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Solve the unknown angle using sine Law and the R or the Resultant Vector is unknown also

To solve the unknown angle and determine the length of the resultant vector (R) in the triangle, we can use the **Law of Sines**.

### Given:
- Angle at \(O\) = \(28^\circ\)
- Side \(A = 11\)
- Side \(B = 13\)
- Opposite angle to \(A\) = \( \text{Unknown angle} \)
- Side \(R\) = Unknown

### Step 1: Identify the known quantities in relation to the Law of Sines

The Law of Sines states:
\[
\frac{A}{\sin \text{Angle opposite to } A} = \frac{B}{\sin \text{Angle opposite to } B} = \frac{R}{\sin \text{Angle opposite to } R}
\]
In this scenario:
- \(A = 11\) and the angle opposite to \(A\) is \( \angle O \), which is \(28^\circ\).
- \(B = 13\) and the angle opposite to \(B\) is \( \angle R \) (which is unknown).
- \(R\) is the unknown side, and the angle opposite to \(R\) is the other unknown angle at \(B\).

### Step 2: Solve for the unknown angle using the Law of Sines
\[
\frac{11}{\sin 28^\circ} = \frac{13}{\sin \text{(Unknown angle at B)}}
\]

Let's denote the unknown angle at \(B\) as \( \angle B \).

So,
\[
\sin \angle B = \frac{13 \times \sin 28^\circ}{11}
\]

Let's calculate this.

The unknown angle at \(B\) is approximately \(33.7^\circ\).

### Step 3: Calculate the third angle (at \(R\))
Since the sum of the angles in a triangle is \(180^\circ\), the angle at \(R\) can be found as follows:
\[
\angle R = 180^\circ - 28^\circ - 33.7^\circ
\]
Let's calculate this angle.

The angle at \(R\) is approximately \(118.3^\circ\).

### Step 4: Use the Law of Sines to find the length of side \(R\)
Now, apply the Law of Sines again to solve for \(R\):
\[
\frac{R}{\sin 118.3^\circ} = \frac{A}{\sin 28^\circ}
\]
Solving for \(R\):
\[
R = \frac{A \times \sin 118.3^\circ}{\sin 28^\circ}
\]
Let's calculate the length of \(R\).The length of the resultant vector \(R\) is approximately \(20.63\) units.

### Summary:
- The unknown angle at \(B\) is approximately \(33.7^\circ\).
- The angle at \(R\) is approximately \(118.3^\circ\).
- The length of the resultant vector \(R\) is approximately \(20.63\) units.

Would you like more details on any of the steps, or do you have any further questions?

### Related Questions:
1. How would the solution change if side \(A\) was longer?
2. What if the angle at \(O\) was larger; how would that affect the lengths and angles?
3. Can you solve for side \(R\) if only sides \(A\) and \(B\) are given, without any angles?
4. How does the Law of Sines compare to the Law of Cosines in solving this problem?
5. What real-life applications can you think of that use the Law of Sines?

### Tip:
Always ensure that the sum of the angles in a triangle equals \(180^\circ\) as a quick check for your calculations.

Learn how to solve for the unknown angle using the Law of Sines and determine the length of the resultant vector (R) in a triangle. This problem involves trigonometric concepts and triangle geometry, suitable for high school students.

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