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