Question

В треугольнике ABC AB=4 см, угол C=30 градусов, угол В=45.НАЙТИ АС

Answer 1

Given Information:

We are given the following information about triangle ABC: - AB = 4 cm - ∠C = 30 degrees - ∠B = 45 degrees

Finding AC:

To find AC, we can use the Law of Cosines, which states that in a 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 know that AB = 4 cm, and we need to find AC. Let's substitute the given values into the equation:

AC^2 = AB^2 + BC^2 - 2 * AB * BC * cos(C)

Since we don't know the value of BC, we need to find it using the Law of Sines. The Law of Sines states that in a triangle with sides a, b, and c, and angles A, B, and C, the following equation holds:

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

Let's use the Law of Sines to find BC:

BC/sin(B) = AB/sin(A)

BC/sin(45) = 4/sin(30)

BC = (4 * sin(45)) / sin(30)

Now that we have the value of BC, we can substitute it back into the equation for AC:

AC^2 = 4^2 + BC^2 - 2 * 4 * BC * cos(30)

Simplifying the equation, we get:

AC^2 = 16 + BC^2 - 8 * BC * cos(30)

Now, we can substitute the value of BC and calculate AC:

AC^2 = 16 + ((4 * sin(45)) / sin(30))^2 - 8 * ((4 * sin(45)) / sin(30)) * cos(30)

AC^2 = 16 + (4 * sin(45))^2 / sin(30)^2 - 8 * (4 * sin(45)) * cos(30) / sin(30)

AC^2 = 16 + 16 / (1/2)^2 - 8 * (4 * sin(45)) * cos(30) / sin(30)

AC^2 = 16 + 16 / (1/4) - 8 * (4 * sin(45)) * cos(30) / sin(30)

AC^2 = 16 + 64 - 8 * (4 * sin(45)) * cos(30) / sin(30)

AC^2 = 80 - 8 * (4 * sin(45)) * cos(30) / sin(30)

AC^2 = 80 - 8 * (4 * (sqrt(2)/2)) * (sqrt(3)/2) / (1/2)

AC^2 = 80 - 8 * 4 * (sqrt(2)/2) * (sqrt(3)/2) / (1/2)

AC^2 = 80 - 8 * 4 * (sqrt(2) * sqrt(3)) / (1/2)

AC^2 = 80 - 8 * 4 * sqrt(6) / (1/2)

AC^2 = 80 - 64 * sqrt(6)

AC^2 = 80 - 64 * sqrt(6)

AC^2 ≈ 80 - 111.23

AC^2 ≈ -31.23

Since the square of a length cannot be negative, it seems that there might be an error in the given information or calculations. Please double-check the values provided for AB, ∠C, and ∠B, and ensure that they are accurate.