https://file.aiquickdraw.com/mathbot/file/17254551536886le37p8c.jpg

The image contains an equation and a set of multiple-choice options. Let's extract and solve the equation step by step.

### 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.

Would you like more details or any follow-up questions?

### 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.

Explore how to solve the trigonometric equation sin^2(π/6) × (x^2 + 8x - 3) = cos(π/3) × (x^2/2) with a step-by-step approach. This problem covers key trigonometric and algebraic concepts and is suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation