Viral trigonometry exponential equation- how to find x

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To solve for x, we'll use trigonometric identities and algebraic manipulation.
Given equation:
(sin(4x)/2) + (cos(4x)/3) = 1/5
Multiply both sides by 6 to eliminate fractions:
3sin(4x) + 2cos(4x) = 6/5
Now, utilize trigonometric identities:
1. sin(a) = cos(π/2 - a)
2. cos(a) = sin(π/2 - a)
Rewrite cos(4x) in terms of sin:
2sin(π/2 - 4x) + 3sin(4x) = 6/5
Combine like terms using sum-to-product identities:
(2 + 3)sin(4x + π/4)cos(π/4) = 6/5
Simplify:
5sin(4x + π/4) = 6/5
sin(4x + π/4) = 6/25
Now, solve for 4x + π/4:
4x + π/4 = arcsin(6/25)
4x = arcsin(6/25) - π/4
x = (arcsin(6/25) - π/4)/4
Numerical solution:
x ≈ 0.1027 (radians)
x ≈ 5.88°