Solving Trigonometric Expressions: A Step-by-Step Guide

Hey guys! Ever stumbled upon a trig problem that seems a bit intimidating? Don't worry; we've all been there! Let's break down the expression 2sin²(20°) + 2sin²(70°) - 3tan(40°)tan(50°) step by step. This isn't just about finding the answer; it's about understanding the why behind each move. We'll use trigonometric identities, angle relationships, and some clever substitutions to simplify this expression. So, grab your pens, and let's dive in! Remember, practice makes perfect, and the more you work through these problems, the more comfortable you'll become with the concepts. Trigonometry is all about patterns and relationships, and once you start recognizing them, the problems become much more manageable. This problem combines several key trigonometric concepts, which is why it's a great example to learn from. We'll focus on applying the double-angle formulas, complementary angle identities, and the relationship between sine, cosine, and tangent. These tools are fundamental in solving a wide array of trigonometric problems. It's like having a toolkit for your math adventures! The goal here is not only to find the solution but also to explain each step clearly, so you grasp the underlying principles. We're building a solid foundation for future math challenges. Believe me, understanding these fundamentals will give you a huge advantage in more complex problems. Now, let's tackle this step-by-step and unravel the mystery of this expression! We'll start by simplifying the parts of the expression, combining like terms, and, finally, arriving at the solution. This is a journey of discovery, and you're going to love the feeling of solving a seemingly complex problem! Get ready to flex your math muscles; it's going to be a fun ride!

Step 1: Utilizing the Pythagorean Identity and Angle Relationships

Alright, let's kick things off by taking a closer look at 2sin²(20°) + 2sin²(70°). Notice something cool? We have sin²(20°) and sin²(70°). These angles are complementary (they add up to 90 degrees). This is our secret weapon. We can rewrite sin(70°) using the cofunction identity: sin(θ) = cos(90° - θ). So, sin(70°) = cos(90° - 70°) = cos(20°). Now, the expression becomes 2sin²(20°) + 2cos²(20°). See where we are going? Using the Pythagorean identity, sin²(θ) + cos²(θ) = 1, we can simplify this part of the expression beautifully. We can factor out the 2, getting 2[sin²(20°) + cos²(20°)]. And what's inside the brackets? Yep, it equals 1! So, this part of the expression simplifies to 2 * 1 = 2. This is a massive simplification, and it's the key to unlocking the rest of the problem. Remember, spotting these relationships is crucial in trigonometry. It's like finding a hidden shortcut on a map; it makes the whole journey easier. Take a moment to appreciate how a simple identity can transform the problem. By understanding the fundamentals and recognizing the patterns, we can dramatically simplify complex expressions. This step highlights how important knowing your trigonometric identities is. It's a core concept that helps us move forward quickly and efficiently. So, always keep those identities in mind. They are the keys to success in trigonometry. We’ve now simplified a major portion of the equation, making the overall problem more manageable.

Let's Break Down the First Part

Let's zoom in on the first part, 2sin²(20°) + 2sin²(70°). As we already mentioned, the key is to recognize the relationship between 20 degrees and 70 degrees. They are complementary angles. This allows us to use the identity sin(90° - θ) = cos(θ). Applying this, we can rewrite sin(70°) as cos(20°). The expression then becomes 2sin²(20°) + 2cos²(20°). Now, we factor out a 2: 2[sin²(20°) + cos²(20°)]. Using the Pythagorean identity sin²(θ) + cos²(θ) = 1, this simplifies to 2 * 1 = 2. This transforms a seemingly complicated part of the expression into a simple '2'. Now, we have successfully reduced the first part of the equation. This highlights the power of trigonometric identities and understanding complementary angles. This approach makes the overall equation more manageable and easier to solve. Excellent work, guys! You're making great progress.

Step 2: Dealing with the Tangent Terms

Now, let's tackle the tangent part of the expression: -3tan(40°)tan(50°). Again, we can use complementary angles. Note that 40° and 50° are also complementary. We know that tan(θ) = cot(90° - θ). Also, recall that cot(θ) = 1/tan(θ). So, we can rewrite tan(50°) as cot(40°), which is equal to 1/tan(40°). This means that tan(40°)tan(50°) = tan(40°) * (1/tan(40°)) = 1. Therefore, -3tan(40°)tan(50°) simplifies to -3 * 1 = -3. This is another crucial simplification. It transforms another complicated part of the equation into a simple number. By recognizing the relationship between tan and cot of complementary angles, we can easily simplify this portion. The trick is to always look for those connections and relationships. This demonstrates how knowing trigonometric identities can help you simplify complex problems efficiently. You'll notice that the key is to break down each part of the expression and apply the appropriate formulas. The goal is always to get the expression into its simplest form. With each simplification, we're getting closer to the solution, and it's exciting to see it all come together! We are now one step closer to solving this problem. Keep it up, you are doing great!

Simplifying the Tangent Expressions

Let's dig a little deeper into the tangent terms. We're looking at -3tan(40°)tan(50°). Here, the key insight is recognizing that 40° and 50° are complementary angles. This means that tan(50°) can be expressed in terms of tan(40°). Using the property tan(90° - θ) = cot(θ), we can rewrite tan(50°) as cot(40°). Recall that cot(θ) is the reciprocal of tan(θ), i.e., cot(θ) = 1/tan(θ). So, cot(40°) = 1/tan(40°). Now the expression becomes -3tan(40°) * (1/tan(40°)). The tan(40°) terms cancel out, leaving us with -3 * 1 = -3. This step elegantly simplifies the tangent part of the expression. The use of complementary angles and reciprocal identities makes this simplification possible. Always remember to look for those angle relationships and identities. This will help you enormously in solving these types of problems. With this simplification, the equation is now in a much simpler form and is ready for the next step. Great job, guys! Keep up the good work.

Step 3: Putting It All Together

We've simplified the expression into two parts: 2sin²(20°) + 2sin²(70°) = 2 and -3tan(40°)tan(50°) = -3. Now, we combine these results. The original expression 2sin²(20°) + 2sin²(70°) - 3tan(40°)tan(50°) becomes 2 - 3. Therefore, the final answer is -1. And there you have it! We successfully simplified and solved the trigonometric expression. It’s a satisfying feeling, right? Remember, it’s all about breaking down the problem into manageable steps, recognizing those key trigonometric identities, and applying them correctly. With practice, you'll become more and more proficient at these problems. The beauty of trigonometry lies in its elegant solutions. Sometimes, problems that initially seem complex can be solved with a few clever steps. This example perfectly illustrates that. We started with an expression that might have looked daunting at first, but we broke it down using basic identities and angle relationships. Now you know how to solve this type of problem. Remember that practice is key. Keep exploring, keep learning, and keep practicing. You are all doing great!

Combining the Results and Finding the Answer

Let's put it all together now. We have simplified two parts of the expression: 2sin²(20°) + 2sin²(70°) = 2 and -3tan(40°)tan(50°) = -3. The original expression was 2sin²(20°) + 2sin²(70°) - 3tan(40°)tan(50°). Substituting the simplified values, we get 2 - 3. This simple arithmetic gives us the final answer: -1. Congratulations, you've solved the expression! This is a great example of how to break down a complex trigonometric expression. By using trigonometric identities and angle relationships, the problem simplifies beautifully, leading to the final answer. Solving it step-by-step makes the process more accessible. Always remember to look for patterns, use the correct formulas, and practice. You are well on your way to mastering trigonometry. It's all about consistent effort and applying the right techniques.

Conclusion: Key Takeaways

So, what have we learned today? We’ve successfully solved a trigonometric expression by applying some fundamental trigonometric identities and angle relationships. Key takeaways: Always look for complementary angles, because they often provide a way to simplify the expression. Remember and utilize the Pythagorean identity, sin²(θ) + cos²(θ) = 1. Understand the relationships between tangent and cotangent, and how they relate to complementary angles. Practice is key! The more you practice, the better you'll become at recognizing patterns and applying the right formulas. Keep an eye out for these connections, and you'll find that many trig problems become much easier to solve. Also, do not be afraid to break complex problems into smaller, more manageable steps. This makes the entire process less daunting. Always remember to double-check your work, especially when dealing with multiple steps and identities. This will help you avoid minor mistakes. And finally, celebrate your successes! Solving these problems is an accomplishment, and you should be proud of your work. Remember, mathematics is a journey of exploration and discovery. Keep exploring, keep learning, and enjoy the process! You've got this!

Recap of Key Points

Let's summarize the key takeaways from solving this trigonometric expression. First, always look for complementary angles. They are your friends in simplifying trig problems. Recognizing that 20° and 70°, as well as 40° and 50°, are complementary angles was essential. Second, the Pythagorean identity sin²(θ) + cos²(θ) = 1 is a powerful tool. It helped us simplify the sine terms. Third, understanding the relationships between tangent and cotangent is crucial. Remember that tan(θ) = cot(90° - θ) and cot(θ) = 1/tan(θ). Fourth, practice regularly. The more you practice, the easier it becomes to spot these relationships and apply the right formulas. The key is to consistently work through problems and reinforce your understanding. Lastly, always break down complex problems into smaller, manageable steps. This reduces the chance of making mistakes and makes the overall process less intimidating. By following these key takeaways, you can confidently approach and solve similar trigonometric expressions. Keep up the great work, and enjoy the journey of learning and discovery!