- Identify the smallest angle as the one opposite the shortest side (length 4).
- Use the Law of Cosines formula: \(\cos(\alpha) = \frac{b^2 + c^2 - a^2}{2bc}\).
- Calculate the cosine: \(\cos(\alpha) = \frac{5^2 + 6^2 - 4^2}{2 \cdot 5 \cdot 6} = 0.75\).
- Find the angle \(\alpha = \arccos(0.75) \approx 41^\circ\).
- The final answer is \(\boxed{41^{\circ}}\).
Explanation
- Identify the smallest angle and its opposite side.
In any triangle, the smallest angle is always opposite the shortest side. Given a triangle with side lengths of 4, 5, and 6, the shortest side is 4. Let the side lengths be denoted as \(a = 4\), \(b = 5\), and \(c = 6\). Let \(\alpha\) be the angle opposite to side \(a\), which will be the smallest angle of the triangle.
- Apply the Law of Cosines to solve for \(\cos(\alpha)\).
The Law of Cosines relates the side lengths of a triangle to the cosine of one of its angles:
\[a^2 = b^2 + c^2 - 2bc \cos(\alpha)\]
Substituting the known values \(a = 4\), \(b = 5\), and \(c = 6\):
\[4^2 = 5^2 + 6^2 - 2(5)(6) \cos(\alpha)\]
\[16 = 25 + 36 - 60 \cos(\alpha)\]
\[16 = 61 - 60 \cos(\alpha)\]
Rearranging the equation to solve for \(\cos(\alpha)\):
\[60 \cos(\alpha) = 61 - 16\]
\[60 \cos(\alpha) = 45\]
\[\cos(\alpha) = \frac{45}{60} = 0.75\]
- Calculate the angle and round to the nearest whole degree.
To find the angle \(\alpha\), we calculate the arccosine of \(0.75\):
\[\alpha = \arccos(0.75)\]
Using calculation, \(\alpha \approx 41.4096^\circ\).
Rounding this value to the nearest whole degree gives \(\alpha \approx 41^\circ\). This corresponds to option (d).
Examples
The Law of Cosines is incredibly useful in real-world scenarios, such as in surveying and navigation. Imagine you are standing at a point and need to find the angle between two landmarks that are a known distance from you, and you also know the distance between the two landmarks themselves. By measuring these distances, you can easily calculate the angle between them using the Law of Cosines, allowing you to plot your position accurately on a map or determine the precise layout of a construction site. This formula acts as a bridge between simple distance measurements and the geometry required for precise navigation or architectural design.