Rounded to the nearest hundredth of a meter, find the length of segment AB (the distance across the swamp) in the following figure: 52 C 31 m
Rounded to the nearest hundredth of a meter, find the length of segment AB (the distance across the swamp) in the following figure: 52 C 31 m
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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![**Finding the Length of Segment AB**
In this educational module, you will learn how to find the length of segment AB, which represents the distance across a swamp, using trigonometric principles.
First, examine the given figure:
- The figure is a triangle labeled \( \triangle ABC \).
- The base \( BC \) of the triangle measures 31 meters.
- The interior angle at vertex \( C \) is given as 52 degrees.
- The segment \( AB \) is the distance we need to determine.
To find the length of segment \( AB \), we will apply the Law of Cosines. The Law of Cosines states:
\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]
Where:
- \( c \) is the length of the side opposite the angle \( C \),
- \( a \) and \( b \) are the lengths of the other two sides,
- \( \cos(C) \) is the cosine of the given angle.
In our case:
- Side \( c \) is \( AB \) (the segment we need to find),
- Side \( a \) is \( AC \), which we don’t have but will be inferred based on our necessities,
- Side \( b \) is \( BC = 31 \) meters,
- Angle \( C = 52 \) degrees.
Since we don't know the length of side \( AC \), a more straightforward approach would be to use basic trigonometry given the additional information or context. However, with the limited information, using the law of cosine and edge trigonometrical application/breakdown specifically helps in this manner.
Thus, the formula simplifies to finding needed triangle metrics to utilize in cosine application/rendered table or calculator:
To confirm with angles you can also break down into sine application from relation derivations if perpendicular height also can be inferred.
Finally, rounding the cosine resultant calculation to the nearest hundredth will give the segment needed across swamp estimation.
**Overall visual summary: The triangle presented and contextual constraints inferences helps us define crucial distance across geographical constraints easily preparing students using trigonometric breakdown dependencies.**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd24861ef-92b0-4cd8-8c54-2ac2200498ec%2F12487786-5c14-4765-965f-505c572cf35a%2Few576vc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding the Length of Segment AB**
In this educational module, you will learn how to find the length of segment AB, which represents the distance across a swamp, using trigonometric principles.
First, examine the given figure:
- The figure is a triangle labeled \( \triangle ABC \).
- The base \( BC \) of the triangle measures 31 meters.
- The interior angle at vertex \( C \) is given as 52 degrees.
- The segment \( AB \) is the distance we need to determine.
To find the length of segment \( AB \), we will apply the Law of Cosines. The Law of Cosines states:
\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]
Where:
- \( c \) is the length of the side opposite the angle \( C \),
- \( a \) and \( b \) are the lengths of the other two sides,
- \( \cos(C) \) is the cosine of the given angle.
In our case:
- Side \( c \) is \( AB \) (the segment we need to find),
- Side \( a \) is \( AC \), which we don’t have but will be inferred based on our necessities,
- Side \( b \) is \( BC = 31 \) meters,
- Angle \( C = 52 \) degrees.
Since we don't know the length of side \( AC \), a more straightforward approach would be to use basic trigonometry given the additional information or context. However, with the limited information, using the law of cosine and edge trigonometrical application/breakdown specifically helps in this manner.
Thus, the formula simplifies to finding needed triangle metrics to utilize in cosine application/rendered table or calculator:
To confirm with angles you can also break down into sine application from relation derivations if perpendicular height also can be inferred.
Finally, rounding the cosine resultant calculation to the nearest hundredth will give the segment needed across swamp estimation.
**Overall visual summary: The triangle presented and contextual constraints inferences helps us define crucial distance across geographical constraints easily preparing students using trigonometric breakdown dependencies.**
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