Sin³ x/3 - Sin² x/3 Cosx/3- 3Sin x/3Cos² x/3 +3Cos³ x/3=0

Предыдущий
вопрос Следующий
вопрос

To solve this trigonometric equation, we will first recognize that we can re-write the equation using the sine and cosine of one-third of the angle x.

Let y = x/3. Then the original equation becomes:

sin^3 y - sin^2 y cos y - 3 sin y cos^2 y + 3 cos^3 y = 0

Next, we can use the trigonometric identity cos^2 y = 1 - sin^2 y to simplify the equation:

sin^3 y - sin^2 y (1 - sin^2 y) - 3 sin y (1 - sin^2 y) + 3 (1 - sin^2 y)^2 = 0

Expand the terms and apply the trigonometric identity sin^2 y + cos^2 y = 1:

sin^3 y - sin^2 y + sin^4 y - 3 sin y + 3 sin^3 y - 3 sin^5 y + 3 - 6 sin^2 y + 3 sin^4 y = 0

Now, combine like terms to simplify the equation:

-6 sin^2 y + 4 sin^3 y - 3 sin^5 y + 3 - sin y = 0

Solving this equation involves finding the value of y (which corresponds to x/3), and then determining the corresponding values of x. Since this equation is quite complex, it may not be analytically solvable without the aid of a graphing calculator or numerical methods.