Finding The Smallest Angle Angle Q In A Triangle With Sides 4, 5, And 6

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To determine the measure of ∠Q\angle Q, the smallest angle in a triangle with sides of lengths 4, 5, and 6, we will employ the Law of Cosines. This fundamental trigonometric principle allows us to relate the lengths of the sides of any triangle to the cosine of one of its angles. The Law of Cosines is particularly useful when we have information about all three sides of a triangle and wish to find the measure of an angle. It's a cornerstone in solving triangles where the basic trigonometric ratios (SOH CAH TOA) don't directly apply, such as in non-right triangles. Understanding and applying the Law of Cosines is not only crucial for solving this specific problem but also for a wide range of geometric and trigonometric problems. Its versatility makes it an indispensable tool in various fields, including engineering, physics, and computer graphics. Mastering this law opens doors to solving complex geometric problems, enabling us to calculate angles and distances in scenarios where direct measurement is impossible or impractical. Thus, our journey to find ∠Q\angle Q begins with a deep dive into the Law of Cosines, ensuring we grasp its essence and application.

Law of Cosines: A Key to Unlocking Angles

The Law of Cosines is a formula that relates the sides and angles in any triangle. If we denote the sides of a triangle as a, b, and c, and the angles opposite these sides as A, B, and C respectively, the Law of Cosines can be expressed in three different forms:

  • a2=b2+c2−2bcâ‹…cos(A)a^2 = b^2 + c^2 - 2bc \cdot cos(A)
  • b2=a2+c2−2acâ‹…cos(B)b^2 = a^2 + c^2 - 2ac \cdot cos(B)
  • c2=a2+b2−2abâ‹…cos(C)c^2 = a^2 + b^2 - 2ab \cdot cos(C)

These formulas are essentially variations of the same principle, each allowing us to solve for a different side or angle depending on the information we have. In our case, we are given the lengths of all three sides (4, 5, and 6) and we are looking for the smallest angle. The smallest angle in a triangle is always opposite the shortest side. This is a fundamental property of triangles – the longer the side, the larger the opposite angle, and vice versa. Therefore, since the side with length 4 is the shortest, ∠Q\angle Q must be opposite this side. Recognizing this relationship is crucial for efficiently solving the problem and avoiding unnecessary calculations. By understanding this connection, we can directly apply the Law of Cosines to find the angle opposite the shortest side, streamlining our approach and ensuring we arrive at the correct solution.

Identifying the Smallest Angle

In our triangle with sides of lengths 4, 5, and 6, the side with length 4 is the shortest side. As mentioned earlier, the smallest angle in a triangle is always opposite the shortest side. Therefore, ∠Q\angle Q is opposite the side with length 4. This principle stems from the geometric properties of triangles, where the relationship between side lengths and angles is directly proportional. A shorter side implies a smaller opposing angle, and a longer side corresponds to a larger opposing angle. This concept is not only vital for solving this particular problem but also serves as a fundamental rule in triangle geometry. Grasping this relationship allows us to quickly identify the smallest angle in any triangle, provided we know the lengths of its sides. It simplifies the process of solving triangles and helps in visualizing the geometric proportions within them. In various practical applications, from architecture to navigation, this principle aids in accurate calculations and spatial reasoning. Therefore, recognizing that ∠Q\angle Q is opposite the side of length 4 is our crucial first step in applying the Law of Cosines to find its measure.

Applying the Law of Cosines to Find ∠Q\angle Q

To find the measure of ∠Q\angle Q, we will use the Law of Cosines. Let's denote the sides of the triangle as follows:

  • a = 4 (the side opposite ∠Q\angle Q)
  • b = 5
  • c = 6

We will use the form of the Law of Cosines that allows us to solve for an angle: a2=b2+c2−2bc⋅cos(A)a^2 = b^2 + c^2 - 2bc \cdot cos(A), where A is the angle opposite side a. In our case, A is ∠Q\angle Q. Substituting the values, we get:

42=52+62−2(5)(6)⋅cos(Q)4^2 = 5^2 + 6^2 - 2(5)(6) \cdot cos(Q)

Now, let's simplify the equation:

16=25+36−60⋅cos(Q)16 = 25 + 36 - 60 \cdot cos(Q)

16=61−60⋅cos(Q)16 = 61 - 60 \cdot cos(Q)

Next, we isolate the term with cos(Q):

60⋅cos(Q)=61−1660 \cdot cos(Q) = 61 - 16

60â‹…cos(Q)=4560 \cdot cos(Q) = 45

Now, divide both sides by 60 to solve for cos(Q):

cos(Q)=4560cos(Q) = \frac{45}{60}

cos(Q)=34cos(Q) = \frac{3}{4}

This step-by-step application of the Law of Cosines demonstrates how a seemingly complex geometric problem can be broken down into simpler algebraic steps. By carefully substituting the known side lengths and isolating the cosine of the angle we wish to find, we arrive at a manageable equation. The key here is the accurate substitution of values and meticulous algebraic manipulation. This process not only allows us to find the cosine of the angle but also reinforces the understanding of how the Law of Cosines works in practice. The subsequent steps will involve finding the inverse cosine to determine the actual angle measure, further highlighting the interplay between trigonometric functions and their applications in solving real-world geometric problems.

Finding the Angle QQ

Now that we have cos(Q)=34cos(Q) = \frac{3}{4}, we need to find the angle Q whose cosine is 34\frac{3}{4}. To do this, we use the inverse cosine function, also known as arccos or cos−1cos^{-1}:

Q=cos−1(34)Q = cos^{-1}(\frac{3}{4})

Using a calculator, we find:

Q≈41.41°Q ≈ 41.41°

Since we need to round the measure to the nearest whole degree, we round 41.41° to 41°.

Therefore, the measure of ∠Q\angle Q, the smallest angle in the triangle, is approximately 41 degrees. This final step involves the practical application of trigonometric functions and the use of a calculator to obtain a numerical solution. The inverse cosine function is a crucial tool in trigonometry, allowing us to find an angle when we know its cosine. The process of rounding to the nearest whole degree is a common practice in many applications, providing a level of precision that is often sufficient for practical purposes. This entire process, from applying the Law of Cosines to finding the inverse cosine and rounding the result, showcases the interconnectedness of mathematical concepts and their utility in solving real-world problems.

Conclusion

In conclusion, the measure of ∠Q\angle Q, the smallest angle in a triangle whose sides have lengths 4, 5, and 6, rounded to the nearest whole degree, is 41 degrees. We arrived at this solution by applying the Law of Cosines, a powerful tool for solving triangles when we know the lengths of all three sides. The Law of Cosines allowed us to relate the side lengths to the cosine of the angle, and by using the inverse cosine function, we were able to find the angle's measure. This problem highlights the importance of understanding trigonometric principles and their applications in geometry. The ability to solve such problems is crucial in various fields, including engineering, architecture, and navigation. Furthermore, this exercise demonstrates the step-by-step approach to problem-solving in mathematics, from identifying the appropriate formula to performing the necessary calculations and arriving at the final answer. The process reinforces the importance of precision, attention to detail, and a solid understanding of fundamental concepts in mathematics.