The law of cosines is used to find the measure of [tex]$\angle Q$[/tex].
[tex]$\begin{array}{l}
24^2=20^2+34^2-2(20)(34) \cos (Q) \\
576=400+1156-(1360) \cos (Q) \\
576=1556-(1360) \cos (Q) \\
-980=-(1360) \cos (Q)
\end{array}$[/tex]
To the nearest whole degree, what is the measure of [tex]$\angle Q$[/tex]?
A. [tex]$44^{\circ}$[/tex]
B. [tex]$49^{\circ}$[/tex]
C. [tex]$54^{\circ}$[/tex]
D. [tex]$59^{\circ}$[/tex]
Answer (2)
The measure of ∠ Q using the provided equation simplifies to approximately 4 4 ∘ . Therefore, the correct answer is A. 4 4 ∘ .
;
Simplify the given equation from the Law of Cosines.
Solve for cos ( Q ) , obtaining cos ( Q ) = 1360 980 .
Find the angle Q by taking the inverse cosine (arccos) of the result.
Round the value of Q to the nearest whole degree, resulting in 4 4 ∘ .
Explanation
Problem Analysis We are given the equation derived from the law of cosines: 2 4 2 = 2 0 2 + 3 4 2 − 2 ( 20 ) ( 34 ) cos ( Q ) We need to find the measure of angle Q to the nearest whole degree.
Simplifying the Equation First, let's simplify the equation: 576 = 400 + 1156 − 1360 cos ( Q ) 576 = 1556 − 1360 cos ( Q ) − 980 = − 1360 cos ( Q )
Solving for cos(Q) Now, solve for cos ( Q ) :
cos ( Q ) = − 1360 − 980 = 1360 980 cos ( Q ) = 68 49
Finding Angle Q To find the angle Q , we take the inverse cosine (arccos) of 68 49 :
Q = arccos ( 68 49 ) Using a calculator, we find that: Q ≈ 44.00 7 ∘
Final Answer Rounding to the nearest whole degree, we get: Q ≈ 4 4 ∘ Therefore, the measure of angle Q is approximately 4 4 ∘ .
Examples
The Law of Cosines is a fundamental concept in trigonometry that helps us solve problems involving triangles when we know either all three sides or two sides and the included angle. For example, imagine you are a surveyor trying to determine the angle between two property lines. You know the lengths of the two property lines and the distance between their endpoints. By using the Law of Cosines, you can calculate the angle between the property lines, which is crucial for accurate land measurements and boundary determination.
