Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section: Chapter Questions
Problem 75RE
Related questions
Question
![**Title:** Understanding the Law of Cosines
**Description:**
The Law of Cosines is a powerful tool in trigonometry that allows you to find unknown sides or angles in a triangle when you have sufficient information. Below is an example problem related to the Law of Cosines.
**Instructions:**
Use the Law of Cosines to solve the triangle. Round your answers to two decimal places.
**Given:**
- Side \( b = 5 \)
- Side \( a = 8 \)
- Angle \( C = 112^\circ \)
**Diagram:**
A triangle \( ABC \) is shown with:
- Side \( a \) opposite to Angle \( A \)
- Side \( b \) opposite to Angle \( B \)
- Side \( c \) (unknown) opposite to Angle \( C \)
- Angle \( C = 112^\circ \) is between sides \( a \) and \( b \)
**Formulas to Use:**
The Law of Cosines formula is given by:
\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]
1. To find side \( c \):
\[ c = \sqrt{a^2 + b^2 - 2ab \cdot \cos(C)} \]
2. To find Angle \( A \):
\[ \cos(A) = \frac{b^2 + c^2 - a^2}{2bc} \]
\[ A = \cos^{-1} \left(\frac{b^2 + c^2 - a^2}{2bc}\right) \]
3. To find Angle \( B \):
\[ \cos(B) = \frac{a^2 + c^2 - b^2}{2ac} \]
\[ B = \cos^{-1} \left(\frac{a^2 + c^2 - b^2}{2ac}\right) \]
**Tasks:**
1. Calculate the length of side \( c \).
2. Calculate the measure of Angle \( A \).
3. Calculate the measure of Angle \( B \).
For accurate results, use a calculator with trigonometric functions and ensure to switch it to degree mode if it has different settings (degree/radian).
**Note:**
For further assistance, use the "Need Help?" section or "Read It" feature provided.
---
This structured approach helps you](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F61a624e8-9805-4882-bd7c-8b439f9ba0a9%2F7bfb996e-5bb2-4f04-ab14-8876dd5c97fd%2Fx3khgdwg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title:** Understanding the Law of Cosines
**Description:**
The Law of Cosines is a powerful tool in trigonometry that allows you to find unknown sides or angles in a triangle when you have sufficient information. Below is an example problem related to the Law of Cosines.
**Instructions:**
Use the Law of Cosines to solve the triangle. Round your answers to two decimal places.
**Given:**
- Side \( b = 5 \)
- Side \( a = 8 \)
- Angle \( C = 112^\circ \)
**Diagram:**
A triangle \( ABC \) is shown with:
- Side \( a \) opposite to Angle \( A \)
- Side \( b \) opposite to Angle \( B \)
- Side \( c \) (unknown) opposite to Angle \( C \)
- Angle \( C = 112^\circ \) is between sides \( a \) and \( b \)
**Formulas to Use:**
The Law of Cosines formula is given by:
\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]
1. To find side \( c \):
\[ c = \sqrt{a^2 + b^2 - 2ab \cdot \cos(C)} \]
2. To find Angle \( A \):
\[ \cos(A) = \frac{b^2 + c^2 - a^2}{2bc} \]
\[ A = \cos^{-1} \left(\frac{b^2 + c^2 - a^2}{2bc}\right) \]
3. To find Angle \( B \):
\[ \cos(B) = \frac{a^2 + c^2 - b^2}{2ac} \]
\[ B = \cos^{-1} \left(\frac{a^2 + c^2 - b^2}{2ac}\right) \]
**Tasks:**
1. Calculate the length of side \( c \).
2. Calculate the measure of Angle \( A \).
3. Calculate the measure of Angle \( B \).
For accurate results, use a calculator with trigonometric functions and ensure to switch it to degree mode if it has different settings (degree/radian).
**Note:**
For further assistance, use the "Need Help?" section or "Read It" feature provided.
---
This structured approach helps you
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