Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Describe the methods, processes, and equations used to determine the missing side and angles.
attached image
![**Solve the following triangle:**
The diagram shows a triangle with the following information:
- One side is labeled as 2.3 cm.
- Another side is labeled as 1.1 cm.
- One angle is labeled as 105°.
**Task:**
Describe the methods, processes, and equations used to determine the missing side and angles.
**Explanation:**
1. **Triangle Description:** The given triangle has two known sides and one known angle, which is 105°. This is an oblique triangle (non-right triangle).
2. **Method to Solve:**
- **Law of Cosines:** Since two sides and the included angle are known, the Law of Cosines can be applied to find the third side. The formula is:
\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]
where \( a \) and \( b \) are the known sides, \( C \) is the known angle (105°), and \( c \) is the unknown side.
- **Law of Sines:** Once the third side is found, the Law of Sines can be used to find the remaining angles. The formula is:
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]
where \( A \) and \( B \) are the unknown angles corresponding to the known and found sides.
3. **Process:**
- Use the Law of Cosines to calculate the length of the third side.
- Apply the Law of Sines to find one of the other angles.
- Calculate the remaining angle using the fact that the sum of angles in a triangle is 180°.
4. **Example Calculation:**
Start by calculating the third side using the Law of Cosines. Then use the Law of Sines to find another angle, remembering to adjust for angles that could be obtuse. Finally, find the last angle.
By following these steps, you will solve the triangle and determine all side lengths and angles.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F692b4bb9-1b89-4fb4-be46-e3686210292a%2F875ad8ef-986a-409e-aaf5-254d7be8a8f5%2Fza9lyp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Solve the following triangle:**
The diagram shows a triangle with the following information:
- One side is labeled as 2.3 cm.
- Another side is labeled as 1.1 cm.
- One angle is labeled as 105°.
**Task:**
Describe the methods, processes, and equations used to determine the missing side and angles.
**Explanation:**
1. **Triangle Description:** The given triangle has two known sides and one known angle, which is 105°. This is an oblique triangle (non-right triangle).
2. **Method to Solve:**
- **Law of Cosines:** Since two sides and the included angle are known, the Law of Cosines can be applied to find the third side. The formula is:
\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]
where \( a \) and \( b \) are the known sides, \( C \) is the known angle (105°), and \( c \) is the unknown side.
- **Law of Sines:** Once the third side is found, the Law of Sines can be used to find the remaining angles. The formula is:
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]
where \( A \) and \( B \) are the unknown angles corresponding to the known and found sides.
3. **Process:**
- Use the Law of Cosines to calculate the length of the third side.
- Apply the Law of Sines to find one of the other angles.
- Calculate the remaining angle using the fact that the sum of angles in a triangle is 180°.
4. **Example Calculation:**
Start by calculating the third side using the Law of Cosines. Then use the Law of Sines to find another angle, remembering to adjust for angles that could be obtuse. Finally, find the last angle.
By following these steps, you will solve the triangle and determine all side lengths and angles.
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