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

Understanding how to determine angles within a triangle based on its side lengths is a fundamental concept in trigonometry. This article delves into the process of finding the measure of the smallest angle in a triangle given its side lengths. Specifically, we will focus on a triangle with sides of length 4, 5, and 6. We'll walk through the steps of applying the Law of Cosines, which is a critical tool for solving such problems. The goal is to not only calculate the angle but also to understand the underlying principles that make this calculation possible. By the end of this exploration, you'll be equipped with the knowledge to tackle similar geometric problems with confidence. This involves a blend of understanding trigonometric principles and applying them practically to geometric shapes. Let's embark on this mathematical journey to uncover the measure of the smallest angle, rounded to the nearest whole degree, in our specified triangle.

Applying the Law of Cosines

When tackling a triangle where we know the lengths of all three sides but none of the angles, the Law of Cosines becomes our primary tool. This law provides a direct relationship between the sides of a triangle and the cosine of one of its angles. The Law of Cosines is a generalization of the Pythagorean theorem, which only applies to right triangles. The Law of Cosines, however, can be used for any type of triangle, whether it's acute, obtuse, or right. This makes it an incredibly versatile tool in trigonometry and geometry. The formula essentially states that for any triangle with sides a, b, and c, and an angle γ opposite side c, the following equation holds true: c² = a² + b² - 2abcos(γ). Understanding this formula is the key to unlocking many geometric problems. It allows us to relate the lengths of the sides to the angles, providing a bridge between these two fundamental aspects of a triangle. Mastering the Law of Cosines opens doors to solving a wide range of problems, from navigation and surveying to engineering and physics. In our specific case, to find the smallest angle, we need to first identify which angle is indeed the smallest. Remember that in any triangle, the smallest angle is always opposite the shortest side. Therefore, in our triangle with sides 4, 5, and 6, the smallest angle will be opposite the side with length 4. This is a crucial observation that will guide our application of the Law of Cosines. Once we've identified the smallest angle, we can set up the equation with the appropriate values and solve for the cosine of that angle. This will then allow us to find the angle itself using the inverse cosine function.

Identifying the Smallest Angle

In any triangle, a fundamental property dictates that the smallest angle is invariably opposite the shortest side. This is a crucial principle in geometry that helps us relate the sides and angles of a triangle. When we examine our triangle with sides of lengths 4, 5, and 6, this principle immediately points us to the side of length 4. Since this is the shortest side, the angle opposite it will be the smallest angle in the triangle. Let's denote this smallest angle as ∠Q. Now that we have identified ∠Q as the angle we need to find, we can proceed with applying the Law of Cosines. This initial step of identifying the smallest angle is vital because it ensures that we are targeting the correct angle for our calculation. If we were to mistakenly apply the Law of Cosines to an angle opposite a longer side, we would end up calculating a different, larger angle. Therefore, this careful identification step is a cornerstone of solving the problem correctly. Understanding this relationship between side lengths and opposite angles is not only crucial for this specific problem but also for a broader understanding of triangle geometry. It's a concept that frequently appears in various geometric proofs and problem-solving scenarios. By internalizing this principle, we can approach triangle-related problems with a more intuitive and efficient mindset. This initial step sets the stage for the rest of our calculation, ensuring we are on the right track to find the measure of the smallest angle, ∠Q.

Calculating the Angle Using the Law of Cosines

With the smallest angle, ∠Q, identified as the angle opposite the side of length 4, we can now apply the Law of Cosines to find its measure. Recall the Law of Cosines formula: c² = a² + b² - 2abcos(γ), where c is the side opposite angle γ, and a and b are the other two sides. In our case, let's assign c = 4 (the side opposite ∠Q), a = 5, and b = 6. Substituting these values into the Law of Cosines formula gives us: 4² = 5² + 6² - 2 * 5 * 6 * cos(∠Q). This equation now allows us to solve for cos(∠Q). First, let's simplify the equation: 16 = 25 + 36 - 60 * cos(∠Q). Combining the constants on the right side, we get: 16 = 61 - 60 * cos(∠Q). Next, we isolate the term with cos(∠Q): 60 * cos(∠Q) = 61 - 16. This simplifies to: 60 * cos(∠Q) = 45. Now, we can solve for cos(∠Q) by dividing both sides by 60: cos(∠Q) = 45 / 60. Reducing the fraction, we have: cos(∠Q) = 3 / 4, or 0.75. This value represents the cosine of the angle ∠Q. To find the actual measure of ∠Q in degrees, we need to take the inverse cosine (also known as arccosine) of 0.75. This is typically done using a calculator. The inverse cosine function is denoted as cos^-1 or arccos. So, ∠Q = cos^-1(0.75). Using a calculator, we find that ∠Q ≈ 41.41 degrees. However, the problem asks us to round the measure to the nearest whole degree. Therefore, rounding 41.41 degrees to the nearest whole degree gives us 41 degrees. Thus, the measure of the smallest angle, ∠Q, in the triangle with sides 4, 5, and 6 is approximately 41 degrees. This completes our calculation and provides the answer to the problem.

Step-by-step Calculation

To further clarify the process of calculating the angle using the Law of Cosines, let's break down the steps in a more detailed manner. This will provide a clear and concise guide for anyone looking to replicate the calculation or apply it to similar problems. We start with the Law of Cosines formula: c² = a² + b² - 2abcos(γ). As we've established, c represents the side opposite the angle we want to find (in our case, ∠Q), and a and b are the other two sides. Step 1: Substitute the known values into the formula. We have c = 4, a = 5, and b = 6. Plugging these into the formula, we get: 4² = 5² + 6² - 2 * 5 * 6 * cos(∠Q). Step 2: Simplify the equation. Calculate the squares: 16 = 25 + 36 - 2 * 5 * 6 * cos(∠Q). Then, multiply the constants: 16 = 25 + 36 - 60 * cos(∠Q). Step 3: Combine like terms. Add 25 and 36: 16 = 61 - 60 * cos(∠Q). Step 4: Isolate the term with cos(∠Q). Subtract 61 from both sides: 16 - 61 = -60 * cos(∠Q). This simplifies to: -45 = -60 * cos(∠Q). Divide both sides by -60: -45 / -60 = cos(∠Q). This gives us: cos(∠Q) = 0.75. Step 5: Find the angle using the inverse cosine function. Use a calculator to find the inverse cosine of 0.75: ∠Q = cos^-1(0.75). This results in: ∠Q ≈ 41.41 degrees. Step 6: Round to the nearest whole degree. As requested in the problem, round 41.41 degrees to the nearest whole number: ∠Q ≈ 41 degrees. This step-by-step approach clearly outlines the process of using the Law of Cosines to find an angle in a triangle when all three sides are known. By following these steps, you can confidently solve similar problems and gain a deeper understanding of trigonometric principles.

Final Answer

After carefully applying the Law of Cosines and performing the necessary calculations, we arrive at the solution for the measure of the smallest angle in the triangle with sides 4, 5, and 6. We identified that the smallest angle, ∠Q, is opposite the side with length 4. By substituting the side lengths into the Law of Cosines formula and solving for cos(∠Q), we found that cos(∠Q) = 0.75. Then, using the inverse cosine function, we determined that ∠Q ≈ 41.41 degrees. Finally, as per the problem's requirement, we rounded this measure to the nearest whole degree. Therefore, the measure of the smallest angle, ∠Q, in the triangle is approximately 41 degrees. This concludes our exploration of finding the smallest angle in a triangle given its side lengths. The process involved understanding the relationship between side lengths and angles, applying the Law of Cosines, and using trigonometric functions to calculate the angle. This problem serves as a valuable example of how trigonometric principles can be used to solve geometric problems. The final answer, 41 degrees, represents a precise and accurate solution to the question posed. This exercise not only provides the answer but also reinforces the understanding of key concepts in trigonometry and geometry. It highlights the importance of careful calculation, attention to detail, and the correct application of mathematical principles.