Calculate F(x) = Sin²(x) + 2cos(x) At X = -π/4

Hey guys! Today, we're diving into a fun math problem where we need to figure out the numerical value of a trigonometric function. Specifically, we're dealing with the function f(x) = sin²(x) + 2cos(x), and we need to calculate its value when x is -π/4. Don't worry, it sounds more complicated than it actually is. We'll break it down step by step so everyone can follow along. Let's get started!

Understanding the Function f(x) = sin²(x) + 2cos(x)

Before we jump into plugging in values, let's make sure we understand what this function actually means. The function f(x) = sin²(x) + 2cos(x) combines two basic trigonometric functions: sine and cosine. The notation sin²(x) simply means (sin(x))², so we're squaring the sine of x. The function also includes 2 times the cosine of x. To solve this, we need to know the values of sin(-π/4) and cos(-π/4). Remember, these are standard values that often come up in trigonometry, so it's helpful to have them in your toolkit. The key here is to recall our unit circle and the special angles. Trigonometric functions can seem daunting, but breaking them down into their core components makes them much easier to handle. We'll use our knowledge of the unit circle and trigonometric identities to find the exact values needed. Understanding the function is the first big step, and once we've got that down, the rest is just plugging in and simplifying. So, let's take a closer look at finding the sine and cosine of -π/4.

Breaking Down sin²(x) and 2cos(x)

To really understand our function, let's take a closer look at each part. First, we have sin²(x), which as we mentioned, is just (sin(x))². This means we need to find the sine of x and then square it. Sine corresponds to the y-coordinate on the unit circle. Next, we have 2cos(x). This means we need to find the cosine of x and then multiply it by 2. Cosine corresponds to the x-coordinate on the unit circle. Now, why is understanding this so important? Well, when we substitute x = -π/4, we're essentially asking: what is the y-coordinate squared, plus two times the x-coordinate at the angle -π/4 on the unit circle? Thinking of it this way makes the problem much more visual and intuitive. We aren't just blindly plugging in numbers; we're connecting the algebra to a geometric representation. This is a powerful strategy in mathematics – visualizing the problem often leads to a clearer understanding and makes the solution process much smoother. So, with this visual in mind, let's proceed to find the sine and cosine of our specific angle.

Importance of the Unit Circle

The unit circle is your best friend when dealing with trigonometric functions! It's a circle with a radius of 1 centered at the origin of a coordinate plane. The beauty of the unit circle is that for any angle, the x-coordinate of the point where the terminal side of the angle intersects the circle is the cosine of the angle, and the y-coordinate is the sine of the angle. This makes it super easy to find sine and cosine values for common angles like 0, π/6, π/4, π/3, π/2, and their multiples. For our problem, we're interested in -π/4. This angle is in the fourth quadrant, and it's a reference angle of π/4 (45 degrees). Remembering the coordinates for π/4, which are (√2/2, √2/2), helps us find the coordinates for -π/4. Since we're in the fourth quadrant, the x-coordinate is positive, and the y-coordinate is negative. Therefore, the coordinates for -π/4 are (√2/2, -√2/2). This is crucial information because it tells us that cos(-π/4) = √2/2 and sin(-π/4) = -√2/2. Mastering the unit circle is a game-changer in trigonometry. It allows you to quickly recall values and solve problems without having to memorize endless tables. So, if you're ever stuck on a trig problem, always go back to the unit circle!

Calculating sin(-π/4) and cos(-π/4)

Okay, let's get down to the specifics. We need to find the values of sin(-π/4) and cos(-π/4). As we touched on earlier, -π/4 is an angle in the fourth quadrant. Think about it on the unit circle: if you start at the positive x-axis and rotate clockwise by π/4 radians (which is 45 degrees), you land in the fourth quadrant. Now, recall the special right triangle with angles 45-45-90. The sides are in the ratio 1:1:√2. When we put this into the unit circle context, we get the coordinates (√2/2, √2/2) for π/4 in the first quadrant. But since -π/4 is in the fourth quadrant, the y-coordinate is negative. So, we have: sin(-π/4) = -√2/2 and cos(-π/4) = √2/2. These are essential values to remember. Knowing these values will save you a lot of time and effort in trigonometry problems. If you're not quite comfortable with this, practice drawing the unit circle and labeling the coordinates for the key angles. The more you practice, the more these values will become second nature. Trust me, it's worth the effort! Now that we have these values, we can plug them into our function and calculate the final answer.

Substituting x = -π/4 into f(x)

Alright, we've done the groundwork, and now it's time for the fun part: substitution! We have our function f(x) = sin²(x) + 2cos(x), and we know that x = -π/4. We also figured out that sin(-π/4) = -√2/2 and cos(-π/4) = √2/2. Now we just need to plug these values into the function. So, we get: f(-π/4) = (-√2/2)² + 2(√2/2). See how we're just replacing the sin(-π/4) and cos(-π/4) with their respective values? This is a crucial step in solving any function problem. It's like following a recipe: once you have all the ingredients (the values), you just need to follow the instructions (the function) to get the final result. Make sure you're comfortable with this substitution process, as it's a fundamental skill in mathematics. Now that we've substituted, we can move on to simplifying the expression.

Simplifying the Expression

Okay, guys, let's simplify this expression. We have f(-π/4) = (-√2/2)² + 2(√2/2). First, let's tackle the (-√2/2)². Remember, squaring a negative number gives you a positive number. So, (-√2/2)² = (-√2/2) * (-√2/2) = (√2 * √2) / (2 * 2) = 2/4 = 1/2. Next, we have 2(√2/2). The 2 in the numerator and the 2 in the denominator cancel each other out, leaving us with just √2. So, now our expression looks like this: f(-π/4) = 1/2 + √2. This is a much simpler form, but we can leave it like this, or write it as a single fraction. If we want to write it as a single fraction, we need a common denominator, which is 2. So, we can rewrite √2 as (2√2)/2. Then, we have f(-π/4) = 1/2 + (2√2)/2 = (1 + 2√2)/2. And there you have it! We've simplified the expression to its final form. Remember, simplification is key in math. It not only makes the answer easier to understand but also reduces the chances of making mistakes. Now, let's state our final answer.

Final Answer: f(-π/4) = (1 + 2√2)/2

We made it! After breaking down the function, finding the values of sin(-π/4) and cos(-π/4), substituting them into the function, and simplifying the expression, we've arrived at our final answer. The numerical value of the function f(x) = sin²(x) + 2cos(x) when x = -π/4 is (1 + 2√2)/2. This might seem like a lot of steps, but each step is manageable when you break it down. The key takeaway here is the importance of understanding the unit circle, knowing your trigonometric values, and following a systematic approach to problem-solving. So, the next time you encounter a similar problem, remember the steps we took today, and you'll be well on your way to solving it. Great job, guys! You tackled a trig function head-on and came out victorious. Keep practicing, and you'll become a math whiz in no time!