Math Snap

STEP 1

1. The equation $3 \sin 2x - 1 = 0$ is a trigonometric equation.
2. We are solving for $x$ within the interval $[0, 2\pi]$.
3. The function $\sin 2x$ has a period of $\pi$.

STEP 2

1. Isolate the trigonometric function $\sin 2x$.
2. Solve the equation $\sin 2x = \frac{1}{3}$.
3. Find all solutions for $2x$ within the interval $[0, 4\pi]$.
4. Solve for $x$ by dividing the solutions for $2x$ by 2.
5. Ensure all solutions for $x$ fall within the interval $[0, 2\pi]$.

STEP 3

First, isolate $\sin 2x$ by adding 1 to both sides and then dividing by 3:
$$3 \sin 2x - 1 = 0$$ $$3 \sin 2x = 1$$ $$\sin 2x = \frac{1}{3}$$

STEP 4

Solve the equation $\sin 2x = \frac{1}{3}$. We need to find the general solutions for $2x$:
The general solution for $\sin \theta = a$ is given by:
$$\theta = \arcsin(a) + 2k\pi \quad \text{or} \quad \theta = \pi - \arcsin(a) + 2k\pi$$ For $\sin 2x = \frac{1}{3}$, we have:
$$2x = \arcsin\left(\frac{1}{3}\right) + 2k\pi$$ $$2x = \pi - \arcsin\left(\frac{1}{3}\right) + 2k\pi$$

STEP 5

Find all solutions for $2x$ within the interval $[0, 4\pi]$:
1. Calculate $\arcsin\left(\frac{1}{3}\right)$. Let's denote it as $\alpha$.
2. The solutions for $2x$ are:
- $2x = \alpha + 2k\pi$
- $2x = \pi - \alpha + 2k\pi$
For $k = 0$:
- $2x = \alpha$
- $2x = \pi - \alpha$
For $k = 1$:
- $2x = \alpha + 2\pi$
- $2x = \pi - \alpha + 2\pi$
Check which solutions fall within $[0, 4\pi]$.

STEP 6

Calculate specific values for $2x$ and check the interval:
1. $2x = \alpha$
2. $2x = \pi - \alpha$
3. $2x = \alpha + 2\pi$
4. $2x = \pi - \alpha + 2\pi$
Assuming $\alpha = \arcsin\left(\frac{1}{3}\right)$, compute the approximate values and ensure they fall within $[0, 4\pi]$.

STEP 7

Solve for $x$ by dividing each valid $2x$ solution by 2:
1. $x = \frac{\alpha}{2}$
2. $x = \frac{\pi - \alpha}{2}$
3. $x = \frac{\alpha + 2\pi}{2}$
4. $x = \frac{\pi - \alpha + 2\pi}{2}$

Ensure all solutions for $x$ fall within the interval $[0, 2\pi]$:
Verify each calculated $x$ value is within the interval $[0, 2\pi]$.
The solutions for $x$ are approximately:
$$x = \frac{\arcsin\left(\frac{1}{3}\right)}{2}, \frac{\pi - \arcsin\left(\frac{1}{3}\right)}{2}, \frac{\arcsin\left(\frac{1}{3}\right) + 2\pi}{2}, \frac{\pi - \arcsin\left(\frac{1}{3}\right) + 2\pi}{2}$$