If θ is an acute angle and 4cosθ+4 
3

 sinθ=8sinθcosθ+2 
3

 , then which one among the following is correct?
(A) sinθ= 
2
3

 

 
(B) cosθ= 
2
1

 
(C) θ=30 
∘
 
(D) tanθ= 
3

 

We are given the equation involving trigonometric terms:

\[
4 \cos \theta + 4\sqrt{3} \sin \theta = 8 \sin \theta \cos \theta + 2\sqrt{3}
\]

We need to determine which one of the following is correct based on the equation:

(A) \(\sin \theta = \frac{2}{3}\)

(B) \(\cos \theta = \frac{1}{2}\)

(C) \(\theta = 30^\circ\)

(D) \(\tan \theta = \sqrt{3}\)

### Step-by-Step Solution:

1. **Simplify the given equation:**

   \[
   4 \cos \theta + 4\sqrt{3} \sin \theta = 8 \sin \theta \cos \theta + 2\sqrt{3}
   \]

   We will try to move all the terms involving \(\theta\) on one side and the constants on the other side.

2. **Rearrange the equation:**

   Move all terms involving \(\theta\) to the left-hand side:

   \[
   4 \cos \theta + 4\sqrt{3} \sin \theta - 8 \sin \theta \cos \theta = 2\sqrt{3}
   \]

3. **Substitute potential values:**

   Let's test each option.

   - **Option B: \(\cos \theta = \frac{1}{2}\)**

     If \(\cos \theta = \frac{1}{2}\), then \(\theta = 60^\circ\).

     We know:

     \[
     \sin 60^\circ = \frac{\sqrt{3}}{2}, \quad \cos 60^\circ = \frac{1}{2}
     \]

     Substitute these values into the original equation:

     \[
     4 \times \frac{1}{2} + 4\sqrt{3} \times \frac{\sqrt{3}}{2} = 8 \times \frac{\sqrt{3}}{2} \times \frac{1}{2} + 2\sqrt{3}
     \]

     Simplifying:

     \[
     2 + 4 \times 3 = 4\sqrt{3} + 2\sqrt{3}
     \]

     \[
     2 + 12 = 6\sqrt{3}
     \]

     But this is not a valid identity. So, Option B is incorrect.

   - **Option D: \(\tan \theta = \sqrt{3}\)**

     If \(\tan \theta = \sqrt{3}\), then \(\theta = 60^\circ\), which is the same as tested in Option B, and that doesn't work either.

   - **Option C: \(\theta = 30^\circ\)**

     If \(\theta = 30^\circ\), then:

     \[
     \sin 30^\circ = \frac{1}{2}, \quad \cos 30^\circ = \frac{\sqrt{3}}{2}
     \]

     Substituting these values into the equation:

     \[
     4 \times \frac{\sqrt{3}}{2} + 4\sqrt{3} \times \frac{1}{2} = 8 \times \frac{1}{2} \times \frac{\sqrt{3}}{2} + 2\sqrt{3}
     \]

     Simplifying:

     \[
     2\sqrt{3} + 2\sqrt{3} = 2\sqrt{3} + 2\sqrt{3}
     \]

     This is a valid identity. Hence, **Option C is correct**: \(\theta = 30^\circ\).

Thus, the correct answer is:

\[
\boxed{\theta = 30^\circ}
\]

---

Would you like further details or have any questions? Here are some related questions:

1. What are the values of \(\sin\), \(\cos\), and \(\tan\) for common angles like \(30^\circ\), \(45^\circ\), and \(60^\circ\)?
2. How can you derive the trigonometric identities used in this problem?
3. What is the relationship between \(\sin\) and \(\cos\) for complementary angles?
4. How can you solve more complex trigonometric equations?
5. What are some common mistakes to avoid when solving trigonometric identities?

**Tip:** When solving trigonometric equations, always check for multiple possible solutions within the given range, especially if the problem involves acute angles or a restricted domain.

If θ is an acute angle and 4cosθ + 4√3 sinθ = 8sinθcosθ + 2√3, then which one among the following is correct? (A) sinθ = 2/3 (B) cosθ = 1/2 (C) θ = 30° (D) tanθ = √3

Solve the trigonometric equation 4cosθ + 4√3 sinθ = 8sinθcosθ + 2√3 and determine the correct value for θ. This problem covers key trigonometric concepts, including identity simplifications and angle relationships. Suitable for students in Grades 10-12.

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