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:$cosA = \frac{a+2}{2a}$

Step-by-step 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

$\frac{a}{sin A} =\frac{a+2}{sin 2A}$

Cross multiply to get

$\frac{sin 2A}{sin A } =\frac{a+2}{a}$

Since sin 2A = 2sin A cos A

we get

$2cosA = \frac{a+2}{a} \\cosA = \frac{a+2}{2a}$