Question

Помогите, пожалуйста! У прямоугольным треугольнику один с катетов ровный 4 см, а синус противоположного яму угла ровный 0,8. Найдите другой катет и гипотенузу этого треугольника. Заранее спасибо!

Answer 1

Определение: синус угла = отношению противолежащего катета к гипотенузе
отсюда можно найти гипотенузу,
а по т.Пифагора и второй катет)) он =3
это "египетский" треугольник: катеты 3 и 4, гипотенуза 5

Answer 2

Finding the Other Leg and Hypotenuse of a Right Triangle

To find the other leg and hypotenuse of a right triangle, we can use the given information about one leg and the sine of the opposite angle.

Let's denote the given leg as a (which is 4 cm) and the sine of the opposite angle as sin(θ) (which is 0.8).

We can use the following trigonometric relationship to find the other leg (b) and the hypotenuse (c):

sin(θ) = b / c

Since we know the value of sin(θ) (0.8), we can rearrange the equation to solve for b:

b = sin(θ) * c

Now, we need to find the value of c (the hypotenuse). To do this, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides:

c^2 = a^2 + b^2

Substituting the value of b from the previous equation, we get:

c^2 = a^2 + (sin(θ) * c)^2

Simplifying the equation, we have:

c^2 = a^2 + (sin(θ))^2 * c^2

Now, we can solve this equation to find the value of c.

Let's calculate the values of b and c using the given information:

Given: - Length of one leg (a) = 4 cm - Sine of the opposite angle (sin(θ)) = 0.8

Using the equation b = sin(θ) * c, we can substitute the values:

b = 0.8 * c

Now, let's substitute the value of b in the Pythagorean theorem equation:

c^2 = a^2 + (sin(θ))^2 * c^2

Substituting the values:

c^2 = 4^2 + (0.8 * c)^2

Simplifying further:

c^2 = 16 + 0.64 * c^2

Moving all terms to one side:

0.36 * c^2 = 16

Dividing both sides by 0.36:

c^2 = 44.44

Taking the square root of both sides:

c ≈ 6.66

Now, we can substitute the value of c in the equation b = 0.8 * c to find b:

b = 0.8 * 6.66 ≈ 5.33

Therefore, the other leg (b) is approximately 5.33 cm and the hypotenuse (c) is approximately 6.66 cm.

Please note that the values are approximate due to rounding.

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