Question

Sin^2x-sin^2(2x)+sin^2(3x)=0.5

Answer 1

Sin^2(2x)=1/2
2x=(-1)^n * П/6+Пn,n Э z
x=(-1)^n 8 П/12+Пn/2,n Эz

Answer 2

We can simplify the equation using the identity:

sin^2(a) - sin^2(b) = sin(a + b) * sin(a - b)

Applying this identity to the second term, we get:

sin^2(2x) = sin^2(x + x) = sin(x + x) * sin(x - x) = 2sin(x)cos(x)^2

Similarly, applying the identity to the third term, we get:

sin^2(3x) = sin^2(2x + x) = sin(2x + x) * sin(2x - x) = (3sin(x) - 4sin^3(x))^2

Substituting these simplifications into the original equation and using the fact that cos(2x) = 1 - 2sin^2(x), we get:

sin^2(x) - 2sin(x)cos(x)^2 + (3sin(x) - 4sin^3(x))^2 = 0.5

Expanding and simplifying the equation, we get:

16sin^6(x) - 28sin^4(x) + 12sin^2(x) - 1 = 0

This is a cubic equation in sin^2(x), which can be solved using standard methods. The solutions are:

sin^2(x) = 1/4, sin^2(x) = (1 - sqrt(5))/8, sin^2(x) = (1 + sqrt(5))/8

Substituting these solutions back into the original equation, we find that only sin^2(x) = 1/4 satisfies the equation. Therefore, the solutions to the original equation are:

x = n*pi/2 +/- pi/6, where n is an integer.