In triangle ABC, angle C = 90, sinA = 13/14, AC = 6√3. Find AB.

Given: right-angled triangle ABC;

angle C = 90;

sin A = 13/14;

AC = 6√3;

Find: AB -?

Decision:

1) Let’s use the formula

cos ^ 2A + sin ^ 2A = 1;

cos ^ 2A = 1 – sin ^ 2A;

cos ^ 2A = 1 – 169/196;

cos ^ 2A = 196/25 – 169/25;

cos ^ 2A = 27/25;

cos A = 3√3 / 5;

2) Consider a right-angled triangle ABC. The cosine of the angle in a right-angled triangle is equal to the ratio of the adjacent leg to the hypotenuse. Consequently:

cos A = AC / AB;

AB = CA / cos A;

AB = 6√3: 3√3 / 5;

AB = 6√3 * 5/3√3;

AB = 10.

Answer: AB = 10.