Question

Найдите косинус угла между диагональю куба и плоскостью одной из его граней

Answer 1

Диагональ куба - это гипотенуза в прямоугольном треугольнике,
у которого один катет - диагональ грани, d = a√2, а второй катет
высота h = а, где а - это сторона куба.
D = √((a√2)^2 + a^2) = √(2a^2 + a^2) = √(3a^2) = a√3
cos A = d/D = (a√2) / (a√3) = √2/√3 = √6/3

Answer 2

Finding the Cosine of the Angle Between the Cube's Diagonal and One of Its Faces

To find the cosine of the angle between the diagonal of a cube and one of its faces, we can use trigonometry. Let's denote the length of the cube's diagonal as d and the length of one of its edges as a.

The diagonal of a cube can be calculated using the Pythagorean theorem. Since a cube has all sides equal, the diagonal forms a right triangle with two sides of length a. Therefore, the length of the diagonal can be calculated as:

d = sqrt(2) * a.

Now, let's consider one of the cube's faces. The diagonal of the face is equal to the length of the edge, which is a.

To find the cosine of the angle between the diagonal of the cube and one of its faces, we can use the formula:

cos(angle) = a / d.

Substituting the values we have:

cos(angle) = a / (sqrt(2) * a).

Simplifying the expression:

cos(angle) = 1 / sqrt(2).

Therefore, the cosine of the angle between the diagonal of a cube and one of its faces is 1 / sqrt(2).