In triangle ABC, angle C = 90 degrees, angle A = 30, AC = 34√3. Find AB.

In triangle ABC, we find the hypotenuse AB, if the values are known:

Angle C = 90 °;
Angle A = 30 °;
AC = 34√3.
Decision:

Consider a triangle ABC.

cos a = AC / AB (the ratio of the adjacent leg to the angle A to the hypotenuse);

From here we will express AC.

AB = AC / cos A;

Substitute the known values of the leg and angle into the formula and find the hypotenuse AB.

AB = 34√3 / cos 30 = 34√3 / (√3 / 2) = 34√3 * 2 / √3 = 34 * 2 * √3 / √3 = 34 * 2 = 30 * 2 + 4 * 2 = 60 + 8 = 68.

This means that the hypotenuse of the triangle ABC is equal to AB = 68.

Answer: AB = 68.