The two sides of the triangle are 4 cm and 10 cm, and the sine of the angle between them is 4/5. Find the third side of the triangle.

From the trigonometric identity (cos a) ^ 2 + (sin a) ^ 2 = 1 we can find (cos a) ^ 2 = 1- (sin a) ^ 2.
Knowing that sin a = 4/5 = 0.8, we find (cos a) ^ 2 = 1-0.8 ^ 2 = 1-0.64 = 0.36; cos a = √0.36 = 0.6.
According to the cosine theorem, the square of any side of a triangle is equal to the sum of the squares of the other two sides minus twice the product of those sides by the cosine of the angle between them.
Find the unknown side of the triangle: a ^ 2 = 4 ^ 2 + 10 ^ 2-2 * 4 * 10 * cos a = 16 + 100-80 * 0.6 = 68.
a = √68 = 2√17, which is approximately 8.246 cm.