Bob is standing 25 feet from a lamppost that is to his left and 30 feet from a lamppost that is to his right. The distance between the two lampposts is 20 feet. What is the measure of the angle formed from the line from each lamppost to Bob? Approximate to the nearest degree.
[tex]
\begin{array}{c}
20^2=25^2+30^2-2(25)(30) \cos (A) \\
400=625+900-(1500) \cos (A) \\
3400=1525-(1500) \cos (A) \\
4.1125=-(1500) \cos (A) \\
\text { degrees }
\end{array}
[/tex]
Answer (2)
The angle formed by the lines from each lamppost to Bob is approximately 41 degrees. This is calculated using the Law of Cosines with the known distances from Bob to the lampposts and between the lampposts. We find \cos(A) and then use the inverse cosine to determine the angle A.
;
Apply the Law of Cosines: c 2 = a 2 + b 2 − 2 ab cos ( A ) .
Rearrange the formula to solve for cos ( A ) : cos ( A ) = 2 ab a 2 + b 2 − c 2 .
Substitute the given values: cos ( A ) = 2 ( 25 ) ( 30 ) 2 5 2 + 3 0 2 − 2 0 2 = 0.75 .
Find the angle A by taking the inverse cosine: A = a rccos ( 0.75 ) ≈ 41 degrees .
The measure of the angle is 41
Explanation
Problem Analysis Let's analyze the problem. We are given the distances from Bob to two lampposts and the distance between the lampposts. We need to find the angle formed by the lines connecting Bob to each lamppost. This is a perfect setup for using the Law of Cosines.
Applying the Law of Cosines Let a = 25 feet, b = 30 feet, and c = 20 feet. Let A be the angle opposite side c . According to the Law of Cosines: c 2 = a 2 + b 2 − 2 ab cos ( A ) We want to find angle A , so we need to rearrange the formula to solve for cos ( A ) :
Rearranging the Formula and Substituting Values Rearranging the Law of Cosines formula: 2 ab cos ( A ) = a 2 + b 2 − c 2 cos ( A ) = 2 ab a 2 + b 2 − c 2 Now, substitute the given values: cos ( A ) = 2 ( 25 ) ( 30 ) 2 5 2 + 3 0 2 − 2 0 2
Calculating cos(A) Calculate the value of cos ( A ) :
cos ( A ) = 1500 625 + 900 − 400 cos ( A ) = 1500 1125 cos ( A ) = 0.75
Finding Angle A Now, find the angle A by taking the inverse cosine (arccos) of 0.75: A = a rccos ( 0.75 ) Using a calculator, we find that: A ≈ 41.4096 degrees Rounding to the nearest degree, we get: A ≈ 41 degrees
Final Answer Therefore, the measure of the angle formed by the lines connecting Bob to each lamppost is approximately 41 degrees.
Examples
The Law of Cosines is not just for abstract triangles! Imagine you're designing a triangular garden. You know the lengths of the three sides, but you need to determine the angle at one of the corners to properly place a sprinkler. By using the Law of Cosines, you can calculate the angle and ensure your sprinkler covers the entire garden efficiently. This principle applies in various fields like architecture, surveying, and navigation, where determining angles and distances is crucial.
