Calculate F(x) = Sin²(x) + 2cos(x) For X = Π/4 Numerical Value

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Hey guys! Let's dive into a fun math problem today. We're going to calculate the numerical value of the function f(x) = sin²(x) + 2cos(x) when x = π/4. This might sound a bit intimidating at first, but trust me, we'll break it down step by step and it'll be super clear. So, grab your calculators (or your mental math hats!) and let's get started!

Understanding the Function

Before we jump into plugging in the value, let's quickly understand what the function f(x) = sin²(x) + 2cos(x) actually represents. This function combines two trigonometric functions: sine (sin) and cosine (cos). Remember, these functions relate angles of a right triangle to the ratios of its sides. The sin²(x) part means we're squaring the sine of x, and 2cos(x) means we're multiplying the cosine of x by 2. When we add these two parts together, we get the value of our function f(x) for a specific angle x. Think of it like a machine: you put in an angle x, and the machine spits out a number f(x). In this case, our input angle is x = π/4. This angle, π/4, is a special angle in trigonometry, equivalent to 45 degrees. Knowing this is crucial because we have well-known values for sine and cosine at this angle. When approaching trigonometric functions, it's super helpful to visualize the unit circle. The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. Angles are measured counterclockwise from the positive x-axis. The sine of an angle corresponds to the y-coordinate of the point where the angle intersects the unit circle, and the cosine corresponds to the x-coordinate. So, for x = π/4 (45 degrees), we're looking at the point on the unit circle that's halfway between the x and y axes. The coordinates of this point are (√2/2, √2/2). This means that cos(π/4) = √2/2 and sin(π/4) = √2/2. Got it? Great! Now we're ready to substitute these values into our function and see what we get. Remember, understanding the basics of sine and cosine and their graphical representation using the unit circle is key to solving more complex trigonometric problems. So, always keep that unit circle in mind, guys! It's your best friend in trigonometry!

Step-by-Step Calculation

Okay, now that we've got a handle on the function and the special angle, let's roll up our sleeves and calculate the numerical value. Our mission is to find f(π/4), which means we need to substitute x = π/4 into our function f(x) = sin²(x) + 2cos(x). First things first, let's write down the function with the substitution: f(π/4) = sin²(π/4) + 2cos(π/4). Now, remember from our previous discussion that sin(π/4) = √2/2 and cos(π/4) = √2/2. These are crucial values to memorize, guys, as they pop up frequently in trigonometry problems. So, let's plug these values into our equation: f(π/4) = (√2/2)² + 2(√2/2). Awesome! We're getting closer. Now, we need to simplify the expression. Let's start with the first term, (√2/2)². Squaring a fraction means squaring both the numerator and the denominator. So, (√2/2)² = (√2)² / 2² = 2/4 = 1/2. Great! We've simplified the first part. Now, let's move on to the second term, 2(√2/2). Here, we're simply multiplying 2 by the fraction √2/2. We can think of 2 as 2/1, so we have (2/1) * (√2/2). The 2 in the numerator and the 2 in the denominator cancel each other out, leaving us with √2. Fantastic! We've simplified the second part as well. Now, let's put it all together. We have f(π/4) = 1/2 + √2. This is the exact numerical value of the function at x = π/4. If we need a decimal approximation, we can use a calculator to find that √2 ≈ 1.414. So, f(π/4) ≈ 1/2 + 1.414 = 0.5 + 1.414 = 1.914. And there you have it! We've successfully calculated the numerical value of the function f(x) = sin²(x) + 2cos(x) for x = π/4, which is approximately 1.914. See? Not so scary after all! By breaking it down into smaller steps and remembering those key trigonometric values, we made it through. Remember, practice makes perfect, guys! Keep working on these problems, and you'll become trigonometry masters in no time!

Graphical Interpretation

To truly grasp what we've calculated, it's super helpful to visualize it graphically. We found that f(π/4) ≈ 1.914 for the function f(x) = sin²(x) + 2cos(x). What does this actually mean on a graph? Well, imagine plotting the function f(x) = sin²(x) + 2cos(x) on a coordinate plane. The x-axis represents the input values (angles in radians), and the y-axis represents the output values of the function. The graph will look like a wave, oscillating up and down as the angle x changes. Our calculation tells us that at the specific angle x = π/4, the height of the graph, or the y-value, is approximately 1.914. So, if you were to find the point on the graph where x = π/4, the y-coordinate of that point would be close to 1.914. This is a powerful concept, guys! It connects the abstract world of equations and formulas to the visual world of graphs. By understanding the graphical representation, we can gain a deeper intuition about how functions behave. For example, we can see how the function changes as x increases or decreases, where it reaches its maximum and minimum values, and where it crosses the x-axis. In the case of our function, f(x) = sin²(x) + 2cos(x), the cosine term dominates the behavior of the function. Remember, the cosine function oscillates between -1 and 1. So, when we multiply it by 2, we get values between -2 and 2. The sine squared term, sin²(x), is always non-negative and oscillates between 0 and 1. Therefore, the function f(x) will generally follow the pattern of the cosine function, but shifted and scaled. At x = π/4, we're in a region where the cosine function is positive and decreasing, and the sine function is increasing. This interplay between the sine and cosine terms gives us the specific value of approximately 1.914. So, always remember to visualize functions graphically, guys! It's a fantastic way to build your understanding and tackle more complex problems. Think of it as adding another tool to your mathematical toolbox.

Key Takeaways and Further Exploration

Alright guys, we've reached the end of our journey to calculate the numerical value of f(x) = sin²(x) + 2cos(x) for x = π/4. Let's quickly recap the key takeaways from this exercise. First, we understood the function itself, recognizing it as a combination of sine and cosine functions. We emphasized the importance of knowing the values of sine and cosine for special angles, like π/4, which is equivalent to 45 degrees. We recalled that sin(π/4) = √2/2 and cos(π/4) = √2/2. Then, we meticulously followed the steps of the calculation, substituting the value of x, simplifying the expression, and arriving at the exact value of 1/2 + √2, which is approximately 1.914. We also highlighted the power of graphical interpretation, understanding that the value we calculated represents the y-coordinate of the function's graph at x = π/4. This gives us a visual understanding of the function's behavior. But the learning doesn't stop here, guys! There's always more to explore in the world of mathematics. If you're feeling adventurous, here are a few avenues for further exploration:

  • Explore other angles: Try calculating the value of the function for different angles, such as 0, π/2, π, and 3π/2. How does the value of the function change as the angle changes?
  • Graph the function: Use a graphing calculator or online tool to plot the graph of f(x) = sin²(x) + 2cos(x). Observe its shape, its maximum and minimum values, and its periodicity.
  • Investigate trigonometric identities: Learn about trigonometric identities, which are equations that are true for all values of the variables. These identities can be used to simplify trigonometric expressions and solve equations.
  • Explore applications of trigonometric functions: Trigonometric functions have numerous applications in physics, engineering, and other fields. Learn about how they are used to model periodic phenomena, such as waves and oscillations.

So, keep exploring, keep practicing, and keep having fun with math, guys! Remember, math is not just about formulas and calculations; it's about understanding the world around us.