Use the Law of Cosines to solve the triangle. Round your answers to two decimal places. a = B = BO C= Need Help? O O A b=12 Read It 30° Watch It C c=24 a B
Use the Law of Cosines to solve the triangle. Round your answers to two decimal places. a = B = BO C= Need Help? O O A b=12 Read It 30° Watch It C c=24 a B
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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![### Solving Triangles Using the Law of Cosines
#### Problem Statement:
Use the Law of Cosines to solve the triangle. Round your answers to two decimal places.
#### Given:
We have a triangle \(ABC\) with the following known measurements:
- Side \(b = 12\) units
- Side \(c = 24\) units
- Angle \(C = 30^\circ\)
#### Diagram:
The diagram provided shows a triangle \(ABC\) with vertex \(A\) at the bottom left, vertex \(B\) at the bottom right, and vertex \(C\) at the top.
- \(AB\) is the side denoted as \(c = 24\) units.
- \(AC\) is the side denoted as \(b = 12\) units.
- \(BC\) is the unknown side \(a\).
- Angle \(C\) opposite to side \(c\) is \(30^\circ\).
#### Solution:
Using the Law of Cosines, which states:
\[ a^2 = b^2 + c^2 - 2bc \cdot \cos(C) \]
We can solve for side \(a\):
1. Substitute the known values into the equation:
\[ a^2 = 12^2 + 24^2 - 2 \cdot 12 \cdot 24 \cdot \cos(30^\circ) \]
2. Calculate:
\[ a^2 = 144 + 576 - 2 \cdot 12 \cdot 24 \cdot \left(\frac{\sqrt{3}}{2}\right) \]
\[ a^2 = 144 + 576 - 288 \sqrt{3} \]
\[ a^2 \approx 720 - 288 \cdot 1.732 \]
\[ a^2 \approx 720 - 498.816 \]
\[ a^2 \approx 221.184 \]
\[ a \approx \sqrt{221.184} \]
\[ a \approx 14.87 \]
Thus,
\[ a \approx 14.87 \]
Next, we need to find the remaining angles \(A\) and \(B\). We can use the Law of Sines for this:
\[ \sin(A) / a = \sin(C) / c \]
3. Solving for \(A\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F61a624e8-9805-4882-bd7c-8b439f9ba0a9%2F971fe427-67d4-4a00-bb6b-96c05918de67%2F4yht5zo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving Triangles Using the Law of Cosines
#### Problem Statement:
Use the Law of Cosines to solve the triangle. Round your answers to two decimal places.
#### Given:
We have a triangle \(ABC\) with the following known measurements:
- Side \(b = 12\) units
- Side \(c = 24\) units
- Angle \(C = 30^\circ\)
#### Diagram:
The diagram provided shows a triangle \(ABC\) with vertex \(A\) at the bottom left, vertex \(B\) at the bottom right, and vertex \(C\) at the top.
- \(AB\) is the side denoted as \(c = 24\) units.
- \(AC\) is the side denoted as \(b = 12\) units.
- \(BC\) is the unknown side \(a\).
- Angle \(C\) opposite to side \(c\) is \(30^\circ\).
#### Solution:
Using the Law of Cosines, which states:
\[ a^2 = b^2 + c^2 - 2bc \cdot \cos(C) \]
We can solve for side \(a\):
1. Substitute the known values into the equation:
\[ a^2 = 12^2 + 24^2 - 2 \cdot 12 \cdot 24 \cdot \cos(30^\circ) \]
2. Calculate:
\[ a^2 = 144 + 576 - 2 \cdot 12 \cdot 24 \cdot \left(\frac{\sqrt{3}}{2}\right) \]
\[ a^2 = 144 + 576 - 288 \sqrt{3} \]
\[ a^2 \approx 720 - 288 \cdot 1.732 \]
\[ a^2 \approx 720 - 498.816 \]
\[ a^2 \approx 221.184 \]
\[ a \approx \sqrt{221.184} \]
\[ a \approx 14.87 \]
Thus,
\[ a \approx 14.87 \]
Next, we need to find the remaining angles \(A\) and \(B\). We can use the Law of Sines for this:
\[ \sin(A) / a = \sin(C) / c \]
3. Solving for \(A\
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