I have a triangle with 1 angle measuring 110 degrees, 1 side measuring 7 feet, and 1 side measuring 20.68 feet. What are the values of the other angles?

To find the values of the other angles in the triangle, we can use the Law of Cosines. This will help us find the length of the unknown side, and then we can use the Law of Sines to find the remaining angles.

Given:
- Angle \( C = 110^\circ \)
- Side \( a = 7 \) feet
- Side \( b = 20.68 \) feet

First, we use the Law of Cosines to find side \( c \):

\[
c^2 = a^2 + b^2 - 2ab \cos(C)
\]

Substitute the given values into the formula:

\[
c^2 = 7^2 + 20.68^2 - 2 \cdot 7 \cdot 20.68 \cdot \cos(110^\circ)
\]

Calculate the cosine of \( 110^\circ \):

\[
\cos(110^\circ) \approx -0.3420
\]

Now substitute this value into the equation:

\[
c^2 = 49 + 427.0464 - 2 \cdot 7 \cdot 20.68 \cdot (-0.3420)
\]
\[
c^2 = 49 + 427.0464 + 99.26016
\]
\[
c^2 = 575.30656
\]
\[
c \approx \sqrt{575.30656} \approx 23.98 \text{ feet}
\]

Now that we have all three sides, we can use the Law of Sines to find the remaining angles.

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

We already know \( \sin(C) \) where \( C = 110^\circ \):

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

So,

\[
\frac{23.98}{\sin(110^\circ)} = \frac{7}{\sin(A)}
\]

Solving for \( \sin(A) \):

\[
\sin(A) = \frac{7 \cdot \sin(110^\circ)}{23.98}
\]
\[
\sin(A) = \frac{7 \cdot 0.9397}{23.98}
\]
\[
\sin(A) \approx 0.2745
\]

Now, find angle \( A \):

\[
A = \arcsin(0.2745) \approx 15.95^\circ
\]

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

\[
B = 180^\circ - C - A
\]
\[
B = 180^\circ - 110^\circ - 15.95^\circ
\]
\[
B \approx 54.05^\circ
\]

Thus, the values of the other angles are:
- Angle \( A \approx 15.95^\circ \)
- Angle \( B \approx 54.05^\circ \)

Would you like more details or have any questions about the process?

Here are 8 related questions for further exploration:
1. How would the calculation change if the given angle was obtuse?
2. Can the Law of Cosines be used to solve any triangle?
3. How do you derive the Law of Sines from the Law of Cosines?
4. What is the significance of knowing all three angles in a triangle?
5. How can you verify the accuracy of your calculated angles?
6. Can these methods be applied to non-Euclidean geometries?
7. How does the Law of Sines help in determining the height of a triangle?
8. What are some real-world applications of solving triangles using these laws?

**Tip:** Always check your work by ensuring the sum of all angles in the triangle adds up to 180 degrees.

Learn how to solve triangles using the Law of Cosines and Law of Sines. This problem involves finding the values of angles in a triangle given one angle and two sides. Detailed step-by-step solution included.

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