Знайти корені рівняння Sin10xsin2x=sin8xsin4x

Ответ:

Объяснение:

One way to solve this equation is to use the product-to-sum formula that states:

sin(a)sin(b) = (1/2)[cos(a-b)-cos(a+b)]

Using this formula, we can rewrite the left-hand side of the equation as:

sin(10x)sin(2x) = (1/2)[cos(10x-2x)-cos(10x+2x)] = (1/2)[cos(8x)-cos(12x)]

Similarly, the right-hand side of the equation becomes:

sin(8x)sin(4x) = (1/2)[cos(8x-4x)-cos(8x+4x)] = (1/2)[cos(4x)-cos(12x)]

Substituting these expressions back into the original equation, we get:

(1/2)[cos(8x)-cos(12x)] = (1/2)[cos(4x)-cos(12x)]

Simplifying and solving for cos(8x), we get:

cos(8x) = cos(4x)

Using the identity cos(a) = cos(-a), we can also write this as:

cos(8x-4x) = 1

Therefore, 8x-4x = 2πn, where n is an integer.

Solving for x, we get x = πn/2 + πm/4, where m and n are integers.

This means that the solutions to the original equation are all values of x that can be expressed in the form πn/2 + πm/4, where m and n are integers.