Find all solutions of the equation in the interval [0, 2pi). Show all work.

sin(x+(pi/6))-sin(x-((pi/6))=1/2

using the difference formula, you have

sin(x)cos(π/6)+cos(x)sin(π/6) - sin(x)sin(π/6)+cos(x)sin(π/6) = 1/2

2cos(x)sin(π/6) = 1/2
2cos(x)(1/2) = 1/2
cos(x) = 1/2
x = π/3 or 5π/3

http://www.wolframalpha.com/input/?i=sin%28x%2B%28pi%2F6%29%29-sin%28x-%28pi%2F6%29%29%3D1%2F2+for+x+%3D+0+to+2pi

To find the solutions of the equation sin(x+(π/6))-sin(x-((π/6))=1/2 in the interval [0, 2π), we'll follow these steps:

Step 1: Simplify the equation
Step 2: Solve for x in the simplified equation
Step 3: Check the solutions in the given interval

Let's start with Step 1:

sin(x+(π/6))-sin(x-(π/6)) = 1/2

To simplify this equation, we'll use the trigonometric identity for the difference of two sines:

sin(A) - sin(B) = 2 * cos((A + B)/2) * sin((A - B)/2)

Applying this identity, our equation becomes:

2 * cos(x) * sin(π/6) = 1/2

Step 2:

Now we'll solve for x using algebraic manipulation:

cos(x) * sin(π/6) = 1/4

Recall that sin(π/6) = 1/2. Multiplying through both sides by 4 gives us:

4cos(x) = 1

Dividing both sides by 4:

cos(x) = 1/4

To find the solutions for cos(x) = 1/4 in the interval [0, 2π), we need to find the angles whose cosine value is 1/4. Using the inverse cosine function, we find two solutions:

x = arccos(1/4) ≈ 1.3181 rad
x = -arccos(1/4) ≈ -1.3181 rad

But we need to find the solutions in the interval [0, 2π). So, we need to adjust the negative solution:

x = -1.3181 + 2π ≈ 4.9654 rad

So, the solutions in the interval [0, 2π) are approximately:
x ≈ 1.3181 rad
x ≈ 4.9654 rad

Step 3:

Lastly, we will check if these solutions lie in the given interval [0, 2π). Both solutions are within this interval, so they are valid solutions.

Therefore, the solutions to the equation sin(x+(π/6))-sin(x-(π/6))=1/2 in the interval [0, 2π) are approximately:
x ≈ 1.3181 rad
x ≈ 4.9654 rad