Hey everyone! Today, we're diving into the world of trigonometry and tackling a specific expression: 2 tan² 45 cos² 30 sin² 60. Don't worry if it looks a bit intimidating at first; we'll break it down step-by-step to make it super clear and easy to understand. Trigonometry, at its core, is all about the relationships between angles and sides in triangles. And these expressions involve trigonometric functions like tangent (tan), cosine (cos), and sine (sin). Our goal is to find the numerical value of the given expression, and along the way, we'll refresh our knowledge of some important trigonometric values. Ready to get started? Let's go!

Understanding the Basics: Trigonometric Functions

Alright, before we jump into the calculation, let's quickly recap what these trigonometric functions actually mean. Think of them as special ratios that relate the angles of a right-angled triangle to the lengths of its sides.

• Tangent (tan): This function relates the opposite side to the adjacent side of an angle in a right triangle. tan(angle) = Opposite / Adjacent.
• Cosine (cos): This function relates the adjacent side to the hypotenuse of an angle. cos(angle) = Adjacent / Hypotenuse.
• Sine (sin): This function relates the opposite side to the hypotenuse of an angle. sin(angle) = Opposite / Hypotenuse.

Each of these functions takes an angle as input and gives you a ratio as output. These ratios are constant for a given angle, no matter the size of the triangle. Understanding these fundamentals is crucial for solving our expression. In our case, we'll be dealing with specific angles: 45 degrees, 30 degrees, and 60 degrees. Knowing the values of these functions for these angles is key to cracking the problem. It is worth noting that, these trigonometric functions are used in many different fields. From navigation and surveying to physics and engineering, these functions are super important. They allow us to calculate distances, angles, and various other parameters in real-world scenarios. So, understanding them is like having a powerful tool in your mathematical toolkit.

Now that we've refreshed our memory of these functions, let's move on to the next section and learn the exact values that we need to calculate our equation. Get ready, this is going to be good!

Essential Trigonometric Values: The Foundation of the Solution

Okay, guys, here comes the fun part! To solve our expression, we need to know the values of tan, cos, and sin for the angles 45°, 30°, and 60°. Luckily, these values are pretty standard and you'll find them in your textbooks. Some of you might already know them by heart, but if not, no worries; let's list them out.

• tan 45° = 1: This is a super important one to remember! The tangent of 45 degrees is always equal to 1. Think of it this way: in a right-angled triangle with a 45-degree angle, the opposite and adjacent sides are equal in length, making their ratio (the tangent) equal to 1.
• cos 30° = √3 / 2: The cosine of 30 degrees is the square root of 3 divided by 2. You can derive this value using the properties of a 30-60-90 triangle, where the ratio of the sides is fixed.
• sin 60° = √3 / 2: Similarly, the sine of 60 degrees is also the square root of 3 divided by 2. It's interesting to note that sin 60° and cos 30° have the same value. This relationship arises from the complementary nature of sine and cosine: sin(x) = cos(90° - x).

It's a really good idea to memorize these values or at least know how to quickly derive them using the unit circle or special triangles. These values are the building blocks for solving a wide range of trigonometric problems. You'll encounter them so often that you'll eventually have them memorized without even trying. The ability to recall these values instantly will save you a lot of time and effort in exams and when solving problems. Now we have everything we need to calculate our original expression! Let's get to the calculations!

Step-by-Step Calculation: Unraveling the Expression

Alright, buckle up, everyone! We're finally ready to solve 2 tan² 45 cos² 30 sin² 60. We'll break it down into smaller, manageable steps to avoid any confusion. Remember the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

1. Substitute the Values: First, we replace the trigonometric functions with their known values:

   2 * (tan 45°)² * (cos 30°)² * (sin 60°)² becomes

   2 * (1)² * (√3 / 2)² * (√3 / 2)²

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2. Calculate the Squares: Next, we evaluate the squares:

   2 * (1) * (3 / 4) * (3 / 4)

3. Multiply: Finally, we multiply all the numbers together:

   2 * 1 * (3 / 4) * (3 / 4) = 2 * (9 / 16) = 18 / 16 = 9 / 8 = 1.125

So, the answer to the expression 2 tan² 45 cos² 30 sin² 60 is 9/8, or 1.125. Congratulations! You've successfully solved a trigonometric expression. You see, it wasn't so scary after all, right? This process helps you break down complex problems into simpler steps. This method is incredibly useful not just for trigonometry but for any mathematical problem you encounter. By clearly identifying each step and systematically applying the formulas and rules, you can solve even the most complicated-looking equations.

Now that we have the result, let's summarize all that we learned and practiced.

Conclusion: Recap and Key Takeaways

Awesome work, everyone! We've successfully calculated the value of the trigonometric expression 2 tan² 45 cos² 30 sin² 60. Here's a quick recap of what we did:

• We reviewed the basic trigonometric functions: tangent, cosine, and sine. We discussed how they relate angles to the sides of a right-angled triangle.
• We identified and used the key trigonometric values for 45°, 30°, and 60°.
• We substituted the values into the expression and carefully followed the order of operations to arrive at the final answer.

Key takeaways from this exercise:

• Understanding the definitions of trigonometric functions is crucial.
• Knowing the common trigonometric values is essential for quick calculations.
• Breaking down complex expressions into simpler steps makes them easier to solve.
• Practicing regularly helps you become more confident in solving trigonometric problems.

Trigonometry might seem daunting at first, but with a solid grasp of the basics, consistent practice, and a step-by-step approach, you can master it. This method can be applied to solve more complex trigonometric problems, including those involving identities, equations, and applications in real-world scenarios. Keep practicing, keep learning, and you'll find that trigonometry becomes more and more comfortable. Remember, math is like any other skill; the more you practice, the better you get. So, keep at it, and you'll be acing those trigonometry problems in no time! Keep in mind that a good grasp of trigonometry can be useful in many real-life applications, such as surveying, navigation, and even computer graphics. So, keep up the great work and happy calculating!