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Trigonometry Height And Distance

    • Theory
    • Set 1
    • Set 2
    • Set 3
    • Set 4
    • Set 1
    • Set 2
  1. Aptitude
  3. Quantitative
  5. Trigonometry & Height and Distance
  7. Theory

Trigonometry (Height and Distance): Concepts, Formulas, and Clear Examples

Use right-triangle ratios sin, cos, tan for angles of elevation/depression. Angle of elevation from horizontal up; depression down—both measured from horizontal.

Standard angles 30∘,45∘,60∘30^\circ, 45^\circ, 60^\circ30∘,45∘,60∘ use 12,13,3\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{3}}, \sqrt{3}2​1​,3​1​,3​ shortcuts.

Draw ground line and two observations; same vertical height links two triangles.

Primary ratios

sin θ

sin⁡θ=oppositehypotenuse\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}sinθ=hypotenuseopposite​

cos θ

cos⁡θ=basehypotenuse\cos\theta = \frac{\text{base}}{\text{hypotenuse}}cosθ=hypotenusebase​

tan θ

tan⁡θ=oppositebase\tan\theta = \frac{\text{opposite}}{\text{base}}tanθ=baseopposite​

Standard values

  • tan⁡30∘=1/3\tan30^\circ=1/\sqrt{3}tan30∘=1/3​, tan⁡45∘=1\tan45^\circ=1tan45∘=1, tan⁡60∘=3\tan60^\circ=\sqrt{3}tan60∘=3​.
  • sin⁡30∘=1/2\sin30^\circ=1/2sin30∘=1/2, cos⁡60∘=1/2\cos60^\circ=1/2cos60∘=1/2.

Elevation single observer

Height

h=dtan⁡θh = d\tan\thetah=dtanθ

Example — 45° elevation

Distance 40 m, angle 45°. Height = 40tan⁡45∘=4040\tan45^\circ=4040tan45∘=40 m.

Depression equals elevation

From top of tower, depression to object equals elevation from object to top if same horizontal reference.

Two observation points

  • Same height difference: d1tan⁡θ1=d2tan⁡θ2d_1\tan\theta_1 = d_2\tan\theta_2d1​tanθ1​=d2​tanθ2​ or set up with variable height.

Shadow problems

  • Sun altitude θ: shadow length = height / tan θ.
  • Moving sun changes θ over time—two snapshots give two equations.

Step-by-step method for exam questions

  1. Sketch horizontal, vertical, sight line.
  2. Mark angle of elevation/depression from horizontal.
  3. Choose ratio linking known side and unknown height.
  4. For two triangles, equate heights or use baseline difference.
  5. Rationalize surds if needed.
  6. Pick option.

Formula cheat sheet

SituationFormulaWhen to use
Height from distanceh=dtan⁡θh=d\tan\thetah=dtanθAngle of elevation
Shadowshadow=h/tan⁡θ\text{shadow}=h/\tan\thetashadow=h/tanθSun altitude
30-60-901:3:21:\sqrt{3}:21:3​:2Special triangle
Complementarysin⁡(90−θ)=cos⁡θ\sin(90-\theta)=\cos\thetasin(90−θ)=cosθSwitch ratio
Two angles same towerEquate h expressionsTwo distances

Frequently asked questions

Angle from vertical?

Convert to from horizontal: 90∘−from vertical90^\circ-\text{from vertical}90∘−from vertical.

Observer height?

Add eye height to ground-level result if stem gives it.

km vs m?

Convert before tan; trig ratios unitless.

Depression from building?

Alternate interior with elevation at ground point.

Practice on this platform

Use 4 practice sets (10 MCQs, detailed solution after each answer) and 2 timed mocks (20 minutes) on this topic to convert formulas into speed and accuracy.

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