I have a triangle we'll call ABC. Angle A is 110 degrees, and angles B and C are unknown. Side a is 20.68 feet, and side c is 7 feet. Side by is unknown. Use the law of sines to find side b.

To find the length of side \( b \) in triangle \( ABC \) using the Law of Sines, we can use the following formula:

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

Given:
- \( A = 110^\circ \)
- \( a = 20.68 \) feet
- \( c = 7 \) feet
- \( B \) and \( C \) are unknown

First, we need to find angle \( C \). Since the sum of the angles in a triangle is \( 180^\circ \), we can write:

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

Thus,

\[ 110^\circ + B + C = 180^\circ \]

\[ B + C = 70^\circ \]

Next, we'll use the Law of Sines to find angle \( C \). We know that:

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

So,

\[ \frac{20.68}{\sin(110^\circ)} = \frac{7}{\sin(C)} \]

Solving for \( \sin(C) \), we get:

\[ \sin(C) = \frac{7 \cdot \sin(110^\circ)}{20.68} \]

First, calculate \( \sin(110^\circ) \):

\[ \sin(110^\circ) \approx 0.9397 \]

Now substitute this value into the equation:

\[ \sin(C) = \frac{7 \cdot 0.9397}{20.68} \]

\[ \sin(C) \approx \frac{6.5789}{20.68} \]

\[ \sin(C) \approx 0.318 \]

Now, find angle \( C \):

\[ C = \sin^{-1}(0.318) \]

\[ C \approx 18.57^\circ \]

Now that we have \( C \), we can find angle \( B \):

\[ B = 70^\circ - C \]

\[ B \approx 70^\circ - 18.57^\circ \]

\[ B \approx 51.43^\circ \]

Now we can use the Law of Sines to find side \( b \):

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

\[ \frac{b}{\sin(51.43^\circ)} = \frac{20.68}{\sin(110^\circ)} \]

Solving for \( b \), we get:

\[ b = \frac{20.68 \cdot \sin(51.43^\circ)}{\sin(110^\circ)} \]

First, calculate \( \sin(51.43^\circ) \):

\[ \sin(51.43^\circ) \approx 0.7805 \]

Now substitute this value into the equation:

\[ b = \frac{20.68 \cdot 0.7805}{0.9397} \]

\[ b \approx \frac{16.15}{0.9397} \]

\[ b \approx 17.19 \]

Therefore, the length of side \( b \) is approximately 17.19 feet.

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

Here are some related questions for further exploration:

1. How would you verify the solution using the Law of Cosines?
2. What is the area of triangle ABC?
3. How do you find the height of the triangle from vertex A to side a?
4. How do you find the height of the triangle from vertex B to side b?
5. What is the perimeter of triangle ABC?
6. How does the Law of Sines apply to this triangle for other sides?
7. What are the remaining sides and angles if side b was different?
8. How do you solve for side \( b \) using Heron's formula after finding all sides?

**Tip:** When using the Law of Sines, always ensure that the sine values used are within the valid range, and check for possible ambiguous cases in non-right triangles.

Learn how to find the length of side b in triangle ABC using the Law of Sines. Given angle A = 110 degrees, side a = 20.68 feet, and side c = 7 feet. Detailed step-by-step solution with calculations included.

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