Calculate Trigonometric Expression: Step-by-Step Guide

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Hey everyone! Today, we're diving into a fun little math problem. We're going to calculate the value of a trigonometric expression: 2.5 * sin(π/3) + 3 * cos(π/6). Don't worry if this looks a bit intimidating at first – we'll break it down step-by-step, so you'll understand it perfectly. This guide will help you understand the core concepts. So, grab your calculators (or your brains, if you're feeling extra smart!), and let's get started. We'll explore the roles of sine and cosine, and how we can easily deal with angles expressed in radians, so that you can become a trigonometry pro! Trigonometry, at its heart, is all about the relationships between angles and sides in triangles. The sine, cosine, and tangent functions are the foundation of this branch of mathematics. Once you understand them, the world of triangles opens up, and you can solve all sorts of fascinating problems, from calculating the height of a building to navigating the stars.

We will start with the basics. The expression we're tackling today involves two trigonometric functions: sine (sin) and cosine (cos). These are fundamental functions in trigonometry, and they relate angles to the ratios of sides in a right-angled triangle. Sine of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse (the longest side). Cosine of an angle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. Both sine and cosine functions always produce values between -1 and 1. These values are determined by the size of the angle. Now, the cool part: these trigonometric functions have specific values for certain angles, and we'll be using some of these values. You should have already studied and learned the unit circle and the table with trigonometric values, but if not, no worries, we'll quickly revise what you need to know. Remember, the trigonometric values for special angles like 30 degrees (π/6 radians), 60 degrees (π/3 radians), and 90 degrees (π/2 radians) are very common and useful.

So, before we even start, let's refresh our memory on the values of sine and cosine for the angles we'll be using. This is crucial for solving our problem. For example, sin(π/3) which means sine of π/3 radians (or 60 degrees), equals √3 / 2. And cos(π/6) which is cosine of π/6 radians (or 30 degrees), is also √3 / 2. Once you remember these, the rest of the calculation is a piece of cake. Knowing the values of sine and cosine for these common angles makes these calculations much quicker. This is where your study and practice comes into play. You can start practicing some basic trigonometry to make sure you've grasped the fundamentals of calculating values for some specific angles, and then move on to more complicated tasks. Keep in mind that using a calculator in these cases is allowed, but try doing it by hand first, so you get the hang of it and improve your skills.

Breaking Down the Expression: Step by Step

Alright, now that we have all the basics covered, let's get down to business and calculate 2.5 * sin(π/3) + 3 * cos(π/6). We'll go step by step, so even if you're new to this, you'll be able to follow along. The key to solving this kind of problem is to break it down into smaller, manageable parts. So, we'll start with sin(π/3) and cos(π/6), and then we'll put everything together. Remember, the goal is not just to get the answer, but also to understand how the answer is derived and how the sine and cosine functions work. We'll first evaluate the sine function at π/3 radians. As we mentioned earlier, sin(π/3) = √3 / 2. This is a value you should memorize, as it’s used quite frequently in trigonometry problems. If you're using a calculator, make sure it’s in radian mode to get the correct value for the angle. Calculators help us a lot, but understanding the fundamentals will give you an edge. Once you have this value, we will substitute it back into the original expression. Then, we will evaluate the cosine function at π/6 radians. Similarly, cos(π/6) = √3 / 2. Again, this is a value we should know. Now that we have calculated both sine and cosine values, we will use these to solve the problem.

So, now we have all the pieces we need, and it's time to put them together. The expression is 2.5 * sin(π/3) + 3 * cos(π/6). We know that sin(π/3) = √3 / 2 and cos(π/6) = √3 / 2. Let's substitute these values into the expression. So, the original expression transforms into 2.5 * (√3 / 2) + 3 * (√3 / 2). Now, we will simply perform the multiplication and addition to get our final answer. Next, we will perform the multiplication: 2.5 * (√3 / 2) equals 2.5 * √3 / 2, and 3 * (√3 / 2) equals 3 * √3 / 2. We can also write it as 1.25 * √3 + 1.5 * √3. And finally, we will combine the terms: 1.25 * √3 + 1.5 * √3 equals 2.75 * √3. If you want a numerical value, you can use a calculator to find the approximate value of √3 (which is roughly 1.732). Then, multiply 2.75 by 1.732, and you'll get your final answer. The ability to break down the problem step-by-step is an essential skill in mathematics, so kudos to you if you've followed along and understood the procedure.

Step-by-step solution

  1. Identify the values:
    • sin(π/3) = √3 / 2
    • cos(π/6) = √3 / 2
  2. Substitute into the expression: 2. 5 * (√3 / 2) + 3 * (√3 / 2)
  3. Simplify:
    1. 25 * √3 + 1. 5 * √3 = 2. 75 * √3
  4. Approximate the value (using a calculator for √3 ≈ 1.732): 2. 75 * 1.732 ≈ 4.763

Key Takeaways and Tips

So, guys, you've successfully calculated the trigonometric expression! Give yourselves a pat on the back! Here's what you should take away from this exercise: First, understanding the unit circle and the values of sine and cosine for common angles (0, π/6, π/4, π/3, π/2, etc.) is super important. This will speed up your calculations and give you a deeper understanding of trigonometry. Then, the ability to break down complex expressions into smaller parts is a game-changer. This approach applies to many math problems, not just trigonometry. And don't forget to practice, practice, practice! The more you practice, the more comfortable you'll become with trigonometric functions.

Also, here are some extra tips:

  • Make sure your calculator is in the correct mode (radians or degrees) depending on the angle units in the problem.
  • Familiarize yourself with the trigonometric identities, as they can simplify calculations.
  • Use diagrams (like the unit circle) to visualize the angles and their sine and cosine values.

By following these steps and tips, you'll be well on your way to mastering trigonometric calculations. Keep up the good work, and happy calculating!

I hope this guide was helpful. If you have any questions, feel free to ask in the comments. Thanks for reading and happy learning!