https://file.aiquickdraw.com/mathbot/file/1729218399153anyd77p7.jpg

The image presents a triangle \(ABC\), where:

- \( \angle A = 69^\circ \),
- \( a = 11 \) (side opposite \(A\)),
- \( b = 6 \) (side opposite \(B\)).

We are tasked with finding the angle \( \angle B \).

### Step 1: Use the Law of Sines
The Law of Sines states:

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

We can use this relationship to solve for \( \sin(B) \):

\[
\frac{11}{\sin(69^\circ)} = \frac{6}{\sin(B)}
\]

### Step 2: Solve for \( \sin(B) \)

First, calculate \( \sin(69^\circ) \):

\[
\sin(69^\circ) \approx 0.9336
\]

Now substitute into the equation:

\[
\frac{11}{0.9336} = \frac{6}{\sin(B)}
\]

This simplifies to:

\[
11 \times \sin(B) = 6 \times 0.9336
\]

\[
11 \times \sin(B) = 5.6016
\]

Solving for \( \sin(B) \):

\[
\sin(B) = \frac{5.6016}{11} \approx 0.5092
\]

### Step 3: Find \( \angle B \)

Now find \( B \) using the inverse sine function:

\[
B = \sin^{-1}(0.5092) \approx 30.62^\circ
\]

So, \( \angle B \approx 30.62^\circ \).

Would you like further details on this calculation, or do you have any questions?

Here are 5 related questions for practice:

1. What is the remaining angle \( \angle C \) in triangle \(ABC\)?
2. How would you use the Law of Cosines to verify the result?
3. What are the side-length relationships for this triangle if it was a right triangle?
4. Can you use the Law of Sines to find side \(c\)?
5. What happens to angle \(B\) if side \(b\) is increased?

**Tip:** Always check that the sum of angles in any triangle equals \(180^\circ\) to verify your results.

What is the measure of angle B in the triangle, given that angle A is 69°, side a = 11, and side b = 6?

In this problem, we use the Law of Sines to find the measure of angle B in triangle ABC. Given that angle A is 69°, side a = 11, and side b = 6, learn the step-by-step process to solve this using trigonometric functions.

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