Determine whether the Law of Sines or the Law of Cosines is needed to solve the triangle. a = 13, b = 15, c = 10 Law of Sines O Law of Cosines Solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places. (If not possible, enter IMPOSSIBLE in each corresponding answer blank.) A =
Determine whether the Law of Sines or the Law of Cosines is needed to solve the triangle. a = 13, b = 15, c = 10 Law of Sines O Law of Cosines Solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places. (If not possible, enter IMPOSSIBLE in each corresponding answer blank.) A =
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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![### Triangle Problem: Solving Using Law of Sines or Cosines
**Problem:**
Determine whether the Law of Sines or the Law of Cosines is needed to solve the triangle with the given side lengths:
- \( a = 13 \)
- \( b = 15 \)
- \( c = 10 \)
**Choices:**
- Law of Sines
- Law of Cosines
**Task:**
Solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places. (If a triangle is not possible, enter IMPOSSIBLE in each corresponding answer blank.)
**Parameters to Find:**
- Angle \( A = \) [input field]
- Angle \( B = \) [input field]
- Angle \( C = \) [input field]
### Solution Process
1. **Determine the Appropriate Law:**
Given the three sides of the triangle (SSS), it is generally appropriate to use the **Law of Cosines** to find one of the angles. Once one angle is found, the Law of Sines can be used to find another angle, but it is often simpler to continue with the Law of Cosines for accuracy.
2. **Law of Cosines Formula:**
For a triangle with sides \( a \), \( b \), and \( c \), and corresponding angles \( A \), \( B \), and \( C \), the Law of Cosines is given by:
\[
c^2 = a^2 + b^2 - 2ab \cos(C)
\]
Rearranging to solve for \( \cos(C) \):
\[
\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}
\]
Substituting the given values:
\[
\cos(C) = \frac{13^2 + 15^2 - 10^2}{2 \cdot 13 \cdot 15}
\]
Simplify to find angle \( C \).
3. **Finding Angles \( A \) and \( B \):**
Once \( C \) is found, use either the Law of Cosines again or, for simplicity, the Law of Sines:
\[
\frac{\sin(A)}{a} = \frac{\sin(C)}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7fab864d-7419-4b8c-bc50-25971d273c3c%2Fc3850864-b2c1-4edb-a0d0-edd02d309e48%2Fayw5nni_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Triangle Problem: Solving Using Law of Sines or Cosines
**Problem:**
Determine whether the Law of Sines or the Law of Cosines is needed to solve the triangle with the given side lengths:
- \( a = 13 \)
- \( b = 15 \)
- \( c = 10 \)
**Choices:**
- Law of Sines
- Law of Cosines
**Task:**
Solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places. (If a triangle is not possible, enter IMPOSSIBLE in each corresponding answer blank.)
**Parameters to Find:**
- Angle \( A = \) [input field]
- Angle \( B = \) [input field]
- Angle \( C = \) [input field]
### Solution Process
1. **Determine the Appropriate Law:**
Given the three sides of the triangle (SSS), it is generally appropriate to use the **Law of Cosines** to find one of the angles. Once one angle is found, the Law of Sines can be used to find another angle, but it is often simpler to continue with the Law of Cosines for accuracy.
2. **Law of Cosines Formula:**
For a triangle with sides \( a \), \( b \), and \( c \), and corresponding angles \( A \), \( B \), and \( C \), the Law of Cosines is given by:
\[
c^2 = a^2 + b^2 - 2ab \cos(C)
\]
Rearranging to solve for \( \cos(C) \):
\[
\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}
\]
Substituting the given values:
\[
\cos(C) = \frac{13^2 + 15^2 - 10^2}{2 \cdot 13 \cdot 15}
\]
Simplify to find angle \( C \).
3. **Finding Angles \( A \) and \( B \):**
Once \( C \) is found, use either the Law of Cosines again or, for simplicity, the Law of Sines:
\[
\frac{\sin(A)}{a} = \frac{\sin(C)}
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