I have a triangle ABC. Angles A and B are unknown, but angle C is 70 degrees. Side b is 7 feet and side a is 22 feet, but side c is unknown. What is the length of side C. Find it using the law of cosines.

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

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

Given:
- \( a = 22 \) feet
- \( b = 7 \) feet
- \( C = 70^\circ \)

Let's plug these values into the formula:

\[ c^2 = 22^2 + 7^2 - 2 \cdot 22 \cdot 7 \cdot \cos(70^\circ) \]

First, calculate each part step by step:

1. \( 22^2 = 484 \)
2. \( 7^2 = 49 \)
3. \( 2 \cdot 22 \cdot 7 = 308 \)
4. \( \cos(70^\circ) \approx 0.342 \)

Now, substitute these values back into the equation:

\[ c^2 = 484 + 49 - 308 \cdot 0.342 \]

\[ c^2 = 484 + 49 - 105.336 \]

\[ c^2 = 533 - 105.336 \]

\[ c^2 = 427.664 \]

Finally, take the square root to find \( c \):

\[ c = \sqrt{427.664} \]

\[ c \approx 20.68 \]

Therefore, the length of side \( c \) is approximately 20.68 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. What are the remaining angles in the triangle?
2. How would you find the area of triangle ABC?
3. How does the Law of Sines apply to this triangle?
4. What is the perimeter of triangle ABC?
5. How do you verify the solution using the Law of Cosines?
6. How would you solve for side \( c \) using Heron's formula after finding all sides?
7. How do you find the height of the triangle from vertex C to side c?
8. What are some real-world applications of the Law of Cosines?

**Tip:** The Law of Cosines is very useful for finding unknown sides and angles in any triangle, not just right triangles.

Learn how to find the length of side c in triangle ABC using the Law of Cosines. Given side lengths a = 22 feet and b = 7 feet, and angle C = 70 degrees, calculate side c step-by-step with detailed mathematical explanations.

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