Question

Дано: треугольник АВС, а=20,в=13, альфа=67найти: с,гамма, бета

Answer 1

С=√20²+13²-2*20*13-сos(76)=√569-520*cos76≈21см

B=37°
Y=76°

Answer 2

Given Information:

We are given a triangle ABC with side lengths a = 20, b = 13, and angle alpha (α) = 67 degrees. We need to find the lengths of sides c and gamma (γ), as well as the angle beta (β).

Solution:

To find the length of side c, we can use the Law of Cosines, which states that for any triangle with sides a, b, and c, and angle C (opposite side c), the following equation holds:

c^2 = a^2 + b^2 - 2ab * cos(C)

In our case, we have a = 20, b = 13, and angle C = alpha (α) = 67 degrees. Plugging these values into the equation, we can solve for c:

c^2 = 20^2 + 13^2 - 2 * 20 * 13 * cos(67)

To find the value of cos(67), we can use a calculator or a trigonometric table. Let's assume that cos(67) is approximately 0.394. Substituting this value into the equation, we get:

c^2 = 400 + 169 - 2 * 20 * 13 * 0.394

Simplifying further:

c^2 = 400 + 169 - 10.22 * 20 * 13

c^2 = 400 + 169 - 265.6

c^2 = 303.4

Taking the square root of both sides, we find:

c ≈ √303.4

c ≈ 17.43

Therefore, the length of side c is approximately 17.43.

To find the value of angle gamma (γ), we can use the Law of Sines, which states that for any triangle with sides a, b, and c, and angles A, B, and C (opposite sides a, b, and c, respectively), the following equation holds:

sin(A) / a = sin(B) / b = sin(C) / c

In our case, we know the values of a, b, and c, and we need to find the value of angle gamma (γ), which is opposite side c. We can rearrange the equation to solve for sin(C):

sin(C) = c * sin(A) / a

Substituting the known values, we get:

sin(C) = 17.43 * sin(67) / 20

Using a calculator or a trigonometric table, let's assume that sin(67) is approximately 0.921. Substituting this value into the equation, we get:

sin(C) = 17.43 * 0.921 / 20

Simplifying further:

sin(C) ≈ 0.799

To find the value of angle gamma (γ), we can take the inverse sine (arcsin) of 0.799:

gamma (γ) ≈ arcsin(0.799)

Using a calculator or a trigonometric table, we find that gamma (γ) is approximately 53.85 degrees.

Finally, to find the value of angle beta (β), we can use the fact that the sum of the angles in a triangle is always 180 degrees:

beta (β) = 180 - alpha (α) - gamma (γ)

Substituting the known values, we get:

beta (β) = 180 - 67 - 53.85

Simplifying further:

beta (β) ≈ 59.15

Therefore, the value of angle beta (β) is approximately 59.15 degrees.

Summary:

Given a triangle ABC with side lengths a = 20, b = 13, and angle alpha (α) = 67 degrees, we have found the following values: - Length of side c ≈ 17.43 - Angle gamma (γ) ≈ 53.85 degrees - Angle beta (β) ≈ 59.15 degrees