Solve the equation sin(x+3)=sinx, 0=<x=<2pi to 4 decimal places. The product of the two solutions is:?

sin(x+3)=sin(x)
=>
sin(x+3)-sin(x)=0
=>
2cos(x+3/2)sin(3/2)=0
[sum to product formula]
=>
cos(x+3/2)=0
=>
x=(2k+1)pi/2-3/2
=>
x=[(2k+1)pi-3]/2, where k=0, 1
k=0=>x0=pi/2-3/2
k=1=>x1=(3pi/2-3/2)
=>
x0x1=
3(pi-3)(pi-1)/4=
0.2274253...

The sine curve has a period of 2*pi (roughly 6.283)
Your question is looking for an angle x which has the same value as it would have 3 radians later.
Doing it "mechanically":
draw a sine curve which measured 6.283 (inches, cm, whatever, as long as you keep the same units throughout). Measure off 3 units on an edge, then run that instrument along the curve, until the values match.
You will find that x must be just after the positive crest (just slightly more than pi/2 radians) putting the (x+3) angle just before the next positive crest

(there will also be another answer where x is just after the negative minimum dip -- making x slightly more than 3pi/2).

---

Using trig identities:

sin(x+3) = sin(x)cos(3) + cos(x)sin(3)
You want this to equal sin(x)

sin(x) = sin(x)cos(3) + cos(x)sin(3)
(remember that cos(3) is "just a number"-- it should not bother you)
(same thing for sin(3))

sin(x) - sin(x)cos(3) = cos(x)sin(3)
factor out sin(x) on the left

sin(x)(1-cos(3)) = cos(x)sin(3)
square both sides

sin^2(x) (1-cos(3))^2 = cos^2(x)sin^2(3)

since sin^2(x) + cos^2(x) = 1, we can replace
cos^2(x) = 1 - sin^2(x)

We now have

sin^2(x) (1 - cos(3))^2 = (1 - sin^2(x))) sin^2(3)
expand on the right
sin^2(x) (1 - cos(3))^2 = sin^2(3) - sin^2(x)sin^2(3)
solve for sin^2(x)
sin^2(x) (1 - cos(3))^2 - sin^2(x)sin^2(3) = sin^2(3)
factor out sin^2(x)

sin^2(x) [ (1 - cos(3))^2 - sin^2(3) ] = sin^2(3)

sin^2(x) = sin^2(3) / [ (1 - cos(3))^2 - sin^2(3) ]

Looks awful but remember that everything on the right is "just a number"

Square-rooting both sides will give you two results (one positive, the other negative). You will have to add 2*pi to the negative angle x, in order to bring it into the range asked in the question.

sin(x + 3) = sin(x)
sin(x)*cos(3) + cos(x)*sin(3) = sin(x)
cos(x)*sin(3) = sin(x) - sin(x)*cos(3)
sin(x)*(1 - cos(3)) = cos(x)*sin(3)
sin(x)/cos(x) = sin(3) / (1 - cos(3))
tan(x) = sin(3) / (1 - cos(3))
x = pi*n + (pi/2) - 1.5, for any integer n
Since we want 0 <= x <= 2*pi, then we have:
x = (pi/2) - 1.5 or x = (3*pi/2) - 1.5

The product of the two solutions is:
((pi/2) - 1.5)((3*pi/2) - 1.5) = 0.75*pi^2 - 3*pi + 2.25
=~ 0.2274253400476392487379381000686

Since you are looking for decimal approximations, I'll use Newton's Method. We need to get your equation into an expression equal to zero, so:

sin(x + 3) = sin(x)
sin(x + 3) - sin(x) = 0

We can call that our function:

f(x) = sin(x + 3) - sin(x)

We'll need the first derivative.  The second term is simple. The first needs the chain rule:

y = sin(u) and u = x + 3
dy/du = cos(u) and du/dx = 1

dy/dx = dy/du * du/dx
dy/dx = cos(u) * 1
dy/dx = cos(u)
dy/dx = cos(x + 3)

So the first derivative is:

f'(x) = cos(x + 3) - cos(x)

Looking at a graph of the two curves, they meet near 0 and near 3.  So we can use those as the first guesses.

The equation we need is:

x₁ = x₀ - f(x₀) / f'(x₀)

Substitute our expressions to get:

x₁ = x₀ - [sin(x₀ + 3) - sin(x₀)] / [cos(x₀ + 3) - cos(x₀)]

Using x₀ = 0, first, and making sure my calculator is in radians mode:

x₁ = 0 - [sin(0 + 3) - sin(0)] / [cos(0 + 3) - cos(0)]
x₁ = 0 - [sin(3) - 0] / [cos(3) - 1]
x₁ = 0 - (0.14112 - 0) / (-0.989992 - 1)
x₁ = 0 - 0.14112 / (-1.989992)
x₁ = 0 + 0.070915
x₁ = 0.070915

Let's loop this again until we know we have a number good to 4DP:

x₂ = x₁ - [sin(x₁ + 3) - sin(x₁)] / [cos(x₁ + 3) - cos(x₁)]
x₂ = 0.070915 - [sin(0.070915 + 3) - sin(0.070915)] / [cos(0.070915 + 3) - cos(0.070915)]
x₂ = 0.070915 - [sin(3.070915) - sin(0.070915)] / [cos(3.070915) - cos(0.070915)]
x₂ = 0.070915 - (0.0706188 - 0.0708556) / (-0.9975034 - 0.9974866)
x₂ = 0.070915 - (-0.0002368) / (-1.99499)
x₂ = 0.070915 - 0.0001187
x₂ = 0.0707963

Since we only subtracted 0.0001 from the previous guess, this should be good to 4DP.

x₂ = 0.0708

Now let's do the same starting with the guess of 3 for the other answer:

x₁ = x₀ - [sin(x₀ + 3) - sin(x₀)] / [cos(x₀ + 3) - cos(x₀)]
x₁ = 3 - [sin(3 + 3) - sin(3)] / [cos(3 + 3) - cos(3)]
x₁ = 3 - [sin(6) - sin(3)] / [cos(6) - cos(3)]
x₁ = 3 - (-0.279415 - 0.14112) / [0.9601703 - (-0.9899925)]
x₁ = 3 - (-0.420535) / (0.9601703 + 0.9899925)
x₁ = 3 + 0.420535 / 0.969876
x₁ = 3 + 0.433597
x₁ = 3.433597

x₂ = x₁ - [sin(x₁ + 3) - sin(x₁)] / [cos(x₁ + 3) - cos(x₁)]
x₂ = 3.433597 - [sin(3.433597 + 3) - sin(3.433597)] / [cos(3.433597 + 3) - cos(3.433597)]
x₂ = 3.433597 - [sin(6.433597) - sin(3.433597)] / [cos(6.433597) - cos(3.433597)]
x₂ = 3.433597 - [0.1498452 - (-0.287872)] / [0.9887095 - (-0.9576688)]
x₂ = 3.433597 - (0.1498452 + 0.287872) / (0.9887095 + 0.9576688)
x₂ = 3.433597 - 0.4377172 / 1.9463783
x₂ = 3.433597 - 0.224888
x₂ = 3.208709

We'll need at least one more loop:

x₃ = x₂ - [sin(x₂ + 3) - sin(x₂)] / [cos(x₂ + 3) - cos(x₂)]
x₃ = 3.208709 - [sin(3.208709 + 3) - sin(3.208709)] / [cos(3.208709 + 3) - cos(3.208709)]
x₃ = 3.208709 - [sin(6.208709) - sin(3.208709)] / [cos(6.208709) - cos(3.208709)]
x₃ = 3.208709 - [-0.074407 - (-0.067066)] / [0.9972279 - (-0.9977485)]
x₃ = 3.208709 - (-0.074407 + 0.067066) / (0.9972279 + 0.9977485)
x₃ = 3.208709 - (-0.007341) / 1.9949764
x₃ = 3.208709 + 0.007341 / 1.9949764
x₃ = 3.208709 + 0.003680
x₃ = 3.212389

One more should do it:

x₄ = x₃ - [sin(x₃ + 3) - sin(x₃)] / [cos(x₃ + 3) - cos(x₃)]
x₄ = 3.212389 - [sin(3.212389 + 3) - sin(3.212389)] / [cos(3.212389 + 3) - cos(3.212389)]
x₄ = 3.212389 - [sin(6.212389) - sin(3.212389)] / [cos(6.212389) - cos(3.212389)]
x₄ = 3.212389 - [-0.07073718 - (-0.07073722)] / [0.997495 - (-0.997495)]
x₄ = 3.212389 - (-0.07073718 + 0.07073722) / (0.997495 + 0.997495)
x₄ = 3.212389 - 0.00000004 / 1.99499
x₄ = 3.212389 - 0.00000002005
x₄ = 3.212389

Round this to 4DP:

x₄ = 3.2124

Finally, the product of your two roots:

0.0708 * 3.2124 = 0.2274 (rounded to 4DP)

 sin(x + 3 )= sin x, 0 ≤ x ≤ 2π to 4 decimal places. 
 sin(3) cos(x) + cos(3) sin(x) = sin(x)
 Solutions:
 x ≈ 2 (3.1416 n + 0.035398), n element Z
 x ≈ 2 (3.1416 n - 0.035398), n element Z
 The product of the two solutions is
 39.4786 n^2 - 0.00501207

looking at the graph, which sections are parallel from 0 to 2pi?

pi/2 and 3pi/2

go 1.5 (half of 3) on either side of those

sin(pi/2 - 1.5) = sin(pi/2 + 1.5)

x = pi/2 - 1.5
x + 3 = pi/2 + 1.5


sin(x) = sin(x + 3) = 0.07074
x = 3pi/2 - 1.5
x + 3 = 3pi/2 + 1.5

sin(x) = sin(x + 3) = -0.07074

I don't know what you mean by the product of two solutions.
x and x+3 or each x?

how can sin (x+3) ever equal sin x?? I've not taken a math class in forever, so I'm really curious if this is actually what you were asked?

Well, you have it all wrong. Pi. Is equal to 78 divided by 5, and 0 is not less than the equation.