The area of an isosceles triangle is 81√3. The side length is 18. Find the cosine of the angle

The area of an isosceles triangle is 81√3. The side length is 18. Find the cosine of the angle opposite the base of the triangle if the angle is known to be obtuse.

Let us denote by α the value of the angle between the lateral sides of this isosceles triangle.

In the initial data for this task, it is reported that the length of the lateral side of this isosceles triangle is 18, and the area of ​​the triangle is 81√3.

Using the formula for the area of ​​a triangle on two sides and the angle between them, we can compose the following equation:

18 * 18 * sinα / 2 = 81√3,

solving which, we get:

18 * 9 * sinα = 81√3;

sinα = 81√3 / (18 * 9);

sinα = 9√3 / 18;

sinα = √3 / 2.

According to the condition of the problem,

According to the condition of the problem, the angle α is obtuse.

Therefore, cosα <0.

Knowing sinα, we can find cosα:

cosα = -√ (1 – (sinα) ^ 2) = -√ (1 – (√3 / 2) ^ 2) = -√ (1 – 3/4) = -√1 / 4 = -1/2.

Answer: cosα = -1 / 2.