Hey guys! Today, we're diving into the fascinating world of trigonometry to solve a specific equation. We'll break down each step, making it super easy to follow, even if you're just starting with trig. Our mission is to solve the equation: pseisinacosbse = cosasinb, given that a = 30 degrees and b = 45 degrees. Ready? Let's jump in!

Understanding the Problem

Before we start crunching numbers, let's make sure we understand what the problem is asking. The equation pseisinacosbse = cosasinb looks a bit intimidating, but don't worry, we'll simplify it. We need to find if this equation holds true when a is 30 degrees and b is 45 degrees. Essentially, we're verifying if the left-hand side (LHS) of the equation equals the right-hand side (RHS) when we plug in these values. Understanding this is the first key step in tackling any math problem. Make sure you grasp this concept before moving on. A solid understanding will make the entire process smoother. Remember, math is like building blocks; you need a strong foundation to build something amazing!

Evaluating Trigonometric Functions

Alright, let's roll up our sleeves and get our hands dirty with some actual calculations! The first thing we need to do is evaluate the trigonometric functions for the given angles. We're working with a = 30 degrees and b = 45 degrees. So, we need to find the values of sin(30°), cos(30°), sin(45°), and cos(45°). If you remember your special triangles, this should be a breeze. If not, no worries, we'll quickly go through them.

• sin(30°) = 1/2
• cos(30°) = √3/2
• sin(45°) = √2/2
• cos(45°) = √2/2

These are fundamental values in trigonometry and are worth memorizing. They pop up all the time! Knowing these values will drastically speed up your problem-solving. Now that we have these values, we're ready to substitute them back into our original equation. This is where the fun really begins. Keep these values handy; we'll be using them in the next step!

Substituting Values into the Equation

Okay, now for the crucial part: substituting the trigonometric values we found earlier into our equation pseisinacosbse = cosasinb. Remember, we have:

• sin(30°) = 1/2
• cos(30°) = √3/2
• sin(45°) = √2/2
• cos(45°) = √2/2

Let's plug these into the equation:

LHS = sin(a)cos(b) = sin(30°)cos(45°) = (1/2) * (√2/2) = √2/4

RHS = cos(a)sin(b) = cos(30°)sin(45°) = (√3/2) * (√2/2) = √6/4

So, after substitution, our equation becomes: √2/4 = √6/4. This step is all about careful replacement and accurate arithmetic. A small mistake here can throw off the entire solution. Double-check each substitution to ensure you've got it right. Once you're confident, you can move on to simplifying and comparing the two sides.

Simplifying and Comparing

Now that we've substituted the values, let's simplify and compare both sides of the equation. We have:

LHS = √2/4 RHS = √6/4

To determine if the equation holds true, we need to check if LHS = RHS. In this case, it's clear that √2/4 is not equal to √6/4. √2 and √6 are different values, and dividing them by the same number (4) won't make them equal. Therefore, the equation sin(a)cos(b) = cos(a)sin(b) is not true for a = 30 degrees and b = 45 degrees. This step is about using your simplification skills and making a direct comparison. If you're unsure, you can use a calculator to get decimal approximations and compare those. The important thing is to be methodical and precise.

Analyzing the Result

We've crunched the numbers and found that √2/4 ≠ √6/4. So, what does this result actually mean? It tells us that the original equation, sin(a)cos(b) = cos(a)sin(b), does not hold true when a = 30 degrees and b = 45 degrees. This is a crucial finding. It's important to understand that not all trigonometric identities hold true for all values of a and b. In this specific case, the equation is false for the given angles. Understanding the implications of your results is just as important as the calculations themselves. It helps you develop a deeper understanding of the underlying concepts. Always take a moment to reflect on what your answer means in the context of the problem. Sometimes, it's not just about getting the right answer, but understanding why it's the right answer.

Connecting to Trigonometric Identities

Now, let's take a step back and connect this problem to broader trigonometric identities. The equation sin(a)cos(b) = cos(a)sin(b) might look familiar. It's actually part of the sine addition formula, which is: sin(a + b) = sin(a)cos(b) + cos(a)sin(b). Notice the plus sign in the middle? Our original equation was missing that plus sign, which is why it didn't hold true. If we had the correct formula, we could have verified it as follows:

sin(30° + 45°) = sin(75°)

sin(30°)cos(45°) + cos(30°)sin(45°) = (1/2)(√2/2) + (√3/2)(√2/2) = √2/4 + √6/4 = (√2 + √6)/4

So, sin(75°) = (√2 + √6)/4, which is a valid result. This highlights the importance of using the correct trigonometric identities. Small differences in the formula can lead to completely different results. Always double-check your formulas and make sure you're using the right one for the given situation. Understanding the connections between different identities will make you a more confident and proficient problem solver.

Tips for Solving Trigonometric Equations

Before we wrap up, here are some essential tips for solving trigonometric equations:

1. Memorize Key Values: Knowing the sine, cosine, and tangent values for common angles like 0°, 30°, 45°, 60°, and 90° will save you a lot of time and effort.
2. Understand Trigonometric Identities: Familiarize yourself with fundamental identities like the Pythagorean identities, sum and difference formulas, and double-angle formulas. These are your go-to tools for simplifying equations.
3. Simplify Before Substituting: If possible, simplify the equation before plugging in any values. This can make the calculations much easier.
4. Check Your Work: Always double-check your calculations and substitutions to avoid errors. A small mistake can lead to a wrong answer.
5. Practice Regularly: The more you practice, the more comfortable you'll become with solving trigonometric equations. Try a variety of problems to challenge yourself.
6. Use a unit circle: It helps understand the relationships between angles and trigonometric functions.

By following these tips, you'll be well-equipped to tackle any trigonometric equation that comes your way. Remember, practice makes perfect!

Conclusion

So, there you have it! We've successfully tackled the problem of solving the equation pseisinacosbse = cosasinb with a = 30 degrees and b = 45 degrees. We found that the equation does not hold true for these values. We also connected this problem to the sine addition formula and shared some valuable tips for solving trigonometric equations. Remember, trigonometry can seem daunting at first, but with practice and a solid understanding of the fundamentals, you'll become a pro in no time! Keep exploring, keep learning, and most importantly, keep having fun with math! Keep up the great work, and I'll catch you in the next one!