Given Equation: 3sin(x) + 4cos(x) = 5

To find the value of sin(x), we can rearrange the equation using trigonometric identities and solve for sin(x).

Rewriting the Equation:
We know that sin^2(x) + cos^2(x) = 1 (Identity 1)
Dividing the entire equation by 5, we get:

(3/5)sin(x) + (4/5)cos(x) = 1

Now, let's introduce a new variable, let's say θ, such that sin(θ) = (3/5) and cos(θ) = (4/5).
Using the Pythagorean identity sin^2(θ) + cos^2(θ) = 1, we can verify that θ is a valid angle.

Using Trigonometric Identities:
We can rewrite the equation using the values of sin(θ) and cos(θ):

(3/5)sin(x) + (4/5)cos(x) = 1
(3/5)sin(x) + (4/5)cos(x) = sin^2(θ) + cos^2(θ)
(3/5)sin(x) + (4/5)cos(x) = sin^2(θ) + cos^2(θ) + 2sin(θ)cos(θ) - 2sin(θ)cos(θ)
(3/5)sin(x) + (4/5)cos(x) = sin^2(θ) + 2sin(θ)cos(θ) + cos^2(θ) - 2sin(θ)cos(θ)
(3/5)sin(x) + (4/5)cos(x) = sin^2(θ) + cos^2(θ) + 2sin(θ)cos(θ) - 2sin(θ)cos(θ)
(3/5)sin(x) + (4/5)cos(x) = sin^2(θ) + 2sin(θ)cos(θ) + cos^2(θ) - 2sin(θ)cos(θ)
(3/5)sin(x) + (4/5)cos(x) = (sin(θ) + cos(θ))^2 - 2sin(θ)cos(θ)
(3/5)sin(x) + (4/5)cos(x) = (sin(θ) + cos(θ))^2 - 2sin(θ)cos(θ)

Using the Given Values:
Now, substituting the values of sin(θ) = (3/5) and cos(θ) = (4/5):

(3/5)sin(x) + (4/5)cos(x) = (3/5 + 4/5)^2 - 2(3/5)(4/5)
(3/5)sin(x) + (4/5)cos(x) = (7/5)^2 - 2(3/5)(4/5)
(3/5)sin(x) + (4/5)cos(x) = (49/25) - (24/25)
(

3 sin x + 4 cos x= 5 or,(3 sin x + 4 cos x)/5=1 or,sin x =(4 cos x /5)×4/3or, sin x= 3/5