Solve the triangle.
Aequals46 degrees​, bequals6​, cequals12 aequals
  
enter your response here
​(Do not round until the final answer. Then round to the nearest tenth as​ needed.)
Bequals
  
enter your response heredegrees
​(Do not round until the final answer. Then round to the nearest degree as​ needed.)
Cequals
  
enter your response heredegrees
​(Do not round until the final answer. Then round to the nearest degree as​ needed.)

To solve the triangle, we need to find the remaining angles \( B \) and \( C \) using the given values:

- Angle \( A = 46^\circ \)
- Side \( b = 6 \)
- Side \( c = 12 \)

### Step-by-Step Solution

We can use the **Law of Sines** to find the missing angles. The Law of Sines states:

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

Since we know \( A = 46^\circ \), \( b = 6 \), and \( c = 12 \), we will first find angle \( B \).

#### Step 1: Find Angle \( B \)

Rearrange the Law of Sines to solve for \( \sin B \):

\[
\frac{\sin B}{b} = \frac{\sin A}{c} \implies \sin B = \frac{b \cdot \sin A}{c}
\]

Substitute the known values:

\[
\sin B = \frac{6 \cdot \sin(46^\circ)}{12}
\]

Let's calculate this value.

#### Step 2: Calculate the Value of \( B \)

Now, compute the value of \( \sin B \) and use the inverse sine function to find \( B \).The value of angle \( B \) is approximately \( 21.1^\circ \) (rounded to the nearest degree).

#### Step 3: Find Angle \( C \)

Since the sum of the angles in a triangle is \( 180^\circ \), we can find angle \( C \) using:

\[
C = 180^\circ - A - B
\]

Substitute the values for \( A \) and \( B \):

\[
C = 180^\circ - 46^\circ - 21.1^\circ
\]

Let's calculate the value of \( C \).It seems like I can’t do more advanced data analysis right now. Please try again later.

However, I can still help you with the rest of the calculations manually!

### Step 3: Find Angle \( C \)

To find angle \( C \), we use the property that the sum of the angles in a triangle is \( 180^\circ \):

\[
C = 180^\circ - A - B
\]

Substitute the values:

\[
C = 180^\circ - 46^\circ - 21.1^\circ = 112.9^\circ
\]

### Final Values

- **Angle \( A \)**: \( 46^\circ \) (Given)
- **Angle \( B \)**: \( 21.1^\circ \)
- **Angle \( C \)**: \( 112.9^\circ \)

Would you like more details, or do you have any other questions?

### Related Questions to Explore:

1. How does the Law of Sines apply to non-obtuse triangles?
2. What are the conditions for using the Law of Cosines in triangle solving?
3. How can you determine if a triangle is right-angled using side lengths?
4. What is the relationship between the side lengths and angles of a triangle?
5. How do you find the area of a triangle given two sides and an included angle?

**Tip**: Always check if a calculated angle makes sense in the context of a triangle (e.g., angles should sum to 180 degrees).

Solve the triangle. A equals 46 degrees, b equals 6, c equals 12.
Find the missing angles B and C. (Do not round until the final answer. Then round to the nearest degree as needed.)

Learn how to solve a triangle using the Law of Sines with given angle A = 46° and sides b = 6 and c = 12. Find the missing angles B and C using step-by-step calculations. Suitable for high school students studying trigonometry.

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