A tower subtends angles α , 2 α and 3 α , respectively, at points A , B and C all lying on a horizontal line through the foot of the tower. If A B B C = 1 + p cos ( p α ) , then the value of p is

Question

A tower subtends angles α , 2 α and 3 α , respectively, at points A , B and C all lying on a horizontal line through the foot of the tower. If A B B C = 1 + p cos ( p α ) , then the value of p is,

MARKS App · Accepted Answer

Let P M be the tower. Given, ∠ B A P = α , ∠ C B P = 2 α a n d ∠ M C P = 3 α ∴ ∠ A P B = ∠ P B C − ∠ B A P = 2 α − α = α Similarly, ∠ B P C = α ⇒ B P = A B By sine formula in triangle B C P , we have B C sin ( ∠ B P C ) = B P sin ( ∠ B C P ) ⇒ B C sin α = A B sin ( 180 o − 3 α ) ⇒ A B B C = sin 3 α sin α = 3 sin α − 4 sin 3 α sin α = 3 − 4 sin 2 α = 3 − 2 ( 1 − cos 2 α ) = 1 + 2 cos 2 α

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