Question

Один из углов прямоугольного треугольника равен 30 градусов, длина гипотенузы составляет 8 см. Чему равны катеты этого треугольника ? Дано, решение, если можно

Answer 1

Finding the Lengths of the Legs of a Right Triangle

To find the lengths of the legs of a right triangle given that one of the angles is 30 degrees and the length of the hypotenuse is 8 cm, we can use trigonometric ratios.

Let's denote the lengths of the legs as a and b.

Using the trigonometric ratio for the sine of an angle, we have:

sin(30 degrees) = a / 8 cm

Simplifying the equation, we get:

a = 8 cm * sin(30 degrees)

To find the value of sin(30 degrees), we can refer to a trigonometric table or use a calculator. The value of sin(30 degrees) is 0.5.

Substituting this value into the equation, we have:

a = 8 cm * 0.5 = 4 cm

Therefore, one leg of the triangle has a length of 4 cm.

To find the length of the other leg, we can use the trigonometric ratio for the cosine of an angle:

cos(30 degrees) = b / 8 cm

Simplifying the equation, we get:

b = 8 cm * cos(30 degrees)

The value of cos(30 degrees) is √3/2.

Substituting this value into the equation, we have:

b = 8 cm * (√3/2) = 4√3 cm

Therefore, the other leg of the triangle has a length of 4√3 cm.

In summary, the lengths of the legs of the right triangle are:

a = 4 cm

b = 4√3 cm

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