 10\sqrt{3}$.

Step 2: Tower plus flagstaff $= 30\tan 45^\circ = 30$.

Step 3: Flagstaff $= 30 - 10\sqrt{3} \approx 12.68$ m.

Summary table

Exam tips

• Always draw the figure; most errors come from placing the angle at the wrong vertex.
• Convert depression angles to elevation angles at the other point.
• Use the shortcut formulas to check, not to replace, the diagram method.
• If the observer's height is given, subtract it from the object's height first.

For more questions, see the trigonometry shortcuts for NIMCET and the NIMCET Trigonometry chapter.

Common mistakes

• Measuring the angle of depression from the vertical instead of the horizontal.
• Ignoring the observer's eye height when it is given.
• Using sin when the hypotenuse is not involved; tan is usually the right ratio.
• Mixing up the two distances in two-observation problems.

Frequently Asked Questions

What is the angle of elevation?

It is the angle between the horizontal line and the line of sight when the observer looks up at an object.

What is the angle of depression?

It is the angle between the horizontal line and the line of sight when the observer looks down at an object.

Is the angle of depression equal to the angle of elevation?

Yes. The angle of depression from A to B equals the angle of elevation from B to A, because the horizontal lines are parallel.

What is the height if elevations from two points are complementary?

If the points are at distances a and b from the foot on the same line, the height is √(ab).

Which trigonometric ratio is used most in heights and distances?

The tangent ratio, because problems usually link a height with a horizontal distance.