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Solve it using Sine Law for triangles, R is the Resultant Vector 

To solve this vector triangle using the Sine Law, let's analyze the provided information.

### Problem Analysis:

The triangle is labeled as follows:
- \( \angle B = 30^\circ \)
- Side \( A = 6 \) units
- Side \( R = 4 \) units (representing the resultant vector)
- The unknown angle opposite side \( A \) is \( \theta \).

We need to determine \( \theta \) using the Sine Law.

### Sine Law Formula:

The Sine Law is given by:

\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]

For this triangle:
- \( a = 4 \) (opposite \( \angle B = 30^\circ \))
- \( b = 6 \) (opposite \( \theta \))
- \( \angle B = 30^\circ \)

### Step-by-Step Solution:

Using the Sine Law, we have:

\[
\frac{4}{\sin 30^\circ} = \frac{6}{\sin \theta}
\]

Substituting \( \sin 30^\circ = 0.5 \):

\[
\frac{4}{0.5} = \frac{6}{\sin \theta}
\]

This simplifies to:

\[
8 = \frac{6}{\sin \theta}
\]

Now, solving for \( \sin \theta \):

\[
\sin \theta = \frac{6}{8} = 0.75
\]

Finally, \( \theta \) can be determined using the inverse sine function:

\[
\theta = \sin^{-1}(0.75)
\]

Let's compute the exact value.

### Calculation:

I will now calculate the angle \( \theta \).The value of \( \theta \) is approximately \( 48.59^\circ \).

Would you like more details on this solution or have any additional questions?

Here are 5 related questions that can expand on this topic:
1. How would you find the remaining angle in the triangle?
2. What does the resultant vector \( R \) represent in this context?
3. Can you use vector addition to verify the solution?
4. How would the solution change if the triangle was not closed (i.e., it had an open vector diagram)?
5. How can you apply the Sine Law in 3D vector problems?

**Tip:** In vector problems, always double-check the angle measurements by visualizing the vector directions and considering the triangle’s shape.

Explore how to solve a vector triangle using the Sine Law. Given a triangle with angle B = 30 degrees, side A = 6 units, and resultant vector R = 4 units, learn how to find the angle opposite side A using the Sine Law formula. Suitable for high school students studying trigonometry and vectors.

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