Question

Plz… help me to do this sum

A spherical balloon of radius 'r' subtends an angle theta at the eye of the observer. If the angle of elevation of its centre @ , find the height of the centre of the balloon.

Answer 1

$\boxed{\bold{TRIGONOMETRY : }}$

✔ Trigonometry is the study of relationships between the sides and the angles of a triangle. 

✔ The general trigonometric ratios are sin theta, cos theta, tan theta, sec theta, cosec theta and cot theta. 

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We all know a triangle has three parts : base, height and hypotenuse.

✔ Sin theta = height/hypotenuse. 

✔ Cos theta = base/hypotenuse. 

✔ tan theta = height /base. 

✔ cot theta = base/height. 

✔ sec theta = hypotenuse/base. 

✔ cosec theta = hypotenuse/height. 

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$\bold{\huge{SOLUTION \: :}}$

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Let us say that spherical balloon of radius "r" subtends an anagle "α" at the eye of an observer. If the angle of elevation of the centre of the balloon be "β", then the height of the centre of the balloons is calculated in the following procedure :


Let the two angles be α and β respectively.

Similarly, let the total height be "h".

In the above picture,

<QPC = α

So, <QPB = <BPC = α/2

In ∆ PQB, sin α/2 = r/l

=> l = r sin α/2

and in ∆ POB sin β=h/l

h = l sinβ

h = r cosec α/2 sinβ [ANSWER]