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
=
18.8
a=18.8​,
b
=
18.9
b=18.9​, and
c
=
18.2
c=18.2​​, solve for the angle
C
C​​​​​. Enter the degree symbol by typing "deg​" and round your answer to the nearest hundredth of a degree.

We are given a triangle \( ABC \) with the sides \( a = 18.8 \), \( b = 18.9 \), and \( c = 18.2 \). To solve for angle \( C \), we can use the **law of cosines**. The formula for the law of cosines is:

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

Rearranging the formula to solve for \( \cos(C) \):

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

### Step 1: Substitute the given values into the formula.

\[
\cos(C) = \frac{18.8^2 + 18.9^2 - 18.2^2}{2 \cdot 18.8 \cdot 18.9}
\]

### Step 2: Calculate each term.

\[
18.8^2 = 353.44
\]
\[
18.9^2 = 357.21
\]
\[
18.2^2 = 331.24
\]

Substitute these values into the equation:

\[
\cos(C) = \frac{353.44 + 357.21 - 331.24}{2 \cdot 18.8 \cdot 18.9}
\]

\[
\cos(C) = \frac{379.41}{710.64}
\]

\[
\cos(C) \approx 0.5339
\]

### Step 3: Solve for \( C \).

To find \( C \), take the inverse cosine (arccos) of 0.5339:

\[
C = \arccos(0.5339)
\]

\[
C \approx 57.68^\circ
\]

### Final Answer:

\[
C \approx 57.68^\circ \text{deg}
\]

Let me know if you'd like further details or have any questions!

### Related Questions:
1. How do we solve for the other angles in the triangle using the law of cosines?
2. Can the law of sines be used after finding one angle?
3. How can you check if a triangle is valid using side lengths?
4. What happens if one side is significantly larger than the other two in a triangle?
5. How does the law of cosines relate to the Pythagorean theorem in a right triangle?

### Tip:
Always double-check if your triangle is valid by ensuring that the sum of the angles adds up to 180 degrees after finding them.

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 = 18.8, b = 18.9, and c = 18.2, 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 angle C in a triangle using the Law of Cosines with sides a = 18.8, b = 18.9, and c = 18.2. This step-by-step guide covers key trigonometric concepts and the inverse cosine function. Perfect for high school 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

18.8 a=18.8, b

18.9 b=18.9, and c

Solution

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