Question

Two circles of radii 4 cm and 1 cm touch each other externally and 8 is the angle contained by their direct
common tangents. Find the value of
$\sin( \frac{ \alpha }{2} ) + \cos( \frac{ \alpha }{2} )$
need right answer​

Answer 1

Given : Two circles of radii 4 cm and 1 cm touch each other externally and α is the angle contained by their direct common tangents

To Find : Value of Sin( α/2) + Cos( α/2)

Solution:

Ref attached figure :

PC & PE are common tangent to Both circles

α is the angle between PC & PE

=> α/2 is the angle ∠BPD

Let say  BP = x

then AP = 4 + 1 + x = 5 + x

ΔACP ≈ ΔBDP ( as ∠C = ∠D = 90°  & ∠P is coomon) )

=> AC/BD = AP/BP

AC = 4   & BD = 1

=> 4/1 = (5 + x)/x

=> 4x = 5 + x

=> 3x = 5

=> x = 5/3

Sin( α/2)  = 1 /(5/3)  = 3/5

=> Cos( α/2) = 4/5   ( as Cos²x + Sin²x = 1)

Sin( α/2) + Cos( α/2)

= 3/5 + 4/5

= 7/5

= 1.4

Sin( α/2) + Cos( α/2) = 7/5 = 1.4

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