Question

Sin 3 альфа *cos 2 альфа + cos 3 альфа *sin 2 альфа

Answer 1

Sin 3α *Cos 2α+ Cos 3α *Sin 2α = Sin(3α+2α) = Sin5α

Answer 2

To simplify the expression sin(3α) * cos(2α) + cos(3α) * sin(2α), we can use trigonometric identities. Specifically, we can apply the angle sum and difference identities for sine and cosine:

1. sin(α ± β) = sin(α) * cos(β) ± cos(α) * sin(β)
2. cos(α ± β) = cos(α) * cos(β) ∓ sin(α) * sin(β)

Let's start by using these identities:

sin(3α) * cos(2α) + cos(3α) * sin(2α)

Using identity 1 with α = 3α and β = 2α:

sin(3α) * cos(2α) = sin(3α - 2α) = sin(α)

Next, using identity 1 with α = 2α and β = 3α:

cos(3α) * sin(2α) = cos(3α - 2α) = cos(α)

Now, the expression becomes:

sin(α) + cos(α)

Finally, we can use the Pythagorean identity to combine sin(α) and cos(α):

sin^2(α) + cos^2(α) = 1

So,

sin(α) + cos(α) = 1

Therefore, the simplified expression is just 1.