The two sides of a triangle, 2 cm and 17 cm long, form an acute angle

The two sides of a triangle, 2 cm and 17 cm long, form an acute angle, the tangent of which is 8/15. Find the length of the third side of the triangle.

According to the condition of the problem, the two sides of the triangle a and b, respectively 2 cm and 17 cm long, form an acute angle α, the tangent of which is 8/15.
Using the formula cos ^ 2α = 1 / (1 + tg ^ 2α) we find cosα:
cos ^ 2α = 1 / (1 + tg ^ 2α) = 1 / (1 + (8/15) ^ 2) = 1 / (1 + 64/225) = 1 / (225/225 + 64/225) = 1 / (225/225 + 64/225) = 1 / (289/225) = 225/289 = 15 ^ 2/17 ^ 2.
By the hypothesis of the problem, the angle α is acute; therefore, cosα is positive, and hence cosα = 15/17.
Find the length of the third side from the triangle using the cosine theorem:
c ^ 2 = a ^ 2 + b ^ 2 – 2 * a * b * cosα = 2 ^ 2 + 17 ^ 2 – 2 * 2 * 17 * (15/17) = 4 + 289 – 60 = 233.
c = √233.
Answer: The length of the third side from the triangle is √233.