Solving Right Triangle ABC A Step-by-Step Guide

Hey guys! Today, we're diving into a classic geometry problem: solving a right triangle. Specifically, we'll be tackling triangle ABC, which is a right triangle at angle B. We're given that angle C is 39 degrees and side 'a' (opposite angle A) is 18 cm. Our mission, should we choose to accept it, is to find all the missing angles and sides. Buckle up, because we're about to embark on a trigonometric adventure!

Understanding the Basics of Right Triangles

Before we jump into the calculations, let's quickly refresh the fundamentals of right triangles. Remember, a right triangle is a triangle that has one angle measuring exactly 90 degrees. This special angle is what gives right triangles their unique properties. In our case, angle B is the right angle, making it 90 degrees. The side opposite the right angle is called the hypotenuse, which is the longest side of the triangle. The other two sides are called legs or cathetus. Now, in a right triangle, the trigonometric functions—sine (sin), cosine (cos), and tangent (tan)—become our best friends. These functions relate the angles of the triangle to the ratios of its sides. Understanding these relationships is crucial for solving any right triangle problem. We'll be using these functions extensively in this guide, so let's make sure we're on the same page. Also, remember the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b): a² + b² = c². This theorem is another powerful tool in our arsenal. Knowing the basics sets the stage for solving the triangle effectively. We can visualize the triangle and the relationships between its sides and angles. This helps in identifying the correct trigonometric function to use for finding the unknown sides and angles. So, let's keep these fundamentals in mind as we proceed to solve triangle ABC. These basics are the building blocks for more complex problems in trigonometry and geometry, ensuring a solid foundation for future mathematical endeavors. The more comfortable you are with these basics, the easier it will be to tackle more challenging problems.

Finding Angle A

Let's kick things off by finding angle A. Remember, the sum of the angles in any triangle is always 180 degrees. Since we know angle B is 90 degrees (it's a right triangle) and angle C is 39 degrees, we can easily find angle A. We use the formula: Angle A + Angle B + Angle C = 180 degrees. Plugging in the values, we get: Angle A + 90 degrees + 39 degrees = 180 degrees. Simplifying the equation, we have: Angle A + 129 degrees = 180 degrees. Now, subtract 129 degrees from both sides to isolate Angle A: Angle A = 180 degrees - 129 degrees. Therefore, Angle A = 51 degrees. Ta-da! We've found our first missing angle. Knowing that the sum of angles in a triangle is 180 degrees is a fundamental concept that helps in solving many geometry problems. This simple calculation is a cornerstone of triangle geometry, making it easier to deduce unknown angles when others are known. With Angle A now known, we've taken a significant step towards fully solving triangle ABC. This step demonstrates how basic principles can lead to crucial solutions in more complex problems. Now that we have two angles, finding the third becomes straightforward, and this foundational knowledge will serve us well as we move forward. Mastering these basic principles allows us to approach more challenging problems with confidence. So, with Angle A safely in our grasp, let's move on to the next step in our trigonometric quest. We are making great progress, guys, and the remaining steps will build upon this initial success.

Calculating Side b (Adjacent to Angle A)

Now that we've conquered Angle A, it's time to find the missing sides. Let's start with side 'b', which is adjacent to angle A and opposite angle C. To find side 'b', we can use the trigonometric function tangent (tan). Remember, the tangent of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. In our case, tan(A) = opposite/adjacent = a/b. We know that Angle A is 51 degrees and side 'a' is 18 cm. So, we can set up the equation: tan(51 degrees) = 18 cm / b. To solve for 'b', we need to rearrange the equation: b = 18 cm / tan(51 degrees). Using a calculator, we find that tan(51 degrees) is approximately 1.2349. Now, plug that value into our equation: b = 18 cm / 1.2349. Performing the division, we get: b ≈ 14.57 cm. Great! We've found the length of side 'b'. Using the tangent function to relate sides and angles is a key technique in solving right triangles. This step demonstrates the power of trigonometric ratios in bridging the gap between angles and side lengths. With side 'b' now known, our understanding of triangle ABC deepens, bringing us closer to a complete solution. The application of the tangent function here is a perfect example of how trigonometric principles can be applied in practical geometric problems. The accuracy of our calculations is enhanced by a clear understanding of these trigonometric functions. We're making excellent progress, and each step builds upon the previous, showcasing the elegance of trigonometry in action. Onward to the next side!

Finding Side c (Hypotenuse)

Time to tackle the hypotenuse, side 'c'! We have a couple of options here. We could use the Pythagorean theorem, since we now know sides 'a' and 'b'. Alternatively, we can use another trigonometric function, such as sine or cosine. Let's go with sine, just to mix things up. Remember, the sine of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse. We can use Angle A again: sin(A) = opposite/hypotenuse = a/c. We know that Angle A is 51 degrees and side 'a' is 18 cm. So, we can set up the equation: sin(51 degrees) = 18 cm / c. To solve for 'c', we rearrange the equation: c = 18 cm / sin(51 degrees). Using a calculator, we find that sin(51 degrees) is approximately 0.7771. Now, plug that value into our equation: c = 18 cm / 0.7771. Performing the division, we get: c ≈ 23.16 cm. Fantastic! We've found the length of side 'c', the hypotenuse. Choosing the right trigonometric function is crucial for efficient problem-solving in trigonometry. This step illustrates the versatility of sine, cosine, and tangent in finding unknown sides. With side 'c' now determined, we have successfully found all the sides of triangle ABC. The sine function, with its elegant relationship between angles and side ratios, has proven to be a powerful tool. Our ability to apply trigonometric principles effectively is evident in this successful calculation. We're nearing the finish line, guys, and the completion of this triangle is a testament to our trigonometric skills. Let's keep this momentum going!

Summarizing the Solution

Alright, let's take a moment to summarize what we've found. We started with a right triangle ABC, where angle B is 90 degrees, angle C is 39 degrees, and side 'a' is 18 cm. Through the power of trigonometry and a bit of algebraic manipulation, we've successfully found all the missing angles and sides. Here's a quick rundown: Angle A: 51 degrees Side b: Approximately 14.57 cm Side c: Approximately 23.16 cm. So, there you have it! We've completely solved triangle ABC. Summarizing the solution provides a clear overview and reinforces understanding of the problem-solving process. This recap highlights the interconnectedness of the steps we took, demonstrating how each calculation built upon the previous. By consolidating our findings, we gain a sense of accomplishment and solidify our grasp of the concepts involved. The final summary brings closure to the problem, emphasizing the importance of clear and organized results. We can see how the initial conditions led to the final solution, showcasing the beauty and precision of mathematics. This thorough review ensures that we fully comprehend the solution and can confidently apply these methods to future problems. A well-presented summary is the hallmark of effective problem-solving, leaving no room for ambiguity. Congratulations, guys! We've navigated the trigonometric terrain and emerged victorious. Let's celebrate our mathematical prowess!

Real-World Applications of Solving Right Triangles

You might be wondering,