how did it come to 9 for B1? it's for solving The Ambiguous Case SSA for triangle.
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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Question
how did it come to 9 for B1? it's for solving The Ambiguous Case SSA for
![**Triangle Angle Calculation Using the Law of Sines**
**Date and Time:** 11/18/23, 11:52 PM
**Section:** 6-01-Vida Lumod
The problem discusses calculating unknown angles in a triangle using trigonometric principles.
**Details:**
- **Initial Consideration:** The second possible angle, \( B_2 \approx 171^\circ \), is disregarded because \( A = 40^\circ \) leads to \( A + B_2 > 180^\circ \).
- **Angle Calculations:** Using \( B_1 \approx 9^\circ \), find angle \( C \) by applying the concept that the sum of the angles in a triangle equals \( 180^\circ \).
\[
A + B + C = 180^\circ
\]
Therefore:
\[
C = 180^\circ - A - B_1 \approx 180^\circ - 40^\circ - 9^\circ \approx 131^\circ
\]
- **Applying the Law of Sines:** To determine side \( c \):
\[
\frac{a}{\sin A} = \frac{c}{\sin C}
\]
By substituting the known values:
\[
\frac{4}{\sin 40^\circ} = \frac{c}{\sin 131^\circ}
\]
Solving for \( c \):
\[
c \approx 4.7
\]
Results are rounded to the nearest tenth.
- **Conclusion:** There is one triangle with solution \( B \approx 9^\circ \), \( C \approx 131^\circ \), and \( c \approx 4.7 \).
- **Answer Verification:** Previously calculated answers were recorded as: 0.1607 and 41.
This solution uses trigonometric laws to validate the existence and measurements of the triangle formed by given angles.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F22ced223-6d4a-4602-8c5b-53f3c81a3102%2Ff6635e37-1cfe-4f00-93ba-63a0b323eaaa%2Fpsgejsia_processed.png&w=3840&q=75)
Transcribed Image Text:**Triangle Angle Calculation Using the Law of Sines**
**Date and Time:** 11/18/23, 11:52 PM
**Section:** 6-01-Vida Lumod
The problem discusses calculating unknown angles in a triangle using trigonometric principles.
**Details:**
- **Initial Consideration:** The second possible angle, \( B_2 \approx 171^\circ \), is disregarded because \( A = 40^\circ \) leads to \( A + B_2 > 180^\circ \).
- **Angle Calculations:** Using \( B_1 \approx 9^\circ \), find angle \( C \) by applying the concept that the sum of the angles in a triangle equals \( 180^\circ \).
\[
A + B + C = 180^\circ
\]
Therefore:
\[
C = 180^\circ - A - B_1 \approx 180^\circ - 40^\circ - 9^\circ \approx 131^\circ
\]
- **Applying the Law of Sines:** To determine side \( c \):
\[
\frac{a}{\sin A} = \frac{c}{\sin C}
\]
By substituting the known values:
\[
\frac{4}{\sin 40^\circ} = \frac{c}{\sin 131^\circ}
\]
Solving for \( c \):
\[
c \approx 4.7
\]
Results are rounded to the nearest tenth.
- **Conclusion:** There is one triangle with solution \( B \approx 9^\circ \), \( C \approx 131^\circ \), and \( c \approx 4.7 \).
- **Answer Verification:** Previously calculated answers were recorded as: 0.1607 and 41.
This solution uses trigonometric laws to validate the existence and measurements of the triangle formed by given angles.
![### Solving an SSA Triangle Using the Law of Sines
The Law of Sines is a fundamental rule in trigonometry that relates the angles and sides of a triangle. Given the measures of angles \( A \), \( B \), and \( C \), and the lengths of the sides \( a \), \( b \), and \( c \) opposite these angles, the Law of Sines states:
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]
#### Determining the Angle \( B \)
**Step 1:** Use the Law of Sines to find angle \( B \).
\[
\frac{a}{\sin A} = \frac{b}{\sin B}
\]
Apply the values given:
\[
\frac{4}{\sin 40^\circ} = \frac{1}{\sin B}
\]
**Solve for \( \sin B \):**
- Cross-multiply to solve for \( \sin B \):
\[
\sin B \approx 0.1607
\]
- Round to four decimal places.
**Step 2:** Identify possible angles for \( B \).
There are two potential angles \( B \) between \( 0^\circ \) and \( 180^\circ \) for which \( \sin B \approx 0.1607 \):
- \( B_1 \approx 9^\circ \) (smaller angle rounded to the nearest degree).
- \( B_2 \approx 171^\circ \) (larger angle rounded).
#### Calculating Angle \( C \)
The possibility of \( B_2 \approx 171^\circ \) is not feasible because:
- Given \( A = 40^\circ \), the sum \( A + B_2 \) would exceed \( 180^\circ \).
Apply angle sum in a triangle:
\[
A + B + C = 180^\circ
\]
- Thus, angle \( C \):
\[
C = 180^\circ - A - B_1 \approx 180^\circ - 40^\circ - 9^\circ = 131^\circ
\]
#### Finding Side \( c \)
Using the Law of Sines again to find \( c \):
\[
\frac{a}{\sin A](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F22ced223-6d4a-4602-8c5b-53f3c81a3102%2Ff6635e37-1cfe-4f00-93ba-63a0b323eaaa%2Fcwxo3r_processed.png&w=3840&q=75)
Transcribed Image Text:### Solving an SSA Triangle Using the Law of Sines
The Law of Sines is a fundamental rule in trigonometry that relates the angles and sides of a triangle. Given the measures of angles \( A \), \( B \), and \( C \), and the lengths of the sides \( a \), \( b \), and \( c \) opposite these angles, the Law of Sines states:
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]
#### Determining the Angle \( B \)
**Step 1:** Use the Law of Sines to find angle \( B \).
\[
\frac{a}{\sin A} = \frac{b}{\sin B}
\]
Apply the values given:
\[
\frac{4}{\sin 40^\circ} = \frac{1}{\sin B}
\]
**Solve for \( \sin B \):**
- Cross-multiply to solve for \( \sin B \):
\[
\sin B \approx 0.1607
\]
- Round to four decimal places.
**Step 2:** Identify possible angles for \( B \).
There are two potential angles \( B \) between \( 0^\circ \) and \( 180^\circ \) for which \( \sin B \approx 0.1607 \):
- \( B_1 \approx 9^\circ \) (smaller angle rounded to the nearest degree).
- \( B_2 \approx 171^\circ \) (larger angle rounded).
#### Calculating Angle \( C \)
The possibility of \( B_2 \approx 171^\circ \) is not feasible because:
- Given \( A = 40^\circ \), the sum \( A + B_2 \) would exceed \( 180^\circ \).
Apply angle sum in a triangle:
\[
A + B + C = 180^\circ
\]
- Thus, angle \( C \):
\[
C = 180^\circ - A - B_1 \approx 180^\circ - 40^\circ - 9^\circ = 131^\circ
\]
#### Finding Side \( c \)
Using the Law of Sines again to find \( c \):
\[
\frac{a}{\sin A
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