In a triangle, the two sides are 5cm and 8cm, and the sine of the angle between them is 0.8. Find the third side

To solve this task, we will use the cosine theorem, that is? the square of any side of the triangle is equal to the sum of the squares of the other two sides without the double product of these sides by the cosine of the angle between them;

a ^ 2 = b ^ 2 + c ^ 2 – 2 a b cos a

And since, by condition, we are given the sine of this angle, we first express the cosine through the sine by the formula:

sin ^ 2 a + cos ^ 2 a = 1; cos a = √ (1 – sin ^ 2 a) = √ (1 – 0.64) = √0.36 = 06;

We substitute the obtained result and the initial data into the cosine theorem;

a = 5 ^ 2 + 8 ^ 2 – 2 * 5 * 8 * 0.6 = √41.