In the interval [0, 5π], the number of solutions of the equation $3~sin^{2}x-7~sin~x+2=0$ is:

Question

Accepted Answer

To solve the equation $3 \sin^2 x - 7 \sin x + 2 = 0$ in the interval $[0, 5\pi]$, we first treat it as a quadratic equation in terms of $\sin x$. Let $y = \sin x$. The equation becomes:

$$3y^2 - 7y + 2 = 0.$$

We solve this quadratic equation using the quadratic formula:

$$y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},$$

where $a = 3$, $b = -7$, and $c = 2$. Plugging in these values, we get:

$$y = \frac{7 \pm \sqrt{(-7)^2 - 4 \cdot 3 \cdot 2}}{2 \cdot 3} = \frac{7 \pm \sqrt{49 - 24}}{6} = \frac{7 \pm \sqrt{25}}{6}.$$

This simplifies to:

$$y = \frac{7 \pm 5}{6}.$$

Thus, the solutions for $y$ are:

$$y = \frac{12}{6} = 2 \quad \text{and} \quad y = \frac{2}{6} = \frac{1}{3}.$$

Since $\sin x$ must be in the range $[-1, 1]$, $y = 2$ is not a valid solution. Therefore, we only consider $y = \frac{1}{3}$.

Now, we need to find the values of $x$ such that $\sin x = \frac{1}{3}$ in the interval $[0, 5\pi]$. The general solution for $\sin x = \frac{1}{3}$ is:

$$x = \arcsin\left(\frac{1}{3}\right) + 2k\pi \quad \text{or} \quad x = \pi - \arcsin\left(\frac{1}{3}\right) + 2k\pi,$$

where $k$ is an integer.

Let's denote $\theta = \arcsin\left(\frac{1}{3}\right)$. Then the solutions in the interval $[0, 5\pi]$ are:

1. $x = \theta + 2k\pi$
2. $x = \pi - \theta + 2k\pi$

We need to find the values of $k$ such that these solutions lie within $[0, 5\pi]$.

For $x = \theta + 2k\pi$:

- When $k = 0$, $x = \theta$.
- When $k = 1$, $x = \theta + 2\pi$.
- When $k = 2$, $x = \theta + 4\pi$.

For $x = \pi - \theta + 2k\pi$:

- When $k = 0$, $x = \pi - \theta$.
- When $k = 1$, $x = \pi - \theta + 2\pi = 3\pi - \theta$.
- When $k = 2$, $x = \pi - \theta + 4\pi = 5\pi - \theta$.

Thus, we have a total of 6 solutions: $\theta$, $\theta + 2\pi$, $\theta + 4\pi$, $\pi - \theta$, $3\pi - \theta$, and $5\pi - \theta$.

Therefore, the number of solutions in the interval $[0, 5\pi]$ is 6, which corresponds to Option 3.

**Explanation for Incorrect Options:**

1. **Option 1 (0 solutions):** This is incorrect because we have found 6 valid solutions within the given interval.

2. **Option 2 (5 solutions):** This is incorrect because we have explicitly identified 6 solutions, not 5.

3. **Option 4 (10 solutions):** This is incorrect because the calculation shows only 6 solutions, not 10.