Height and Distance Questions, Formulas and Shortcut Tricks

Height and Distance Solved Examples - Page 2
Height and Distance Important Questions - Page 3
Height and Distance Video Lecture - Page 4

Height and Distance Formulas

One of the important applications of trigonometry is in finding the height and distances of the point which are not directly measurable. This is done with the help of trigonometric ratios.

DEFINITIONS

(A) Angle of Elevation
Let
O: Position of the object
A: Position of the height
Here, object (A) is at height level than observer (O)
OX: Reference line (or horizontal line)
OA: Line of sight (or line of observation)
Then
$\theta$ = $\angle$ AOX =angle of elevation

Angle of Depression

Let
O: Position of the observer
B: Position of the object
Here, object (B) is at lower level than the observer (O)
OX: Reference line (or horizontal line)
OB: Line of sight (or line of observation)
Then
$\beta$ = $\angle$ BOX =angle of depression

RESULTS USEFUL IN FINDING HEIGHTS AND DISTANCE

(A) In a right-angle triangle ABC,*

Sin= $\theta$ = $\frac{p}{h}$
cos= $\theta$ = $\frac{b}{h}$
tan= $\theta$ = $\frac{p}{b}$

(B) In any triangle ABC,

$\frac{a}{sin A}=\frac{b}{sin B}=\frac{c}{sin C}$ [ Sin Rule ]

$\Rightarrow \:\frac{Length\:\:of\:\:any\:\:side}{Sine\:\:of\:\:the\:\:angle\:\:opposite\:\:to\:\:the\:\:side}$ = constant (of each side of the angle)

(C) In any angle ABC

If $\frac{AD}{DC}=\frac{m}{n}$ and
$\angle$ BAD = $\alpha$
$\angle$ CAD = $\beta$
$\angle$ ADC = $\theta$
Then, (m + n) cot $\theta$ =m cot $\alpha$ -n cot $\beta$

(D) In a right - angle triangle ABC,

If DE $\left |\right |$ AB, then
$\frac{AB}{DE}=\frac{BC}{DC}$

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