Solving Right Triangle ABC A Step-by-Step Guide

Hey guys! Let's dive into solving a classic geometry problem: finding the missing sides and angles of a right triangle. Specifically, we're tackling right triangle ABC, where the right angle is at vertex C. We know one side (the opposite side) measures 5 cm, and the angle opposite that side is 55 degrees. And to make things easier, we have the sine, cosine, and tangent values for 55 degrees: sin(55°) = 0.81, cos(55°) = 0.57, and tan(55°) = 1.42. So, grab your calculators, and let's get started!

1. Understanding the Problem: Setting the Stage for Success

Before we jump into calculations, let's make sure we're all on the same page. Visualizing the problem is key! Imagine a triangle, ABC, with a little square in the corner at C – that signifies our 90-degree right angle. The side opposite this right angle is the hypotenuse, the longest side of the triangle. Now, we're given that the side opposite the 55-degree angle (let's call this angle A) measures 5 cm. This is crucial information because it allows us to use trigonometric ratios to find the other sides. Remember SOH CAH TOA? This is our golden ticket! It reminds us that Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, and Tangent = Opposite/Adjacent. By identifying the knowns (opposite side and angle) and the unknowns (adjacent side and hypotenuse), we can strategically choose which trigonometric ratio to use. A clear understanding of these relationships is the foundation for accurately solving the triangle. We need to find the length of the adjacent side (AC), the length of the hypotenuse (AB), and the measure of the remaining angle (angle B). With these pieces of the puzzle, we'll have completely solved the right triangle ABC. This initial setup, the careful labeling, and the visualization – it's all part of the process, guys. Don't skip it! It'll save you time and frustration in the long run. We are not just crunching numbers; we are building a mental model of the problem, and that's how you become a true problem-solver in math and beyond. So, let’s keep this image in our minds as we move forward and see how the trigonometric ratios help us unveil the unknown sides and angles of this triangle. The beauty of trigonometry lies in these interconnected relationships, and once we grasp them, solving triangles becomes almost like second nature.

2. Finding the Adjacent Side: Tangent to the Rescue

Okay, so we know the opposite side (5 cm) and the angle (55 degrees). We need to find the adjacent side. Which trigonometric ratio connects these three? Ding ding ding! It's the tangent (TOA: Tangent = Opposite / Adjacent). We know tan(55°) = 1.42, and we know the opposite side is 5 cm. So we can set up a simple equation: 1.42 = 5 / Adjacent. Now, a little bit of algebraic maneuvering – multiply both sides by the Adjacent side, then divide both sides by 1.42 – and we get: Adjacent = 5 / 1.42. Plug that into your calculator, and you'll find the adjacent side is approximately 3.52 cm. Easy peasy, right? Guys, this is where the power of SOH CAH TOA really shines. By remembering those simple ratios, you can unlock the secrets of any right triangle. This step highlights how crucial it is to correctly identify which sides and angles are known and which are unknown. It's like being a detective, using the clues to solve the mystery! The tangent function acted as our magnifying glass in this case, allowing us to zoom in on the relationship between the opposite and adjacent sides. And now, with the length of the adjacent side revealed, we're one step closer to completely unraveling this triangular puzzle. Don't underestimate the importance of these fundamental relationships. They are the building blocks for more advanced concepts in trigonometry and beyond. Think of each trigonometric function – sine, cosine, and tangent – as a special tool in your mathematical toolbox. Knowing when and how to use each tool is what transforms you from a novice to a master problem solver. So, let's take a moment to appreciate the tangent function and its ability to help us find missing side lengths. It's a powerful ally in our quest to conquer right triangles!

3. Calculating the Hypotenuse: Sine Steps Up

Now that we've found the adjacent side, let's tackle the hypotenuse. We still know the opposite side (5 cm) and the angle (55 degrees). Which ratio links the opposite side and the hypotenuse? You guessed it – the sine (SOH: Sine = Opposite / Hypotenuse). We know sin(55°) = 0.81, and the opposite side is 5 cm. So our equation is: 0.81 = 5 / Hypotenuse. Just like before, we rearrange the equation to solve for the Hypotenuse: Hypotenuse = 5 / 0.81. Crunch those numbers, and you'll find the hypotenuse is approximately 6.17 cm. Awesome! We've now found two sides of our right triangle. See how these trigonometric relationships start to build upon each other? Each piece of information we uncover helps us unlock another piece of the puzzle. This is the elegance of mathematics at work, guys. The sine function played a crucial role in revealing the hypotenuse. By utilizing the known angle and the opposite side, the sine function provided a direct path to calculating the longest side of the triangle. This is a perfect example of how understanding the properties of trigonometric ratios can simplify complex problems. Instead of relying on guesswork or complicated formulas, we used a fundamental relationship to achieve our goal. This approach is not only efficient but also provides a deeper understanding of the underlying mathematical principles. Remember, the hypotenuse is always the side opposite the right angle, and it's the longest side in the triangle. This fact, combined with the sine function, allowed us to confidently determine its length. So, let's celebrate another successful calculation and move on to finding the final piece of the puzzle – the missing angle! We're on a roll, guys, and the feeling of accomplishment you get from solving these problems is just the beginning. There's a whole world of mathematical exploration waiting for you, and these fundamental skills are your passport to adventure.

4. Unveiling the Missing Angle: Angle Sum Property to the Rescue

We've found two sides, but we're not quite done yet! We still need to find the missing angle. Remember, the angles in any triangle always add up to 180 degrees. We know one angle is 90 degrees (the right angle) and another is 55 degrees. So, to find the third angle (let's call it angle B), we simply subtract the known angles from 180: Angle B = 180 - 90 - 55. That means angle B is 35 degrees. Boom! We've now found all three angles and all three sides of the right triangle. This simple yet powerful concept of the angle sum property is a cornerstone of geometry. It's a fundamental truth that applies to all triangles, not just right triangles. By understanding this principle, we were able to effortlessly calculate the missing angle without resorting to complex trigonometric functions. This illustrates a crucial point in problem-solving: always look for the simplest, most direct route to the answer. Sometimes, the most elegant solutions are the ones that require the least amount of effort. Think of the angle sum property as a shortcut, a handy tool that can save you time and energy. Guys, it's this kind of thinking – this ability to identify the most efficient path – that separates a good problem solver from a great one. So, let's add this strategy to our repertoire and remember to always consider the basic geometric principles before diving into more complicated calculations. With the missing angle now in our possession, we can confidently declare that we have completely solved the right triangle ABC. We've determined all three sides and all three angles, and we've done so by utilizing a combination of trigonometric ratios and fundamental geometric principles. This is a triumph worth celebrating!

5. Putting It All Together: The Grand Finale

Let's recap! We started with a right triangle ABC, a known opposite side of 5 cm, and a known angle of 55 degrees. Using the tangent function, we found the adjacent side to be approximately 3.52 cm. Then, using the sine function, we calculated the hypotenuse to be approximately 6.17 cm. Finally, using the angle sum property of triangles, we determined the missing angle to be 35 degrees. We've successfully solved the triangle! This entire process, from visualizing the problem to the final calculations, demonstrates the power and elegance of trigonometry. We've used SOH CAH TOA as our guide, the angle sum property as our shortcut, and our algebraic skills to manipulate equations. And the result? A complete understanding of the right triangle ABC. But this isn't just about finding numbers; it's about developing a problem-solving mindset. It's about breaking down complex problems into smaller, manageable steps. It's about choosing the right tools for the job. And it's about celebrating the satisfaction of a problem well solved. So, guys, let's take a moment to appreciate the journey we've taken. We've not only solved a right triangle, but we've also strengthened our problem-solving muscles. And that's a skill that will serve us well in all areas of life. Remember, math isn't just about formulas and equations; it's about logical thinking, critical analysis, and creative problem-solving. And with each problem we solve, we're honing these skills and becoming more confident in our abilities. So, keep exploring, keep questioning, and keep solving! The world of mathematics is vast and fascinating, and there's always something new to discover. And who knows, maybe the next triangular challenge you face will be even more exciting and rewarding! Now, let’s go and conquer some more mathematical mountains, guys! The possibilities are endless, and the journey is just beginning.