The sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one. Determine the sides of the triangle.

Explanation

Let a-1, a+1`((a-1) int N)` be the sides of the given triangle and `theta` is the smallest angle and `2theta` is largest angle. Obviously the third angle is `180^(@)-3theta`
`therefore` Sine rule,
`(sintheta)/(sin theta)= (a+1)/(1-1)`
`rArr costheta = (a+1)/(2(a-1))`..........(i)
Also, `(sintheta)/(a-1) =(3sin theta-4sin^(3)theta)/(a)`
`rArr a/(a-1) = 3-4sin^(2)theta`
`rArr sin^(2)theta = (2(a-3))/(4(a-1))`...........(ii)
`therefore sin^(2)theta + cos^(2)theta=1`
`therefore sin^(2)theta + cos^(2)theta=1`
`therefore ((2a-3)/(4(a-1))) + ((a+1)/(2(a-1)))^(2)=1` [By (i) and (ii)]
`rArr 2a^(2) - 5a+3 + a^(2)+2a+1 = 4a^(2) -8a+4`
`rArr a^(2)-5a=0`
`rArr a=5 {therefore a ne 0}`
Hence sides of a triangle are 4,5,6