Trigonometric Equation: Solve Tan5° Tan10° Tan80° Tan85°

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Trigonometric Equation: Solve tan5° tan10° tan80° tan85°

Hey guys! Today, let's dive into a fascinating trigonometric problem. We're going to solve the equation: tan5° * tan10° * tan80° * tan85° = ?. This might look intimidating at first, but don't worry, we'll break it down step by step. Trigonometry can be a bit tricky, but with the right approach, it becomes much more manageable. This specific problem involves using trigonometric identities and understanding the relationships between different angles. We'll explore how to use complementary angles and product-to-sum formulas to simplify the equation and arrive at the solution. So, grab your calculators and let's get started!

Understanding the Problem

In this trigonometric problem, our main goal is to find the value of the expression tan5° * tan10° * tan80° * tan85°. To solve this, we need to leverage our knowledge of trigonometric identities and angle relationships. Remember, trigonometric functions like tangent (tan) have specific properties that we can exploit. The key here is to recognize that some angles are complementary, meaning they add up to 90 degrees. This will allow us to use the identity tan(90° - θ) = cot(θ), where cotangent (cot) is the reciprocal of tangent (i.e., cot(θ) = 1/tan(θ)). By pairing the angles strategically, we can simplify the expression significantly. We'll also need to be mindful of how these identities interact when multiplied together. This problem isn't just about plugging in values; it's about understanding the relationships between trigonometric functions and applying the right identities to simplify the expression. So, let's move on to the next section where we'll start applying these concepts.

Applying Trigonometric Identities

To crack this problem, we'll use some clever trigonometric identities. The key one here is the relationship between tangent and cotangent of complementary angles. Remember that tan(90° - θ) = cot(θ). We also know that cot(θ) = 1/tan(θ).

Let's rewrite our expression: tan5° * tan10° * tan80° * tan85°. Notice that 5° and 85° are complementary angles (5° + 85° = 90°), and so are 10° and 80° (10° + 80° = 90°). This is a crucial observation! Now, let’s use our identity. We can rewrite tan85° as tan(90° - 5°), which equals cot5°. Similarly, tan80° can be rewritten as tan(90° - 10°), which equals cot10°.

So, our expression now looks like this: tan5° * tan10° * cot10° * cot5°. Now, remember that cot(θ) = 1/tan(θ). So, we can rewrite cot5° as 1/tan5° and cot10° as 1/tan10°. Our expression then becomes: tan5° * tan10° * (1/tan10°) * (1/tan5°). See how things are starting to simplify? This is the beauty of using trigonometric identities. Next, we'll see how these terms cancel out, leading us to the final answer.

Simplifying the Expression

Okay, so we've transformed our expression to tan5° * tan10° * (1/tan10°) * (1/tan5°). Now comes the fun part – simplification! Notice that we have tan5° and 1/tan5°, and tan10° and 1/tan10°. These are multiplicative inverses, meaning when you multiply them together, they equal 1. Think of it like this: any number multiplied by its reciprocal gives you 1. So, tan5° * (1/tan5°) = 1 and tan10° * (1/tan10°) = 1.

Our expression now simplifies to 1 * 1. And what is 1 multiplied by 1? It’s simply 1! Therefore, tan5° * tan10° * tan80° * tan85° = 1. How cool is that? We started with a seemingly complex expression, but by using trigonometric identities and simplifying, we found the answer quite easily. This is a classic example of how understanding trigonometric relationships can make problem-solving much more efficient. In the next section, we'll recap the steps we took and highlight the key takeaways from this problem.

Final Answer and Key Takeaways

Alright guys, we've reached the end! We successfully solved the trigonometric equation tan5° * tan10° * tan80° * tan85°. By strategically applying trigonometric identities, we found that the answer is simply 1.

Let's quickly recap the steps we took:

  1. We recognized the complementary angles (5° and 85°, 10° and 80°).
  2. We used the identity tan(90° - θ) = cot(θ) to rewrite tan85° as cot5° and tan80° as cot10°.
  3. We used the identity cot(θ) = 1/tan(θ) to rewrite cot5° as 1/tan5° and cot10° as 1/tan10°.
  4. We simplified the expression by canceling out the reciprocal terms.

The key takeaway here is the power of recognizing and applying trigonometric identities. These identities act as tools that allow us to transform complex expressions into simpler forms. In this case, recognizing the complementary angles and using the related identities was crucial. Another important point is to always look for opportunities to simplify expressions. Often, terms will cancel out or combine in ways that make the problem much easier to solve. So, next time you encounter a trigonometric problem, remember these strategies and see if you can apply them. Trigonometry might seem daunting at first, but with practice and the right techniques, you'll become a pro in no time! Keep practicing, and you'll be amazed at what you can achieve.