Solve the right triangle shown in the figure. Round lengths to one decimal place and express angles to the nearest tenth of a degree. B. b = 180, c= 470 O A. A=67.5°, B= 22.5°, a = 503.3 OB. A=67.5°, B= 22.5°, a = 434.2 O C. A=21.0°, B = 69.0°, a = 434.2 O D. A= 69.0°, B= 21.0°, a = 503.3
Solve the right triangle shown in the figure. Round lengths to one decimal place and express angles to the nearest tenth of a degree. B. b = 180, c= 470 O A. A=67.5°, B= 22.5°, a = 503.3 OB. A=67.5°, B= 22.5°, a = 434.2 O C. A=21.0°, B = 69.0°, a = 434.2 O D. A= 69.0°, B= 21.0°, a = 503.3
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section: Chapter Questions
Problem 75RE
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Question
![### Solving Right Triangles
When solving a right triangle, knowing certain side lengths and angles will allow you to determine the missing measurements using trigonometric ratios and the Pythagorean Theorem.
#### Example Problem
Consider the right triangle shown in the figure. You are tasked with solving for the unknown parts of the triangle, given that:
\[ b = 180 \]
\[ c = 470 \]
where \( b \) is one leg, \( c \) is the hypotenuse, and \( a \) is the other leg of the right triangle. The goal is to find the values of angles \( A \) and \( B \), as well as the length of side \( a \).
#### Diagram
The right triangle is labeled as follows:
- Vertices: \( A \), \( B \), and \( C \) (where \( C \) is the right-angle vertex)
- Side opposite \( A \): \( a \)
- Side opposite \( B \): \( b \)
- Hypotenuse (\( AB \)): \( c \)
```
B
/|
c / |
/ | a
/___|
A C
```
#### Given Data:
- \( b = 180 \)
- \( c = 470 \)
#### Options:
Evaluate the following choices and determine which one correctly solves the triangle:
A. \( A = 67.5^\circ, B = 22.5^\circ, a = 503.3 \) \
B. \( A = 67.5^\circ, B = 22.5^\circ, a = 434.2 \) \
C. \( A = 21.0^\circ, B = 69.0^\circ, a = 434.2 \) \
D. \( A = 69.0^\circ, B = 21.0^\circ, a = 503.3 \)
#### Solution Approach:
1. **Calculate angle \( A \) and \( B \)**:
Use trigonometric relationships in a right triangle (with \( C \) being 90 degrees):
\[
\sin(A) = \frac{a}{c}, \quad \cos(A) = \frac{b}{c}
\]
\[
\angle A + \angle B + \angle C = 180^\circ \quad (\angle C](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad727bdc-baba-4947-a2e3-e934fc451c32%2Fb3803a5f-5250-4982-90d6-e4ee523e1e41%2F0rvey2d_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving Right Triangles
When solving a right triangle, knowing certain side lengths and angles will allow you to determine the missing measurements using trigonometric ratios and the Pythagorean Theorem.
#### Example Problem
Consider the right triangle shown in the figure. You are tasked with solving for the unknown parts of the triangle, given that:
\[ b = 180 \]
\[ c = 470 \]
where \( b \) is one leg, \( c \) is the hypotenuse, and \( a \) is the other leg of the right triangle. The goal is to find the values of angles \( A \) and \( B \), as well as the length of side \( a \).
#### Diagram
The right triangle is labeled as follows:
- Vertices: \( A \), \( B \), and \( C \) (where \( C \) is the right-angle vertex)
- Side opposite \( A \): \( a \)
- Side opposite \( B \): \( b \)
- Hypotenuse (\( AB \)): \( c \)
```
B
/|
c / |
/ | a
/___|
A C
```
#### Given Data:
- \( b = 180 \)
- \( c = 470 \)
#### Options:
Evaluate the following choices and determine which one correctly solves the triangle:
A. \( A = 67.5^\circ, B = 22.5^\circ, a = 503.3 \) \
B. \( A = 67.5^\circ, B = 22.5^\circ, a = 434.2 \) \
C. \( A = 21.0^\circ, B = 69.0^\circ, a = 434.2 \) \
D. \( A = 69.0^\circ, B = 21.0^\circ, a = 503.3 \)
#### Solution Approach:
1. **Calculate angle \( A \) and \( B \)**:
Use trigonometric relationships in a right triangle (with \( C \) being 90 degrees):
\[
\sin(A) = \frac{a}{c}, \quad \cos(A) = \frac{b}{c}
\]
\[
\angle A + \angle B + \angle C = 180^\circ \quad (\angle C
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