Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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![**Using the Law of Cosines to Determine the Largest Angle**
To solve for the largest angle in a triangle given the side lengths, we use the Law of Cosines. For a triangle with sides \(a = 22m\), \(b = 24m\), and \(c = 26m\), find the largest angle. Round your answers to the nearest tenth.
### Options:
- \(C = 68.7^\circ\)
- \(C = 98.3^\circ\)
- \(B = 78.3^\circ\)
- \(B = 88.3^\circ\)
### Steps to Solve:
1. **Identify the largest side:**
- In this triangle, \(c = 26m\) is the longest side.
2. **Apply the Law of Cosines:**
\[
c^2 = a^2 + b^2 - 2ab \cos(C)
\]
Substituting the given values:
\[
26^2 = 22^2 + 24^2 - 2 \cdot 22 \cdot 24 \cdot \cos(C)
\]
Simplify the equation:
\[
676 = 484 + 576 - 1056 \cos(C)
\]
Combine like terms:
\[
676 = 1060 - 1056 \cos(C)
\]
Solve for \(\cos(C)\):
\[
676 - 1060 = -1056 \cos(C)
\]
\[
-384 = -1056 \cos(C)
\]
\[
\cos(C) = \frac{384}{1056}
\]
\[
\cos(C) \approx 0.3636
\]
3. **Find angle \(C\) using inverse cosine:**
\[
C \approx \cos^{-1}(0.3636) \approx 68.7^\circ
\]
### Conclusion:
The largest angle in the triangle with sides \(a = 22m\), \(b = 24m\), and \(c = 26m\) is approximately \(C = 98.3^\circ\), not any of the other provided options.
### Explanation of the Answer:
The correct answer aligns with](https://content.bartleby.com/qna-images/question/ebfa71ca-f003-4e76-b16d-d56ed3281ebe/78111320-1fc3-46a1-b6f9-664ad83e141e/oqnqikh_processed.jpeg)
Transcribed Image Text:**Using the Law of Cosines to Determine the Largest Angle**
To solve for the largest angle in a triangle given the side lengths, we use the Law of Cosines. For a triangle with sides \(a = 22m\), \(b = 24m\), and \(c = 26m\), find the largest angle. Round your answers to the nearest tenth.
### Options:
- \(C = 68.7^\circ\)
- \(C = 98.3^\circ\)
- \(B = 78.3^\circ\)
- \(B = 88.3^\circ\)
### Steps to Solve:
1. **Identify the largest side:**
- In this triangle, \(c = 26m\) is the longest side.
2. **Apply the Law of Cosines:**
\[
c^2 = a^2 + b^2 - 2ab \cos(C)
\]
Substituting the given values:
\[
26^2 = 22^2 + 24^2 - 2 \cdot 22 \cdot 24 \cdot \cos(C)
\]
Simplify the equation:
\[
676 = 484 + 576 - 1056 \cos(C)
\]
Combine like terms:
\[
676 = 1060 - 1056 \cos(C)
\]
Solve for \(\cos(C)\):
\[
676 - 1060 = -1056 \cos(C)
\]
\[
-384 = -1056 \cos(C)
\]
\[
\cos(C) = \frac{384}{1056}
\]
\[
\cos(C) \approx 0.3636
\]
3. **Find angle \(C\) using inverse cosine:**
\[
C \approx \cos^{-1}(0.3636) \approx 68.7^\circ
\]
### Conclusion:
The largest angle in the triangle with sides \(a = 22m\), \(b = 24m\), and \(c = 26m\) is approximately \(C = 98.3^\circ\), not any of the other provided options.
### Explanation of the Answer:
The correct answer aligns with
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