https://file.aiquickdraw.com/mathbot/file/1725822409683y1mc8r61.jpg

The equation from the image appears to be:

\[
\left( \frac{1}{4} \sin^2(x) + \frac{5}{6} \right) \left( \frac{17}{21} \sin(x) - \frac{8}{11} \right) = 0
\]

### Step-by-Step Solution:

The product of two terms is equal to zero. This implies that either of the terms must be zero. Therefore, we will solve for \(x\) by setting each term equal to zero individually.

#### 1. First Equation: 
\[
\frac{1}{4} \sin^2(x) + \frac{5}{6} = 0
\]

To solve this:

\[
\frac{1}{4} \sin^2(x) = -\frac{5}{6}
\]

This equation has no real solutions because \(\sin^2(x)\) is always non-negative, and the right-hand side is negative. Therefore, there are no solutions from this equation.

#### 2. Second Equation:
\[
\frac{17}{21} \sin(x) - \frac{8}{11} = 0
\]

Solve for \(\sin(x)\):

\[
\frac{17}{21} \sin(x) = \frac{8}{11}
\]

Multiply both sides by \(\frac{21}{17}\):

\[
\sin(x) = \frac{8}{11} \times \frac{21}{17} = \frac{168}{187} = \frac{8}{11}
\]

Thus, we have:

\[
\sin(x) = \frac{8}{11}
\]

#### 3. Solving for \(x\):

The general solution for \( \sin(x) = \frac{8}{11} \) is given by:

\[
x = \sin^{-1}\left(\frac{8}{11}\right) + 2n\pi \quad \text{or} \quad x = \pi - \sin^{-1}\left(\frac{8}{11}\right) + 2n\pi
\]

Where \(n\) is any integer.

You can use a calculator to find the inverse sine:

\[
x_1 = \sin^{-1}\left(\frac{8}{11}\right) \approx 0.7954 \text{ radians}
\]

\[
x_2 = \pi - 0.7954 \approx 2.3462 \text{ radians}
\]

Thus, the solutions are:

\[
x = 0.7954 + 2n\pi \quad \text{or} \quad x = 2.3462 + 2n\pi
\]

These represent the general solutions for \(x\).

Would you like more details or have any further questions?

Here are five related questions to explore further:

1. How do we derive the general solution for trigonometric equations?
2. What is the significance of the periodicity of sine in solving these equations?
3. How would the solutions change if the right-hand side was positive in the first equation?
4. Could we solve this equation graphically to confirm the solutions?
5. What other methods exist to solve trigonometric equations involving products?

**Tip:** Always verify that the argument inside the inverse sine function stays within the valid range for real numbers: \([-1, 1]\).

Determine all solutions to the equation (1/4 * sin^2(x) + 5/6)(17/21 * sin(x) - 8/11) = 0.

Learn how to solve a trigonometric equation with sine squared and products using inverse sine function. This problem involves solving sin^2(x) + sin(x) and applying the zero product property. Ideal for students in Grades 10-12.

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