Solving Triangles: Functions, Formulas & Real-World Examples

Hey guys! Let's dive into the fascinating world of triangle calculations. Specifically, we're going to tackle a problem where we need to figure out the properties of a triangle given some initial information. The problem states: Determine the triangles that fulfill the following functions: a = 9cm, b = 12cm, and A = 35 degrees. This might seem a little daunting at first, but trust me, with the right formulas and a little bit of patience, we can crack this! Understanding how to solve triangles is not just a math exercise; it's a skill that has real-world applications in fields like architecture, engineering, and even navigation. So, buckle up, and let's get started on this exciting journey of discovery. We will explore the Law of Sines, the Law of Cosines, and various trigonometric functions, ensuring we're equipped to solve any triangle-related challenge thrown our way. Get ready to flex those brain muscles!

Understanding the Given Information and Setting Up

Alright, first things first, let's break down what the problem gives us. We know:

• a = 9 cm: This is the length of one side of the triangle.
• b = 12 cm: This is the length of another side of the triangle.
• A = 35°: This is the angle opposite side a.

This is a classic scenario where we can use the Law of Sines. The Law of Sines is a fundamental concept in trigonometry that relates the sides of a triangle to the sines of its angles. It states that the ratio of the length of a side to the sine of its opposite angle is constant for all sides and angles in a triangle. In other words, for any triangle, the ratio of a side to the sine of its opposite angle is the same as the ratio of any other side to the sine of its opposite angle. Sounds complex? It's not, I promise! The formula looks like this: a/sin(A) = b/sin(B) = c/sin(C). Before diving in, let's set up the problem properly. It's always a good idea to sketch a triangle and label the known information. This helps you visualize the problem and prevents mistakes. In our case, we have side a, side b, and angle A. Our goal is to find the remaining angles and side c (if a solution exists, of course!). We will use the Law of Sines to find angle B. With angle B found, we can use the angle sum property of a triangle (which states that the sum of all angles in a triangle is 180 degrees) to find angle C, and then, using the Law of Sines again or the Law of Cosines, we'll find side c. Easy peasy!

To make sure you really understand, let's also talk about some common types of triangles. You've got your acute triangles (all angles less than 90°), obtuse triangles (one angle greater than 90°), and right triangles (one angle exactly 90°). Knowing the type of triangle you're dealing with can sometimes help you anticipate the range of possible solutions. For our specific problem, we don't know the exact type initially, but as we calculate the angles, we will be able to determine the type.

Applying the Law of Sines

Okay, time to put the Law of Sines to work. We know a = 9 cm, b = 12 cm, and A = 35°. Using the Law of Sines formula, we can set up the following equation to find angle B: a/sin(A) = b/sin(B). This becomes 9/sin(35°) = 12/sin(B). Now, let's solve for sin(B): sin(B) = (12 * sin(35°))/9. Now, we calculate sin(35°), which is approximately 0.5736. So the equation is sin(B) = (12 * 0.5736)/9. Simplifying gives us sin(B) = 0.7648. To find angle B, we need to take the inverse sine (arcsin) of 0.7648. Using a calculator, we find that B ≈ 49.9°. But wait, there's a little twist! The sine function has a property where it gives the same value for two angles in the range of 0° to 180°. This means there might be another possible value for angle B. To find this second possible value, we subtract our first value from 180°: 180° - 49.9° = 130.1°. Therefore, angle B could also be 130.1°. That means that we might have two potential triangles that satisfy the initial conditions. This is why it’s really essential to understand the basic trigonometric functions and how they relate to triangles.

Exploring the Two Possible Triangle Solutions

So, as we've found out from the Law of Sines, we have two potential solutions. Let's explore each one and see if they make sense.

Solution 1: Angle B ≈ 49.9°

If angle B is approximately 49.9 degrees, we can find angle C using the fact that the sum of angles in a triangle is 180 degrees: C = 180° - A - B = 180° - 35° - 49.9° = 95.1°. Now that we know all three angles, we can use the Law of Sines (or the Law of Cosines) to find side c. Let's use the Law of Sines: a/sin(A) = c/sin(C). So, 9/sin(35°) = c/sin(95.1°). Calculate sin(95.1°), which is approximately 0.9961. Rearranging the equation to solve for c: c = (9 * sin(95.1°))/sin(35°) = (9 * 0.9961)/0.5736 ≈ 15.6 cm. Therefore, one possible triangle has angles A = 35°, B ≈ 49.9°, and C ≈ 95.1°, with sides a = 9 cm, b = 12 cm, and c ≈ 15.6 cm. This triangle is an obtuse triangle because it has one angle greater than 90 degrees.

Solution 2: Angle B ≈ 130.1°

Now, let's consider the second possibility, where angle B is approximately 130.1 degrees. Calculate angle C: C = 180° - A - B = 180° - 35° - 130.1° = 14.9°. With all three angles known, again use the Law of Sines to find side c: a/sin(A) = c/sin(C). Therefore, 9/sin(35°) = c/sin(14.9°). Calculate sin(14.9°), which is approximately 0.2576. Rearranging the formula to solve for c: c = (9 * sin(14.9°))/sin(35°) = (9 * 0.2576)/0.5736 ≈ 4.0 cm. So, the second possible triangle has angles A = 35°, B ≈ 130.1°, and C ≈ 14.9°, with sides a = 9 cm, b = 12 cm, and c ≈ 4.0 cm. This is also an obtuse triangle because angle B is greater than 90 degrees.

Summary and Conclusion

So there you have it, guys! We have successfully determined two triangles that satisfy the given functions. Remember, the key is to understand the formulas, apply them carefully, and be aware of the multiple solutions that can arise when using the Law of Sines. To recap:

• Triangle 1: Angles: A = 35°, B ≈ 49.9°, C ≈ 95.1°; Sides: a = 9 cm, b = 12 cm, c ≈ 15.6 cm.
• Triangle 2: Angles: A = 35°, B ≈ 130.1°, C ≈ 14.9°; Sides: a = 9 cm, b = 12 cm, c ≈ 4.0 cm.

Understanding these types of calculations is not only useful for academic purposes but is also fundamental to many real-world applications. By mastering the fundamentals of trigonometry, we equip ourselves with valuable problem-solving skills that can be applied across many different disciplines. The best way to get good at this is by practicing, so try working through some more similar problems.

Remember to double-check your work, use a calculator carefully, and always draw a diagram to visualize the problem. Keep practicing, and you'll become a triangle-solving pro in no time! Keep exploring, keep learning, and don't be afraid to tackle challenging problems – it's all part of the fun! And that concludes our journey into the world of triangles, for now. Keep those math skills sharp, and I'll catch you next time! Feel free to ask more questions.