Consider a triangle ABC like the one below. Suppose that a=65, b=42, and c=30. (The figure is not drawn to scale.) Solve the triangle. Carry your intermediate computations to at least four decimal places, and round your answers to the nearest tenth. If there is more than one solution, use the button labeled "or". C A = 0, B = -0%, C = a A B ☐or X No solution
Consider a triangle ABC like the one below. Suppose that a=65, b=42, and c=30. (The figure is not drawn to scale.) Solve the triangle. Carry your intermediate computations to at least four decimal places, and round your answers to the nearest tenth. If there is more than one solution, use the button labeled "or". C A = 0, B = -0%, C = a A B ☐or X No solution
Trigonometry (MindTap Course List)
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Chapter3: Additional Topics In Trigonometry
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Problem 40CT: To determine the angle of elevation of a star in the sky, you align the star and the top of the...
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![### Solving a Triangle with Given Side Lengths
Consider a triangle \(ABC\) as shown below. Suppose that \(a = 65\), \(b = 42\), and \(c = 30\) (the figure is not drawn to scale). We need to solve the triangle.
#### Steps to Solve the Triangle
1. **Determine the Angles Using the Law of Cosines**
The Law of Cosines states that for any triangle with sides \(a\), \(b\), and \(c\), and corresponding opposite angles \(A\), \(B\), and \(C\):
\[
\cos(A) = \frac{b^2 + c^2 - a^2}{2bc}
\]
\[
\cos(B) = \frac{a^2 + c^2 - b^2}{2ac}
\]
\[
\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}
\]
2. **Calculate Each Angle**
- Calculate angle \(A\):
\[
\cos(A) = \frac{42^2 + 30^2 - 65^2}{2 \cdot 42 \cdot 30}
\]
- Calculate angle \(B\):
\[
\cos(B) = \frac{65^2 + 30^2 - 42^2}{2 \cdot 65 \cdot 30}
\]
- Calculate angle \(C\):
\[
\cos(C) = \frac{65^2 + 42^2 - 30^2}{2 \cdot 65 \cdot 42}
\]
3. **Convert to Degree**
Use the inverse cosine function to find the angles in degrees. Round the results to the nearest tenth.
#### Diagram Explanation
In the provided triangle diagram:
- Vertex \(A\) is the point where sides \(b\) and \(c\) meet.
- Vertex \(B\) is the point where sides \(a\) and \(c\) meet.
- Vertex \(C\) is the point where sides \(a\) and \(b\) meet.
The sides are labeled as follows:
- Side \(a\) is opposite vertex \(A\).
- Side \(b\)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fec8216c7-9603-414d-880e-1b2c9e8c3a03%2F9a077102-1936-4309-9b70-ff87786fd44a%2Fhulxex_processed.png&w=3840&q=75)
Transcribed Image Text:### Solving a Triangle with Given Side Lengths
Consider a triangle \(ABC\) as shown below. Suppose that \(a = 65\), \(b = 42\), and \(c = 30\) (the figure is not drawn to scale). We need to solve the triangle.
#### Steps to Solve the Triangle
1. **Determine the Angles Using the Law of Cosines**
The Law of Cosines states that for any triangle with sides \(a\), \(b\), and \(c\), and corresponding opposite angles \(A\), \(B\), and \(C\):
\[
\cos(A) = \frac{b^2 + c^2 - a^2}{2bc}
\]
\[
\cos(B) = \frac{a^2 + c^2 - b^2}{2ac}
\]
\[
\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}
\]
2. **Calculate Each Angle**
- Calculate angle \(A\):
\[
\cos(A) = \frac{42^2 + 30^2 - 65^2}{2 \cdot 42 \cdot 30}
\]
- Calculate angle \(B\):
\[
\cos(B) = \frac{65^2 + 30^2 - 42^2}{2 \cdot 65 \cdot 30}
\]
- Calculate angle \(C\):
\[
\cos(C) = \frac{65^2 + 42^2 - 30^2}{2 \cdot 65 \cdot 42}
\]
3. **Convert to Degree**
Use the inverse cosine function to find the angles in degrees. Round the results to the nearest tenth.
#### Diagram Explanation
In the provided triangle diagram:
- Vertex \(A\) is the point where sides \(b\) and \(c\) meet.
- Vertex \(B\) is the point where sides \(a\) and \(c\) meet.
- Vertex \(C\) is the point where sides \(a\) and \(b\) meet.
The sides are labeled as follows:
- Side \(a\) is opposite vertex \(A\).
- Side \(b\)
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