The side of the triangle is 21, and the other two form an angle of 60 and are related as 3: 8.

The side of the triangle is 21, and the other two form an angle of 60 and are related as 3: 8. find the unknown sides and angles of the triangle.

Let the side of the triangle c = 21. Let’s denote the other two sides by a and b.

By the condition of the problem, it is known that the angle between sides a and b is 60 ° and

b / a = 3/8, b = 3/8 * a.

Let’s apply the cosine theorem for a given triangle:

c ^ 2 = a ^ 2 + b ^ 2 – 2 * a * b * cos (60 °),

21 ^ 2 = a ^ 2 + (3/8 * a) ^ 2 – 2 * a * 3/8 * a * 1/2,

21 ^ 2 = a ^ 2 * (1 + (3/8) ^ 2 – 3/8) = a ^ 2 * (1 + 9/64 – 3/8) = a ^ 2 * (64 + 9 -24 ) / 64 = a ^ 2 * 49/64,

21 = a * 7/8,

a = 24. So b = 3/8 * a = 9.

We got that the sides of the triangle are a = 24, b = 9, c = 21.

By the sine theorem, we have:

a / sin (A) = b / sin (B) = c / sin (60 °).

Hence,

sin (A) = a * sin (60 °) / c = 24/21 * √3 / 2 = 12 * √3 / 21, A = arcsin (12 * √3 / 21).

sin (B) = b * sin (60 °) / c = 9/21 * √3 / 2 = 3 * √3 / 14, B = arcsin (3 * √3 / 14).