Question

Sin x +sin 3x+cos x+ cos 3x

Answer 1

Вот это ответ!!! нет

Answer 2

To simplify the expression sin(x) + sin(3x) + cos(x) + cos(3x), we can make use of trigonometric identities. One useful identity is the sum-to-product identity, which states:

sin(a) + sin(b) = 2 * sin((a + b) / 2) * cos((a - b) / 2)

Using this identity, we can rewrite the expression as follows:

sin(x) + sin(3x) + cos(x) + cos(3x) = [sin(x) + cos(x)] + [sin(3x) + cos(3x)] = sqrt(2) * [sin(x + π/4)] + sqrt(2) * [sin(3x + π/4)]

Now we have two terms in the form sin(a), and we can apply the sum-to-product identity again:

sqrt(2) * [sin(x + π/4)] + sqrt(2) * [sin(3x + π/4)] = 2 * sqrt(2) * sin((x + π/4 + 3x + π/4) / 2) * cos((x + π/4 - 3x - π/4) / 2) = 2 * sqrt(2) * sin((4x + π/2) / 2) * cos((-2x) / 2) = 2 * sqrt(2) * sin(2x + π/4) * cos(-x)

Now we can simplify further by using the cosine identity cos(-x) = cos(x):

2 * sqrt(2) * sin(2x + π/4) * cos(-x) = 2 * sqrt(2) * sin(2x + π/4) * cos(x)

So the simplified expression is:

2 * sqrt(2) * sin(2x + π/4) * cos(x)

Note that there may be alternative forms or equivalent expressions depending on the context or specific requirements of the problem you're working on.