In triangle ABC, a = 356, b = 207 and c = 296. Calculate the size of the largest angle. the answer is 88°15′

To find the size of the largest angle in triangle ABC, we will use **the Law of Cosines**. The largest angle will correspond to the largest side, which is \( a = 356 \) in this case.

The Law of Cosines states:
\[
\cos A = \frac{b^2 + c^2 - a^2}{2bc}
\]

Given:
- \( a = 356 \)
- \( b = 207 \)
- \( c = 296 \)

We can now substitute the values into the formula:

\[
\cos A = \frac{207^2 + 296^2 - 356^2}{2 \cdot 207 \cdot 296}
\]

Let’s calculate this step-by-step:

1. Compute the squares:
   - \( 207^2 = 42849 \)
   - \( 296^2 = 87616 \)
   - \( 356^2 = 126736 \)

2. Apply these values to the Law of Cosines formula:
   \[
   \cos A = \frac{42849 + 87616 - 126736}{2 \cdot 207 \cdot 296}
   \]
   \[
   \cos A = \frac{130465 - 126736}{122544}
   \]
   \[
   \cos A = \frac{3729}{122544} \approx 0.0304
   \]

3. Find the angle \( A \) by taking the inverse cosine:
   \[
   A = \cos^{-1}(0.0304)
   \]

   Using a calculator:
   \[
   A \approx 88.25^\circ
   \]

This is approximately \( 88^\circ 15' \) (88 degrees and 15 minutes).

Thus, the largest angle \( A \) is \( 88^\circ 15' \).

Would you like further details or have any questions?

### Related Questions:
1. How can we apply the Law of Sines to check our result?
2. What are the other two angles of the triangle?
3. How would the calculation change if side \( c \) were the largest?
4. How does the Law of Cosines differ from the Law of Sines in application?
5. Can we solve this triangle using a different method such as Heron's formula?

**Tip:** Always identify the largest angle by finding the angle opposite the largest side in any triangle problem.

In triangle ABC, a = 356, b = 207 and c = 296. Calculate the size of the largest angle. The answer is 88°15′.

Learn how to calculate the largest angle in triangle ABC where sides a = 356, b = 207, and c = 296 using the Law of Cosines. This problem involves key trigonometric concepts and detailed calculations, perfect for students in Grades 10-12.

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