Question

9 класс. Тригонометрия. Теоремы синуса и косуниса. АВС треугольник. AB=5см угол В=30 градусам угол С=45 градусам Найти: АВ, ВС, угол А? Заранее спасибо за помощь)

Answer 1

Так как сумма всех углов треугольника равна 180 градусам, то
Угол А+Угол В+Угол С=180
Угол А=180-(Угол В+Угол С)=180-(30+45)=180-75=105 градусов.
По теореме синусов:
AB\sinC=BC/sinA=AC\sinB
5\sin45=AC\sin30
5\(√2\2)=AC\0.5
10\√2=AC\0.5
AC=5\√2=2.5√2

5\sin45=BC\sin105
10\√2=BC\0.97
BC≈9.7\√2≈4.85√2.
Ответ: Угол А=105 градусов;
            AC=2,5√2;
            BC≈4,85√2.
             

Answer 2

Trigonometry: Sine and Cosine Theorems

To solve the given problem, we can use the Sine and Cosine Theorems in trigonometry. These theorems relate the sides and angles of a triangle.

Let's consider triangle ABC, where AB = 5 cm, angle B = 30 degrees, and angle C = 45 degrees. We need to find AB, BC, and angle A.

Applying the Sine Theorem

The Sine Theorem states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. Mathematically, it can be written as:

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

In our case, we can use the Sine Theorem to find the length of side BC (denoted as c) and the measure of angle A.

Let's start by finding the length of side BC (c):

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

Substituting the given values:

BC/sin(45) = 5/sin(A)

Now, we can solve for BC:

BC = (5 * sin(45)) / sin(A)

To find the measure of angle A, we can rearrange the equation:

sin(A) = (5 * sin(45)) / BC

Taking the inverse sine (arcsin) of both sides:

A = arcsin((5 * sin(45)) / BC)

Applying the Cosine Theorem

The Cosine Theorem states that in any triangle, the square of a side is equal to the sum of the squares of the other two sides, minus twice the product of those sides and the cosine of their included angle. Mathematically, it can be written as:

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

In our case, we can use the Cosine Theorem to find the length of side AB (a) and side BC (b).

Let's start by finding the length of side AB (a):

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

Substituting the given values:

5^2 = BC^2 + AC^2 - 2 * BC * AC * cos(30)

Simplifying the equation:

25 = BC^2 + AC^2 - BC * AC * sqrt(3)

Since we already have the value of BC from the Sine Theorem, we can substitute it into the equation:

25 = (5 * sin(45))^2 + AC^2 - (5 * sin(45)) * AC * sqrt(3)

Now, we can solve for AC:

AC = (25 - (5 * sin(45))^2) / ((5 * sin(45)) * sqrt(3))

Finally, we can find the length of side BC (b) using the Cosine Theorem:

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

Substituting the given values:

BC^2 = 5^2 + AC^2 - 2 * 5 * AC * cos(A)

Simplifying the equation:

BC^2 = 25 + AC^2 - 10 * AC * cos(A)

Since we already have the values of BC and AC, we can substitute them into the equation:

BC^2 = 25 + ((25 - (5 * sin(45))^2) / ((5 * sin(45)) * sqrt(3)))^2 - 10 * ((25 - (5 * sin(45))^2) / ((5 * sin(45)) * sqrt(3))) * cos(A)

Now, we can solve for BC:

BC = sqrt(25 + ((25 - (5 * sin(45))^2) / ((5 * sin(45)) * sqrt(3)))^2 - 10 * ((25 - (5 * sin(45))^2) / ((5 * sin(45)) * sqrt(3))) * cos(A))

Calculating the Values

To find the values of AB, BC, and angle A, we can substitute the given values into the equations derived from the Sine and Cosine Theorems.

Using a calculator or a trigonometric table, we can evaluate the expressions to find the final values.

Please note that due to the complexity of the calculations involved, it is recommended to use a calculator or a mathematical software tool to obtain accurate results.

I hope this explanation helps! Let me know if you have any further questions.