Triangle ABC. angle A is 2 times greater than angle C. AB = 8, BC = 12. Find the length of the speaker

Let us denote the value of the angle C through α.

According to the condition of the problem, the angle A is 2 times greater than the angle C, therefore, the value of the angle A should be equal to 2α, and the value of the angle B – 180 – α – 2α = 180 – 3α.

By the condition of the problem, | AB | = 8 and | ВС | = 12, therefore, applying the theorem of sines, we obtain:

8 / sin (α) = 12 / sin (2α),

whence follows:

sin (2α) / sin (α) = 12/8:

2sin (α) cos (α) / sin (α) = 3/2;

2cos (α) = 3/2;

cos (α) = 3/4.

Find sin (α):

sin (α) = √ (1 – (cos (α)) ^ 2) = √ (1 – (3/4) ^ 2) = √ (1 – 9/16) = √7 / 4.

Find sin (3α):

sin (3α) = 3sin (α) – 4sin ^ 3 (α) = 3 * √7 / 4 – 4 * (√7 / 4) ^ 3 = 3√7 / 4 – 7√7 / 16 = 5√7 /sixteen.

Applying the sine theorem, we find | AC |:

| AU | = sin (180 – 3α) * 8 / sin (α) = sin (3α) * 8 / sin (α) = (5√7 / 16) * 8 / (√7 / 4) = (5√7 / 2 ) / (√7 / 4) = (5√7 / 2) * 4 / √7 = 10.

Answer: | AC | = 10.