5sin(2x)+4cos(x)+0

To solve the equation 5sin(2x) + 4cos(x) = 0, we can use trigonometric identities and techniques to simplify and solve for x. Here's how you can proceed:

Step 1: Apply the Pythagorean identity sin^2(x) + cos^2(x) = 1 to the equation:
5sin(2x) + 4cos(x) = 0
5sin(2x) + 4cos(x) = 0
5sin(2x) = -4cos(x)

Step 2: Divide both sides of the equation by cos(x):
(5sin(2x))/cos(x) = -4

Step 3: Simplify using the double-angle identity for sin():
2sin(x)cos(x)/cos(x) = -4
2sin(x) = -4

Step 4: Divide both sides of the equation by 2:
sin(x) = -2

Step 5: Since the value of sin(x) cannot exceed 1 or go below -1, there are no x-values that satisfy sin(x) = -2. Hence, there is no solution to the equation 5sin(2x) + 4cos(x) = 0.

In summary, the equation 5sin(2x) + 4cos(x) = 0 has no solutions.