Question

Допоможіть будь ласка у трикутнику АВС відомо, що <С =90° ВС = 41 см. АС 20 см. Знайдіть косинуси гострих кутів трикутника.​

Answer 1

Ответ:

$cos A =\dfrac{20}{\sqrt{2081} } ;\\\\cos B =\dfrac{41}{\sqrt{2081} } .$

Объяснение:

В треугольнике АВС известно, что ∠С =90°, ВС =41 см, АС =20 см. Найти косинусы острых углов треугольника.

Пусть дан ΔАВС -прямоугольный, так как ∠С =90°. Катеты ВС =41 см, АС =20 см. Найдем гипотенузу АВ по теореме Пифагора: в прямоугольном треугольнике квадрат гипотенузы равен сумме квадратов катетов.

$AB^{2} =AC^{2} +BC^{2} ;\\AB = \sqrt{AC^{2} +BC^{2} } ;\\AB =\sqrt{41^{2} +20^{2} } =\sqrt{1681+400} =\sqrt{2081}$ см.

Косинусом острого угла прямоугольного треугольника называется отношение прилежащего катета к гипотенузе.

$cos A =\dfrac{AC}{AB} ;\\\\cos A =\dfrac{20}{\sqrt{2081} } .$

$cos B =\dfrac{BC}{AB} ;\\\\cos B =\dfrac{41}{\sqrt{2081} } .$

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Answer 2

Triangle ABC

In triangle ABC, we are given that angle C is 90°, BC is 41 cm, and AC is 20 cm. We need to find the cosines of the acute angles of the triangle.

Finding the Cosines of the Acute Angles

To find the cosines of the acute angles of the triangle, we can use the cosine function, which relates the cosine of an angle to the lengths of the sides of a right triangle.

The cosine of an angle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. In triangle ABC, the hypotenuse is AC, and the adjacent sides are BC and AB.

To find the cosine of angle A, we can use the ratio of the length of the adjacent side BC to the length of the hypotenuse AC. Therefore, cos(A) = BC/AC.

To find the cosine of angle B, we can use the ratio of the length of the adjacent side AB to the length of the hypotenuse AC. Therefore, cos(B) = AB/AC.

Calculating the Cosines

Let's calculate the cosines of the acute angles A and B using the given values:

- BC = 41 cm - AC = 20 cm

Using these values, we can calculate the cosines as follows:

- cos(A) = BC/AC = 41/20 - cos(B) = AB/AC

However, we still need to find the length of side AB. To do that, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

In triangle ABC, we have: - AC = 20 cm - BC = 41 cm - AB = ?

Using the Pythagorean theorem, we can calculate the length of side AB as follows:

- AB^2 = AC^2 - BC^2 - AB^2 = 20^2 - 41^2 - AB^2 = 400 - 1681 - AB^2 = -1281

Since the result is negative, it means that the given side lengths do not form a valid right triangle. Therefore, it is not possible to calculate the cosine of angle B in this case.

In summary, we can calculate the cosine of angle A as cos(A) = 41/20, but it is not possible to calculate the cosine of angle B with the given information.