Question

The lengths of the sides of a triangle are consecutive integers, and the largest angle is twice the smallest angle. What is the cosine of the smallest angle

Answer 1

Answer:
`cosA = (a+2)/(2a)`

Explanation:

Given that the lengths of the sides of a triangle are consecutive integers

Since side cannot be negative we can assume that the sides are

a, a+1 and a+2 where a>0

The largest angle is opposite side a+2 and smallest is angle opposite a

Using sine formula and the given information that the largest angle is twice the smallest angle

we get

`(a)/(sin A) =(a+2)/(sin 2A)`

Cross multiply to get

`(sin 2A)/(sin A ) =(a+2)/(a)`

Since sin 2A = 2sin A cos A

we get

`2cosA = (a+2)/(a) \\cosA = (a+2)/(2a)`