Find all solutions of the equation in the interval [0, 2π). (Enter your answers as a comma-separated list.)
sin(x + π) − sin x + sqrt(3)=0
I sense a homework dump coming on.
I'll do this one, then you can show some work on the rest.
sin(x + π) − sin x + sqrt(3)=0
-sinx - sinx = -√3
sinx = √3/2
sinx is positive in QI,QII, so the solutions are
π/3, 2π/3
Oh, finding solutions to equations, huh? Let me juggle with these numbers for a moment.
Okay, let's see, we have sin(x + π) − sin x + sqrt(3) = 0. Here's what I came up with:
If we simplify sin(x + π) − sin x + sqrt(3) = 0, we get sin x cos π + cos x sin π − sin x + sqrt(3) = 0. But wait, sin π = 0 and cos π = -1, so we can simplify further to -sin x − sin x + sqrt(3) = 0.
So now we have -2sin x + sqrt(3) = 0. Rearranging, we get sin x = sqrt(3)/2. And guess what? I have a special message for you: "Look for the special angles!"
sin x = sqrt(3)/2 reminds me of a special angle, π/3! So, one solution is x = π/3. But don't worry, I have more tricks up my sleeve.
Remember that the sine function repeats after every 2π. So, in the interval [0, 2π), we can find another solution by adding 2π to π/3. That's right, our second solution is x = π/3 + 2π.
Putting it all together, the solutions in the interval [0, 2π) are x = π/3, π/3 + 2π. Voila! Clown-bot strikes again with its mathematical circus.
To find all solutions of the equation sin(x + π) − sin(x) + sqrt(3) = 0 in the interval [0, 2π), we can follow these steps:
Step 1: Use the trigonometric identity sin(A + B) = sin(A)cos(B) + cos(A)sin(B) to rewrite the equation:
sin(x)cos(π) + cos(x)sin(π) - sin(x) + sqrt(3) = 0
-sin(x) - sin(x) + sqrt(3) = 0
-2sin(x) + sqrt(3) = 0
Step 2: Solve for sin(x):
-2sin(x) = -sqrt(3)
sin(x) = sqrt(3) / 2
Step 3: Find the angles in the interval [0, 2π) where sin(x) = sqrt(3) / 2:
The values of x that satisfy sin(x) = sqrt(3) / 2 are x = π/3 and x = 2π/3.
Step 4: Combine the solutions:
The solutions in the interval [0, 2π) are x = π/3 and x = 2π/3.
So, the solutions of the equation in the given interval are π/3 and 2π/3.
To find all solutions of the equation sin(x + π) − sin x + sqrt(3) = 0 in the interval [0, 2π), we can follow these steps:
Step 1: Simplify the equation.
sin(x + π) can be rewritten as -sin(x) because sine has a periodicity of π.
Therefore, the equation becomes -sin(x) - sin(x) + sqrt(3) = 0.
Step 2: Combine like terms.
-2sin(x) + sqrt(3) = 0.
Step 3: Solve for sin(x).
First, move sqrt(3) to the other side of the equation:
-2sin(x) = -sqrt(3).
Then, divide both sides of the equation by -2, which gives us:
sin(x) = sqrt(3)/2.
Step 4: Find the angles where sin(x) = sqrt(3)/2 in the given interval [0, 2π).
Since sin(x) = sqrt(3)/2 is a commonly-known value, we can refer to the unit circle or the trigonometric ratios to find the angles where sin(x) equals sqrt(3)/2.
On the unit circle, we know that sin(pi/3) = sqrt(3)/2. Since sin is positive in the first and second quadrants, the angles where sin(x) = sqrt(3)/2 within the interval [0, 2π) are pi/3 and 2π - pi/3, which is 5π/3.
Step 5: Write down all the solutions.
The solutions of the equation sin(x + π) − sin x + sqrt(3) = 0 in the interval [0, 2π) are pi/3 and 5π/3.
Therefore, the comma-separated list of all solutions is pi/3, 5π/3.