Let 
A
B
C
ABC​ be a triangle such that angle 
A
A​ is opposite side 
a
a​, angle 
B
B​ is opposite side 
b
b​, and angle 
C
C​ is opposite side 
c
c​. If ​​
a
=
12.5
a=12.5​, 
b
=
16.9
b=16.9​, and 
c
=
11.1
c=11.1​​, solve for the angle 
C
C​​​​​. Enter the degree symbol by typing "deg​" and round your answer to the nearest hundredth of a degree.


To solve for the angle \( C \) in a triangle with sides \( a = 12.5 \), \( b = 16.9 \), and \( c = 11.1 \), we can use the **Law of Cosines**. The Law of Cosines relates the sides of a triangle to the cosine of one of its angles, and it is given by:

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

### Step 1: Plug the given values into the formula
\[
\cos(C) = \frac{12.5^2 + 16.9^2 - 11.1^2}{2 \times 12.5 \times 16.9}
\]

### Step 2: Calculate the squares of the sides
\[
12.5^2 = 156.25
\]
\[
16.9^2 = 285.61
\]
\[
11.1^2 = 123.21
\]

### Step 3: Substitute these values into the equation
\[
\cos(C) = \frac{156.25 + 285.61 - 123.21}{2 \times 12.5 \times 16.9}
\]
\[
\cos(C) = \frac{318.65}{422.5}
\]
\[
\cos(C) \approx 0.7542
\]

### Step 4: Take the inverse cosine to find \( C \)
\[
C = \cos^{-1}(0.7542)
\]

Let me calculate the angle.It seems like I can’t do more advanced data analysis right now. Please try again later. However, you can use the following steps to complete the calculation manually:

1. Calculate \( \cos^{-1}(0.7542) \) using a calculator or scientific tool.
2. Convert this result into degrees.

Let me know if you'd like further help with this process!

### Expanding Questions:
1. How do we derive the Law of Cosines?
2. Can the Law of Cosines be used to solve for angles other than \( C \)?
3. What happens if all sides of a triangle are equal (equilateral triangle) in the Law of Cosines?
4. How does the Law of Cosines relate to the Law of Sines?
5. What if one of the angles was a right angle? How would the Law of Cosines change?

**Tip:** Always round trigonometric values carefully, as small changes can affect the final result.

Let ABC be a triangle such that angle A is opposite side a, angle B is opposite side b, and angle C is opposite side c. If a = 12.5, b = 16.9, and c = 11.1, solve for the angle C. Enter the degree symbol by typing 'deg' and round your answer to the nearest hundredth of a degree.

Learn how to solve for the angle C in a triangle with sides a = 12.5, b = 16.9, and c = 11.1 using the Law of Cosines. This step-by-step solution covers key trigonometric concepts and is ideal for students in Grades 10-12.

Math Problem Statement

Let A B C ABC be a triangle such that angle A A is opposite side a a, angle B B is opposite side b b, and angle C C is opposite side c c. If a

12.5 a=12.5, b

16.9 b=16.9, and c

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