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

by Admin 81 views

Hey guys! Let's dive into a geometry problem that might seem tricky at first, but with a few key concepts and formulas, we'll crack it in no time. We're going to tackle a problem where we need to find the cosine of the smallest angle in a triangle, given the lengths of its sides. This is a classic application of the Law of Cosines, a fundamental tool in trigonometry and geometry. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into the solution, let's make sure we fully grasp the problem. We have a triangle with sides of lengths 4 cm, 5 cm, and 6 cm. Our mission is to find the cosine of the smallest angle in this triangle. Now, a crucial piece of information here is that the smallest angle is always opposite the shortest side. Why? Because in any triangle, the larger the angle, the longer the side opposite it, and vice versa. So, in our case, the smallest angle is opposite the side with a length of 4 cm. Let's call this angle α{ \alpha }. Our goal is to find cos⁡(α){ \cos(\alpha) }.

To solve this, we'll heavily rely on the Law of Cosines. This law is a generalization of the Pythagorean theorem and is incredibly useful for solving triangles when you know either all three sides or two sides and the included angle. The Law of Cosines states that for any triangle with sides of lengths a, b, and c, and angles A, B, and C opposite those sides, the following equations hold:

  • a2=b2+c2−2bccos⁡(A){ a^2 = b^2 + c^2 - 2bc \cos(A) }
  • b2=a2+c2−2accos⁡(B){ b^2 = a^2 + c^2 - 2ac \cos(B) }
  • c2=a2+b2−2abcos⁡(C){ c^2 = a^2 + b^2 - 2ab \cos(C) }

These formulas might look a bit intimidating at first, but trust me, they're not as scary as they seem. The key is to identify which sides and angles you know and which ones you need to find. In our case, we know all three sides and we want to find the cosine of one of the angles. So, we'll use the Law of Cosines and a little bit of algebraic manipulation to get our answer.

Now, let's apply the Law of Cosines to our specific problem. We'll plug in the side lengths and solve for cos⁡(α){ \cos(\alpha) }. Get ready for some math!

Applying the Law of Cosines

Okay, let's put the Law of Cosines into action! We know the sides of our triangle are 4 cm, 5 cm, and 6 cm. We've identified that the smallest angle, α{ \alpha }, is opposite the side of length 4 cm. So, let's label the sides as follows:

  • a = 4 cm
  • b = 5 cm
  • c = 6 cm

Now, we'll use the Law of Cosines formula that relates side 'a' to angle α{ \alpha }:

a2=b2+c2−2bccos⁡(α){ a^2 = b^2 + c^2 - 2bc \cos(\alpha) }

Our goal is to isolate cos⁡(α){ \cos(\alpha) }. So, let's rearrange the formula:

2bccos⁡(α)=b2+c2−a2{ 2bc \cos(\alpha) = b^2 + c^2 - a^2 }

cos⁡(α)=b2+c2−a22bc{ \cos(\alpha) = \frac{b^2 + c^2 - a^2}{2bc} }

Now comes the fun part – plugging in the values! We have:

cos⁡(α)=52+62−422imes5imes6{ \cos(\alpha) = \frac{5^2 + 6^2 - 4^2}{2 imes 5 imes 6} }

Let's simplify this step by step:

cos⁡(α)=25+36−1660{ \cos(\alpha) = \frac{25 + 36 - 16}{60} }

cos⁡(α)=4560{ \cos(\alpha) = \frac{45}{60} }

We can simplify this fraction further by dividing both the numerator and the denominator by their greatest common divisor, which is 15:

cos⁡(α)=45ext÷1560ext÷15{ \cos(\alpha) = \frac{45 ext{ ÷ } 15}{60 ext{ ÷ } 15} }

cos⁡(α)=34{ \cos(\alpha) = \frac{3}{4} }

Woohoo! We've found the cosine of the smallest angle. It's 3/4. That wasn't so bad, right? The Law of Cosines is a powerful tool, and with a little practice, you'll be able to tackle similar problems with ease.

Analyzing the Solution and Key Concepts

So, we've successfully found that the cosine of the smallest angle in the triangle is 3/4. That matches option 1) in the original problem. But let's take a moment to reflect on what we've done and reinforce some key concepts.

The Law of Cosines was our main weapon in this battle. It allowed us to relate the sides of a triangle to the cosine of one of its angles. Remember, this law is a generalization of the Pythagorean theorem. If the angle in question is a right angle (90 degrees), then its cosine is 0, and the Law of Cosines simplifies to the Pythagorean theorem: a2=b2+c2{ a^2 = b^2 + c^2 }. But the Law of Cosines works for all triangles, not just right triangles.

Another crucial concept we used was the relationship between the size of an angle and the length of the side opposite it. The smallest angle is always opposite the shortest side, and the largest angle is always opposite the longest side. This simple rule helped us identify which angle we needed to focus on.

Understanding the Law of Cosines is vital for solving many geometry problems, especially those involving non-right triangles. It allows you to find missing side lengths or angles when you have enough information. Think of it as a versatile tool in your geometry toolbox. You can use it to calculate distances, determine angles in various shapes, and even in real-world applications like navigation and surveying.

In this problem, we were given all three sides and asked to find the cosine of an angle. But the Law of Cosines can also be used in other scenarios. For example, if you know two sides and the included angle (the angle between those two sides), you can use the Law of Cosines to find the length of the third side. Or, if you know all three angles, you can't directly determine the side lengths (you'd need at least one side length to scale the triangle), but you can use the Law of Sines in conjunction with the Law of Cosines to solve for unknown sides.

Practice Makes Perfect: Similar Problems and Further Exploration

Now that we've conquered this problem, the best way to solidify your understanding is to practice! Try solving similar problems with different side lengths and see if you can apply the Law of Cosines confidently. Here are a few ideas for variations:

  1. Change the side lengths: Try triangles with sides of 3, 4, and 5 cm (a classic right triangle!), or 7, 8, and 9 cm. Can you still find the cosine of the smallest angle? What about the largest angle?
  2. Find a different angle: Instead of finding the cosine of the smallest angle, try finding the cosine of one of the other angles. You'll just need to rearrange the Law of Cosines formula accordingly.
  3. Work backwards: What if you were given the cosine of an angle and two side lengths? Could you find the length of the third side using the Law of Cosines?
  4. Combine with other concepts: Try problems that combine the Law of Cosines with other geometric concepts, such as the area of a triangle (using Heron's formula) or the properties of specific types of triangles (like isosceles or equilateral triangles).

To really level up your geometry skills, I encourage you to explore these variations and practice as much as you can. The more you work with the Law of Cosines, the more comfortable and confident you'll become using it. Remember, geometry is all about seeing patterns and relationships, and the Law of Cosines is a powerful tool for uncovering those connections.

Conclusion: You've Got This!

Alright, guys, we've reached the end of our journey through this geometry problem. We started with a seemingly complex question, but by breaking it down, understanding the Law of Cosines, and applying a bit of algebraic manipulation, we found the answer. The cosine of the smallest angle in the triangle is indeed 3/4!

Remember, math isn't about memorizing formulas; it's about understanding the underlying concepts and applying them creatively. The Law of Cosines is just one tool in your mathematical arsenal. As you continue to explore geometry and trigonometry, you'll discover many more powerful techniques for solving problems and understanding the world around you.

So, keep practicing, keep exploring, and most importantly, keep having fun with math! You've got this! And who knows, maybe the next geometry problem you tackle will be even more exciting.