Cosine of a basic angle of an isosceles triangle is 1⁄2, while its area equals 4√3 cm2. Find the lengths of the sides of this triangle.

To find the lengths of the sides of the isosceles triangle, we can use the cosine rule and the formula for the area of a triangle.

Let's label the two equal sides of the triangle as "a" and the third side as "b". We know that the cosine of one of the base angles is 1/2. Since the triangle is isosceles, the other base angle also has the same cosine value.

Using the cosine rule, we have:
cos(base angle) = (a^2 + a^2 - b^2) / (2 * a * a)
1/2 = (2a^2 - b^2) / (2 * a^2)
1 = 2a^2 - b^2

Since the area of the triangle is given as 4√3 cm², we can use the formula for the area of a triangle:
Area = (1/2) * b * h
4√3 = (1/2) * b * h
8√3 = b * h

Now, we have two equations:
1 = 2a^2 - b^2
8√3 = b * h

To solve these equations, we can substitute h in terms of b using the area formula:
h = (8√3) / b

Substituting h in the cosine equation, we have:
1 = 2a^2 - b^2
1 = 2a^2 - (b * (8√3) / b)^2
1 = 2a^2 - (64 * 3) / b^2
1 = 2a^2 - (192 / b^2)
1 = 2a^2 - (192 / (2a^2 - 1))
Simplifying the equation:
2a^2 - (192 / (2a^2 - 1)) = 1

To solve this equation, we can use numerical methods or graphing calculators to find the values of a.

Once we have the value of a, we can substitute it back into either of the equations to find the value of b.