In `Delta ABC`, If `angle C = 3 angle A, BC = 27, and AB =48`. Then the value of AC is ______

Answer

35

Explanation

Using the sine law, we get
`(27)/(sin theta) = (48)/(sin 3 theta)`, where `theta = angleA`
`rArr (sin 3 theta)/(sin theta) = (16)/(9)`
`rArr 3 - 4 sin^(2) theta = (16)/(9)`
`rArr 4 sin^(2) theta = 3 - (16)/(9) = (11)/(9)`
`rArr sin^(2) theta = (11)/(36)`
Now `(b)/(sin 4 theta) = (27)/(sin theta)`
or `(b)/(27) = (sin 4 theta)/(sin theta)`
`= (2 sin 2 theta cos 2 theta)/(sin theta) = 4 cos theta cos 2 theta`
`= 4 cos theta (1- 2 sin^(2) theta)`
or `(b)/(27) = 4 sqrt(1 - sin^(2) theta) (1 -2 sin^(2) theta)`
`= sqrt(1 - (11)/(36)) (1 - (22)/(26))`
`= 4 ((5)/(6)) ((14)/(36)) or b = 35`