Question

Sinx=sin(-x), x is on interval (π,3π/2).

No solutions on real world, hence we got to take this expression to the complex world. I asked the author if he didin't make mistake, the answer is negative, thus there has to be a rule to make this true.

Answer 1

The equation sin(x) = sin(-x) considers the symmetric properties of the sine function in the complex domain, where equal sines are possible for complex conjugates. The given interval does not yield solutions in the real domain due to the odd nature of the sine function but could have solutions when extended to complex numbers.

Step-by-step explanation:

The equation sin(x) = sin(-x) is based on the symmetric properties of the sine function.

For the given interval of (π,3π/2), we normally expect no solutions for this equation since the sine function is odd, meaning sin(-x) = -sin(x).

However, given that equality is insisted upon in the complex domain, we need to consider that complex conjugates have equal sines due to the properties of trigonometric functions extended to the complex plane.

Typically, in the real domain, solutions to such equations would be at angles where sine reaches the same values during its periodic cycle, but in the given interval, negative and positive x would yield different signs, thus not satisfying the equality. It's important to note that in complex analysis, one can explore solutions for equations that do not hold in the real number system by extending the functions into complex numbers, which have both real and imaginary components.

For a conventional understanding, values of x that result in sin(kx)=±1 can help determine points of maximum and minimum amplitudes referred to as antinodes in wave equations.