Eduprobe
Question:
The equation esinx - e-sinx = 0 has:
No real roots
Exactly four real roots
Infinite number of real roots.
Exactly one real root
Solution:
esinx - e-sinx = 0
esinx = e-sinx
Taking the natural logarithm of both sides:
sinx = -sinx
2sinx = 0
sinx = 0
The general solution for sinx = 0 is x = nπ, where n is an integer. Since there are infinitely many integers, there are infinitely many real roots. Therefore, the equation has an infinite number of real roots.