Question

Гипотенуза АВ наклонена к плоскости бета под углом 45 градусов катет ВС лежит в плоскости бета и составляет с проекцией гипотенузы на плоскость бета угол 30 чему равен косинус угла ВАС ОТВЕТЫ: корень из 10/4 корень из 5/3 3/4 корень из 15/7

Answer 1

Треугольник АВС прямоугольный, косинус угла ВАС равен отношению АС/АВ.
НС - перпендикуляр к ВС, так как это проекция перпендикуляра АС к ВС.
В прямоугольном треугольнике НСВ катет СН =(1/2)*НВ = (1/2)*АН, так как в прямоугольном треугольнике АНВ <ABH=45° (дано) и гипотенуза АВ=НВ√2.
В прямоугольном треугольнике АСН гипотенуза
АС =√(СН²+АН²) = √((1/2)*АН)²+АН²) = √(АН²/4*+АН²) = √[(АН²+4АН²/4)] =АН*√5/2.
АН=ВН. Значит АС= НВ*√5/2.
Итак, АС=НВ*√5/2, АВ=НВ√2 тогда АС/АВ=(НВ*√5/2)/НВ√2 = √5/(2√2) = √10/4
Ответ: Cos(<BAC) = √10/4.

Answer 2

Problem Analysis

We are given a right triangle ABC, where the hypotenuse AB is inclined at an angle of 45 degrees to the plane beta, and the leg BC lies in the plane beta and forms an angle of 30 degrees with the projection of the hypotenuse on the plane beta. We need to find the cosine of the angle VAC.

Solution

To find the cosine of the angle VAC, we need to find the lengths of the sides of the triangle ABC.

Let's assume that the length of AB (the hypotenuse) is x.

Using trigonometric ratios, we can find the lengths of the other sides of the triangle.

The length of BC (the leg) can be found using the sine of the angle between BC and the projection of AB on the plane beta:

BC = x * sin(30 degrees)

The length of AC (the other leg) can be found using the cosine of the angle between AC and the projection of AB on the plane beta:

AC = x * cos(30 degrees)

Now, we can find the cosine of the angle VAC using the cosine rule:

cos(VAC) = (BC^2 + AC^2 - AB^2) / (2 * BC * AC)

Substituting the values we found earlier:

cos(VAC) = (x * sin(30 degrees))^2 + (x * cos(30 degrees))^2 - x^2) / (2 * x * sin(30 degrees) * x * cos(30 degrees))

Simplifying the expression:

cos(VAC) = (sin^2(30 degrees) + cos^2(30 degrees) - 1) / (2 * sin(30 degrees) * cos(30 degrees))

Using the trigonometric identities:

sin^2(30 degrees) = (1 - cos^2(30 degrees))

cos^2(30 degrees) = 1 - sin^2(30 degrees)

Substituting these values:

cos(VAC) = ((1 - cos^2(30 degrees)) + cos^2(30 degrees) - 1) / (2 * sin(30 degrees) * cos(30 degrees))

Simplifying further:

cos(VAC) = 0 / (2 * sin(30 degrees) * cos(30 degrees))

Since the numerator is 0, the cosine of the angle VAC is 0.

Therefore, cos(VAC) = 0.

Answer

The cosine of the angle VAC is 0.