Solve AABC. Round decimal answers to the nearest tenth. A 23 14 'В 25 O A B C~° O
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![### Solving Triangle ABC
To solve triangle \( \triangle ABC \), follow these instructions and calculations.
#### Instructions:
- **Round decimal answers to the nearest tenth.**
#### Given:
- Side \( AC = 23 \)
- Side \( AB = 14 \)
- Side \( CB = 25 \)
#### Steps:
1. **Using the Law of Cosines to find the angles:**
- For angle \( A \):
\[
\cos(A) = \frac{b^2 + c^2 - a^2}{2bc}
\]
Here, \( a = 25 \), \( b = 23 \), \( c = 14 \):
\[
\cos(A) = \frac{23^2 + 14^2 - 25^2}{2 \cdot 23 \cdot 14}
\]
- For angle \( B \):
\[
\cos(B) = \frac{a^2 + c^2 - b^2}{2ac}
\]
Here, \( a = 25 \), \( b = 14 \), \( c = 23 \):
\[
\cos(B) = \frac{25^2 + 14^2 - 23^2}{2 \cdot 25 \cdot 23}
\]
- For angle \( C \):
\[
\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}
\]
Here, \( a = 14 \), \( b = 23 \), \( c = 25 \):
\[
\cos(C) = \frac{14^2 + 23^2 - 25^2}{2 \cdot 14 \cdot 23}
\]
2. **Calculate the actual values for each angle:**
- **Angle \( A \):**
\[
\cos(A) = \frac{529 + 196 - 625}{2 \cdot 23 \cdot 14} = \frac{100}{644} = 0.1553
\]
\[
A = \cos^{-1}(0.1553) \approx 81.0^\circ
\]
- **Angle \( B \):**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7d8e10cc-1afe-4ebd-96e4-e6a65b7f7ecb%2Ff60a9f0c-30d4-49fa-9c19-62942ccf905f%2F76mio0q_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving Triangle ABC
To solve triangle \( \triangle ABC \), follow these instructions and calculations.
#### Instructions:
- **Round decimal answers to the nearest tenth.**
#### Given:
- Side \( AC = 23 \)
- Side \( AB = 14 \)
- Side \( CB = 25 \)
#### Steps:
1. **Using the Law of Cosines to find the angles:**
- For angle \( A \):
\[
\cos(A) = \frac{b^2 + c^2 - a^2}{2bc}
\]
Here, \( a = 25 \), \( b = 23 \), \( c = 14 \):
\[
\cos(A) = \frac{23^2 + 14^2 - 25^2}{2 \cdot 23 \cdot 14}
\]
- For angle \( B \):
\[
\cos(B) = \frac{a^2 + c^2 - b^2}{2ac}
\]
Here, \( a = 25 \), \( b = 14 \), \( c = 23 \):
\[
\cos(B) = \frac{25^2 + 14^2 - 23^2}{2 \cdot 25 \cdot 23}
\]
- For angle \( C \):
\[
\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}
\]
Here, \( a = 14 \), \( b = 23 \), \( c = 25 \):
\[
\cos(C) = \frac{14^2 + 23^2 - 25^2}{2 \cdot 14 \cdot 23}
\]
2. **Calculate the actual values for each angle:**
- **Angle \( A \):**
\[
\cos(A) = \frac{529 + 196 - 625}{2 \cdot 23 \cdot 14} = \frac{100}{644} = 0.1553
\]
\[
A = \cos^{-1}(0.1553) \approx 81.0^\circ
\]
- **Angle \( B \):**
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