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Equation:

$\sin^2\left(\frac{\pi}{6}\right) \times \left(x^2 + 8x - 3\right) = \cos\left(\frac{\pi}{3}\right) \times \frac{x^2}{2}$

Step 1: Calculate Trigonometric Values

• $\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$, so $\sin^2\left(\frac{\pi}{6}\right) = \left(\frac{1}{2}\right)^2 = \frac{1}{4}$.
• $\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$.

Step 2: Substitute These Values into the Equation

$\frac{1}{4} \times \left(x^2 + 8x - 3\right) = \frac{1}{2} \times \frac{x^2}{2}$

Step 3: Simplify the Equation

$\frac{x^2 + 8x - 3}{4} = \frac{x^2}{4}$

Multiply through by 4 to eliminate the fraction: $x^2 + 8x - 3 = x^2$

Step 4: Simplify Further

Subtract $x^2$ from both sides: $8x - 3 = 0$

Step 5: Solve for $x$

$8x = 3$ $x = \frac{3}{8}$

It seems the answer $x = \frac{3}{8}$ is not listed among the choices provided. Let's recheck the steps, or consider whether the trigonometric values or simplification had an error. Please let me know if you'd like to proceed with additional verification or further clarification.

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Related Questions:

1. What are the steps to check if the solution $x = \frac{3}{8}$ is correct by substituting back into the original equation?
2. How does the choice of trigonometric identities impact the solution?
3. Can we derive the value of $x$ using a different method?
4. What happens if we mistakenly use the wrong trigonometric values?
5. How do we interpret the possible multiple-choice answers in the context of this problem?

Tip:

Always verify your trigonometric values and substitutions, especially when dealing with multiple-choice questions, as they may have exact, simplified, or approximate answers.