Triangle Angles: Law Of Cosines Example
Hey guys! Let's dive into a cool math problem where we need to figure out the angles of a triangle. Specifically, we have triangle ABC, and we know the lengths of its sides: a = 12, b = 7, and c = 6. Our mission, should we choose to accept it (and we do!), is to find the angles A, B, and C. This is a classic trigonometry problem, and we're going to use the Law of Cosines to crack it. Trust me; it's not as intimidating as it sounds! We will break it down step by step, making it super easy to follow. First, we will introduce the formulas, then we apply them, and finally, we interpret the results. So, grab your calculators, and let's get started!
Understanding the Law of Cosines
Okay, so what exactly is the Law of Cosines? Well, it's a super handy formula that relates the sides and angles of any triangle. It's especially useful when you know all three sides of a triangle (which we do!) or when you know two sides and the included angle. For our problem, knowing all three sides, it's the perfect tool. The Law of Cosines essentially extends the Pythagorean theorem to non-right triangles. Remember that classic a² + b² = c²? The Law of Cosines builds on that, adding an extra term that accounts for the angle opposite the side we're trying to find. There are three versions of the formula, one for each angle:
- a² = b² + c² - 2bc * cos(A)
- b² = a² + c² - 2ac * cos(B)
- c² = a² + b² - 2ab * cos(C)
See? They all look pretty similar, just with the sides and angles swapped around. Here, 'a', 'b', and 'c' represent the lengths of the sides, and A, B, and C represent the angles opposite those sides. Easy peasy, right? The key thing to remember is that each formula isolates one side and its opposite angle. So, if we want to find angle A, we use the first formula. If we want to find angle B, we use the second, and so on. Now that we have these formulas in our arsenal, we're ready to roll up our sleeves and start plugging in our values. Before we do that, let's make sure we understand why this law works. The Law of Cosines is derived from basic trigonometric principles and the Pythagorean theorem itself. It's essentially a way to correct for the fact that triangles aren't always right-angled. The term - 2bc * cos(A) (and its equivalents in the other formulas) adjusts the Pythagorean theorem based on the cosine of the angle, which tells us how much the triangle deviates from being a right triangle. If angle A was 90 degrees, cos(A) would be 0, and that term would disappear, leaving us with the familiar Pythagorean theorem. Isn't that neat? So, the Law of Cosines is a powerful and versatile tool, and understanding its origins helps us appreciate its elegance and usefulness. Now, let's get back to our specific triangle and start crunching those numbers!
Calculating Angle A
Alright, let's kick things off by calculating angle A. Remember the Law of Cosines formula we just talked about? We're going to use the first one:
a² = b² + c² - 2bc * cos(A)
We know a = 12, b = 7, and c = 6. So, let's plug those values into the formula:
12² = 7² + 6² - 2 * 7 * 6 * cos(A)
Now it's just a matter of simplifying and solving for cos(A). Let's do the math:
144 = 49 + 36 - 84 * cos(A)
Combine the numbers on the right side:
144 = 85 - 84 * cos(A)
Now, let's isolate the term with cos(A) by subtracting 85 from both sides:
59 = -84 * cos(A)
Next, we'll divide both sides by -84 to solve for cos(A):
cos(A) = -59 / 84
cos(A) ≈ -0.7024
We've got cos(A), but we want angle A itself. How do we get there? We use the inverse cosine function, also known as arccos or cos⁻¹:
A = arccos(-0.7024)
Grab your calculators, guys (make sure they're in degree mode!), and punch that in. You should get:
A ≈ 134.6 degrees
There we have it! Angle A is approximately 134.6 degrees. Isn't that satisfying? We've successfully used the Law of Cosines to find one of the angles in our triangle. But we're not done yet! We still need to find angles B and C. Before we move on, let's just take a moment to appreciate what we've done. We started with a formula, plugged in our known values, did some algebra, and used the inverse cosine function to find an angle. This is the core process of solving many trigonometry problems. So, if you've understood this, you're well on your way to becoming a trigonometry whiz! Now, let's move on to calculating angle B.
Calculating Angle B
Okay, now let's tackle angle B. We're going to use the same strategy as before, but this time we'll use the Law of Cosines formula that isolates angle B:
b² = a² + c² - 2ac * cos(B)
Again, we know a = 12, b = 7, and c = 6. Let's plug those values in:
7² = 12² + 6² - 2 * 12 * 6 * cos(B)
Time to simplify and solve for cos(B). Let's get to it:
49 = 144 + 36 - 144 * cos(B)
Combine the numbers on the right side:
49 = 180 - 144 * cos(B)
Now, isolate the term with cos(B) by subtracting 180 from both sides:
-131 = -144 * cos(B)
Divide both sides by -144 to solve for cos(B):
cos(B) = -131 / -144
cos(B) ≈ 0.9097
Awesome! We've got cos(B). To find angle B, we need to use the inverse cosine function again:
B = arccos(0.9097)
Grab your calculators, guys, and let's find that angle:
B ≈ 24.5 degrees
Boom! Angle B is approximately 24.5 degrees. We're on a roll here! We've found two of the angles in our triangle. But there's still one more to go: angle C. Now, we could use the Law of Cosines one more time, but there's a much easier way. Remember that the angles in any triangle add up to 180 degrees? We can use that fact to find angle C without any more complicated calculations. This is a great shortcut, and it's always good to look for these kinds of opportunities to simplify your work. Before we move on to calculating angle C, let's just pause and appreciate the power of the Law of Cosines. We've used it twice now to find angles in a triangle, and each time the process has been the same: plug in the known values, simplify the equation, isolate the cosine of the angle, and then use the inverse cosine function. This is a technique that you can apply to many different trigonometry problems, so it's a valuable skill to have. Okay, let's finish this up and find angle C!
Calculating Angle C
Alright, let's wrap this up by finding angle C. As we just discussed, we can use the fact that the angles in a triangle add up to 180 degrees. This is a super handy trick that will save us from having to use the Law of Cosines again. We already know angle A is approximately 134.6 degrees and angle B is approximately 24.5 degrees. So, we can set up the following equation:
A + B + C = 180
Plug in the values we know:
134.6 + 24.5 + C = 180
Combine the numbers on the left side:
159.1 + C = 180
Now, simply subtract 159.1 from both sides to solve for C:
C = 180 - 159.1
C ≈ 20.9 degrees
And there you have it! Angle C is approximately 20.9 degrees. We've successfully found all three angles of triangle ABC. How cool is that? We started with just the lengths of the sides, and using the Law of Cosines and a little bit of algebra, we were able to determine all the angles. This is a great example of how math can be used to solve real-world problems (or at least, problems involving triangles!). Now that we've found all the angles, let's just take a moment to check our work. One way to do this is to make sure that the angles add up to 180 degrees. Let's add our angles together:
134.6 + 24.5 + 20.9 = 180
They do! This gives us confidence that our calculations are correct. It's always a good idea to double-check your work, especially in math. So, we've calculated all the angles of triangle ABC. We've used the Law of Cosines, the inverse cosine function, and the fact that the angles in a triangle add up to 180 degrees. We've shown that even a seemingly complex problem can be broken down into smaller, manageable steps. Great job, guys!
Conclusion
So, there you have it, guys! We successfully calculated the angles of triangle ABC given the lengths of its sides using the Law of Cosines. We found that angle A is approximately 134.6 degrees, angle B is approximately 24.5 degrees, and angle C is approximately 20.9 degrees. This problem demonstrates the power and versatility of trigonometry, especially the Law of Cosines, in solving geometric problems. We started by understanding the Law of Cosines formula and how it relates the sides and angles of a triangle. We then applied the formula step-by-step, plugging in our known values, simplifying the equations, and using the inverse cosine function to find the angles. We also learned a handy shortcut: the fact that the angles in a triangle add up to 180 degrees. This allowed us to find the third angle much more easily once we had calculated the first two. Throughout the process, we emphasized the importance of breaking down complex problems into smaller, manageable steps, and double-checking our work to ensure accuracy. These are valuable problem-solving skills that can be applied not only in math but in many other areas of life. What's really cool about this problem is that it highlights the connection between algebra and geometry. We used algebraic techniques to manipulate equations and solve for unknowns, and we used geometric principles to understand the relationships between sides and angles in a triangle. This interplay between different branches of mathematics is what makes math so fascinating and powerful. If you've followed along with this problem and understood the steps involved, you've gained a valuable skill in trigonometry. You're now equipped to tackle similar problems involving triangles and the Law of Cosines. Keep practicing, and you'll become a true math whiz! Remember, math isn't just about memorizing formulas and procedures; it's about understanding the underlying concepts and using them to solve problems creatively. So, keep exploring, keep questioning, and keep having fun with math! Until next time, happy calculating!
