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The Law of Cosines is used to find the measure of $\angle Q$.



$\begin{aligned}

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{aligned}$



To the nearest whole degree, what is the measure of $\angle Q$?


A$44^{\circ}$
B$49^{\circ}$
C$54^{\circ}$
D$59^{\circ}$

Verified Answer :

  • Begin with the given equation based on the Law of Cosines: \(24^2 = 20^2 + 34^2 - 2(20)(34) \cos(Q)\).
  • Simplify the equation to find \(\cos(Q) = \frac{980}{1360} = \frac{49}{68}\).
  • Calculate the inverse cosine to find \(Q \approx 43.897^{\circ}\).
  • Round the angle to the nearest whole degree to identify the final answer as \(\boxed{44^{\circ}}\).

Explanation

  1. Setup and Simplify the Law of Cosines Equation

The Law of Cosines is given by the formula \(a^2 = b^2 + c^2 - 2bc \cos(Q)\). Given the equation \(24^2 = 20^2 + 34^2 - 2(20)(34) \cos(Q)\), we first evaluate the squares: \(24^2 = 576\), \(20^2 = 400\), and \(34^2 = 1156\). Substituting these values gives us \(576 = 400 + 1156 - 1360 \cos(Q)\), which simplifies to \(576 = 1556 - 1360 \cos(Q)\). Subtracting 1556 from both sides yields \(-980 = -1360 \cos(Q)\).

  1. Calculate Angle Q

To find the measure of angle \(Q\), we rearrange the equation \(-980 = -1360 \cos(Q)\) to isolate the cosine term:
\[\cos(Q) = \frac{-980}{-1360} = \frac{980}{1360} = \frac{49}{68}\]
Dividing 980 by 1360 gives approximately \(\cos(Q) \approx 0.720588\). To find \(Q\), we calculate the inverse cosine (arccos) of this ratio:
\[Q = \arccos\left(\frac{49}{68}\right) \approx 43.897^{\circ}\]
Rounding this value to the nearest whole degree gives \(44^{\circ}\). Comparing this result with the provided options, it corresponds to option (a).

Examples

The Law of Cosines is essential in fields like surveying and navigation. Imagine you are surveying a triangular plot of land where you know the lengths of all three sides. Using the Law of Cosines, you can easily determine the internal angles of the plot, which is crucial for property boundaries and landscaping plans. For instance, if sides are \(20\text{m}\), \(34\text{m}\), and the opposite side is \(24\text{m}\), the calculated angle of approximately \(44^{\circ}\) helps in drawing the precise blueprint for the land development.

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