Question

Теоремы косинусов и синусов​

Answer 1

Ответ:

Отношение противолежащего углу А катета к гипотинузе называется синусом угла А прямоугольного треугольника и обозначается sinA

Отношение прилежащего углу А катета к гипотинузе называется косинусом угла А прямоугольного треугольника и обозначается cosA.

Answer 2

подробнее все расписано на картинке, если есть вопросы , пишите

Теорема синусов : стороны треугольника пропорциональны синусам противолежащих углов.

Теорема косинусов : квадрат стороны треугольника равен сумме квадратов двух других его сторон минус удвоенное произведение этих сторон на косинус угла между ними.

Answer 3

The Cosine and Sine Theorems

The Cosine and Sine theorems are mathematical theorems that relate the sides and angles of a triangle. They are used to calculate unknown side lengths or angles in a triangle when certain information is known.

Cosine Theorem

The Cosine Theorem, also known as the Law of Cosines, relates the lengths of the sides of a triangle to the cosine of one of its angles. It can be stated as follows:

In a triangle with sides of lengths a, b, and c, and an angle opposite side c, the square of side c is equal to the sum of the squares of sides a and b, minus twice the product of the lengths of sides a and b and the cosine of the angle opposite side c.

Mathematically, the Cosine Theorem can be expressed as:

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

Where: - c is the length of side c - a and b are the lengths of sides a and b, respectively - C is the angle opposite side c

The Cosine Theorem is useful for calculating the length of an unknown side in a triangle when the lengths of the other two sides and the measure of the included angle are known.

Sine Theorem

The Sine Theorem, also known as the Law of Sines, relates the lengths of the sides of a triangle to the sines of its angles. It can be stated as follows:

In a triangle with sides of lengths a, b, and c, and angles A, B, and C opposite sides a, b, and c, respectively, the ratio of the length of a side to the sine of the opposite angle is constant for all sides and angles.

Mathematically, the Sine Theorem can be expressed as:

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

The Sine Theorem is useful for calculating unknown angles or side lengths in a triangle when the lengths of one side and the measures of the angles opposite that side are known.

Example Usage

Here are a couple of examples demonstrating the usage of the Cosine and Sine theorems in programming:

1. Calculating the length of the hypotenuse of a right triangle using the Cosine Theorem in C:

#include 
#include

int main(void) {
    int a, b, c2;
    scanf("%d", &a);
    scanf("%d", &b);
    c2 = a*a + b*b;
    printf("c = %.2f\n", sqrt(c2));
    return 0;
}

2. Calculating the sine of an angle in degrees using the Sine Theorem in C:

#include 
#include

int main(void) {
    double alpha, sin_a, pi = 3.1415926;
    scanf("%lf", &alpha);
    alpha = alpha * pi / 180;
    sin_a = sin(alpha);
    printf("%.2f\n", sin_a);
    return 0;
}

In the first example, the program calculates the length of the hypotenuse (side c) of a right triangle using the lengths of the other two sides (a and b) and the Cosine Theorem.

In the second example, the program calculates the sine of an angle (angle A) in degrees using the Sine Theorem.

Please note that these examples are just for illustration purposes and may need to be adapted to fit specific programming languages or frameworks.

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