In triangle ABC, the angle at the apex A is 60 degrees, AB = 3cm, BC = 4cm. Find the AC.

Let’s build the height of the ВН and in the right-angled triangle ABН determine the length of the legs ВН and AН.

The AH leg is located opposite an angle of 300, then AH = AB / 2 = 3/2 = 1.5 cm.

Cos60 = BH / AB.

BH = AB * Cos60 = 3 * √3 / 2 cm.

In a right-angled triangle BCH, according to the Pythagorean theorem, we determine the length of the leg CH.

CH ^ 2 = BC ^ 2 – BH ^ 2 = 16 – 27/4 = (64 – 27) / 4 = 37/4.

CH = √37 / 2.

Then AC = BH + CH = (3/2) + (√37 / 2) = (3 + √37) / 2 cm.

Answer: The length of the speaker side is (3 + √37) / 2 cm.