Question

Подскажите как выразить a+ a через cos4a

Answer 1

 sin^4(x) + cos^4(x) = sin^{4}(x) + cos^4(x) + 2sin^2(x) cos^2(x)  - 2sin^2(x) cos^2(x) = (sin^2(x) + cos^2(x) )^2 - 2sin^2(x) cos^2(x) = 1 - 2sin^2(x) cos^2(x) = 1 - 1/2 sin^2(2x) = 1 - 1/2 (1/2 (1 - cos(4x))) = 1 - 1/4 (1 - cos(4x)) = 3/4 + 1/4 cos(4x)

При решении пользуемся тем, что cos(4x) = 1 - 2sin^2(2x), поэтому
sin^2(2x) = 1/2 (1 - cos(4x))

Ответ: 3/4 + 1/4 cos(4x)

Answer 2

Expressing a in terms of cos(4a)

To express a in terms of cos(4a), we can use the trigonometric identity: cos(2x) = 1 - 2sin^2(x)

We can use this identity to express cos(4a) in terms of cos(2a), and then further express cos(2a) in terms of a.

First, we express cos(4a) in terms of cos(2a): cos(4a) = 2cos^2(2a) - 1

Then, we express cos(2a) in terms of a: cos(2a) = 1 - 2sin^2(a) cos(2a) = 1 - 2(1 - cos^2(a)) cos(2a) = 1 - 2 + 2cos^2(a) cos(2a) = 2cos^2(a) - 1

Now, we can substitute cos(2a) = 2cos^2(a) - 1 into the expression for cos(4a): cos(4a) = 2(2cos^2(a) - 1) - 1 cos(4a) = 4cos^2(a) - 2 - 1 cos(4a) = 4cos^2(a) - 3

So, a can be expressed in terms of cos(4a) as: a = cos^(-1)((cos(4a) + 3) / 4)

This provides a way to express a in terms of cos(4a).