General Solution of Trigonometric Equation

To find the general solution of the given trigonometric equation, we need to solve the quadratic equation involving the sine function. The equation is:

\(3 \sin^2 x - 7 \sin x + 2 = 0\)

Solving the Quadratic in Sine x

This equation is a quadratic in terms of \(\sin x\). Let's use a substitution to make it clearer. Let \(y = \sin x\).

Substituting \(y\) into the equation, we get:

\(3y^2 - 7y + 2 = 0\)

We can solve this quadratic equation by factoring or using the quadratic formula. Let's factor it:

• We need two numbers that multiply to \((3 \times 2 = 6)\) and add up to \(-7\). These numbers are \(-6\) and \(-1\).

So, we can rewrite the middle term \(-7y\) as \(-6y - y\):

\(3y^2 - 6y - y + 2 = 0\)

Now, group the terms and factor by grouping:

\(3y(y - 2) - 1(y - 2) = 0\)

\((3y - 1)(y - 2) = 0\)

This gives us two possible values for \(y\):

\(3y - 1 = 0 \implies 3y = 1 \implies y = \frac{1}{3}\)

or

\(y - 2 = 0 \implies y = 2\)

Evaluating Possible Solutions for sin x

Now, we substitute back \(\sin x\) for \(y\):

Case 1: \(\sin x = \frac{1}{3}\)

Case 2: \(\sin x = 2\)

We know that the range of the sine function is between \(-1\) and \(1\), inclusive. That is, \(-1 \le \sin x \le 1\).

• For Case 2, \(\sin x = 2\) is not possible because \(2\) is outside the range \([-1, 1]\). Therefore, this case yields no solution.
• For Case 1, \(\sin x = \frac{1}{3}\) is a valid solution because \(\frac{1}{3}\) is within the range \([-1, 1]\).

So, we only need to find the general solution for \(\sin x = \frac{1}{3}\).

Applying the General Solution Formula for Sine

The general solution for an equation of the form \(\sin x = \sin \alpha\) is given by:

\(x = n\pi + (-1)^n \alpha\), where \(n \in \mathbb{Z}\) (i.e., \(n\) is an integer).

In our case, we have \(\sin x = \frac{1}{3}\).

Let \(\alpha = \sin^{-1}\left(\frac{1}{3}\right)\).

Therefore, the general solution for \(\sin x = \frac{1}{3}\) is:

\(x = n\pi + (-1)^n \sin^{-1}\left(\frac{1}{3}\right)\)

This matches one of the provided options.

Summary of Steps

Here's a quick summary of the steps to solve this trigonometric equation:

1. Recognize the equation as a quadratic in \(\sin x\).
2. Solve the quadratic equation for \(\sin x\).
3. Discard any solutions for \(\sin x\) that fall outside the range \([-1, 1]\).
4. Apply the general solution formula for \(\sin x = k\), which is \(x = n\pi + (-1)^n \sin^{-1}(k)\), where \(n\) is an integer.

The final general solution obtained is \(x = n\pi + (-1)^n \sin^{-1}\left(\frac{1}{3}\right)\).