Solving √3 Sin(x/2) + Cos(x) = 1: Solutions & Steps

Hey guys! Let's dive into solving this trigonometric equation: √3 sin(x/2) + cos(x) = 1, where we need to find the solutions for x within the interval 0° ≤ x < 360°. This type of problem often appears in mathematics, especially in trigonometry, and it's crucial to understand the steps involved to tackle it effectively. We will break down each step in detail, making it super easy to follow along. So grab your pencils, and let's get started!

Initial Equation and Strategy

Our main goal is to find the values of x that satisfy the equation √3 sin(x/2) + cos(x) = 1. To do this, we’ll use trigonometric identities to simplify the equation and express it in a form that’s easier to solve. One common strategy is to try and get the equation in terms of a single trigonometric function. In this case, we'll aim to use the double-angle formula for cosine. Remember, the cosine double-angle formula is a key player here: cos(2θ) = 1 - 2sin²(θ). This identity will help us relate cos(x) to sin(x/2), which is exactly what we need to simplify our equation. Think of it like having a secret weapon in our math arsenal!

Let’s kick things off by rewriting cos(x) using this double-angle identity. Since x is 2*(x/2), we can substitute θ with x/2 in our formula. This will give us a direct link between cos(x) and sin(x/2), making our equation much more manageable. We need to be strategic here and think about how each step brings us closer to isolating the variable x. It’s like planning a route on a map; we need to know where we are going to get there efficiently. Now, let’s apply this identity and see how it transforms our equation.

Applying the Double-Angle Formula

Let's replace cos(x) with 1 - 2sin²(x/2) in our original equation. This is a critical step, as it allows us to express the entire equation in terms of sin(x/2). By doing this, we're essentially turning a complex equation into a more manageable one. So, our equation now becomes:

√3 sin(x/2) + 1 - 2sin²(x/2) = 1

Notice how the '1' on both sides of the equation conveniently cancels out. This simplifies things further, making our next steps even clearer. It’s always satisfying when we can eliminate terms like this because it means less clutter and easier calculations. Our equation now looks like this:

√3 sin(x/2) - 2sin²(x/2) = 0

Now, we have a quadratic-like equation in terms of sin(x/2). It's starting to look a lot like something we can solve! The next step is to factor out the common term, which will help us isolate the possible values for sin(x/2). Factoring is a powerful tool in algebra, and it's going to be our best friend here. Let's move on to factoring and see what solutions we can uncover.

Factoring and Finding Solutions for sin(x/2)

Now that we have √3 sin(x/2) - 2sin²(x/2) = 0, let's factor out sin(x/2). This is a crucial step because it allows us to break down the equation into simpler parts. By factoring, we get:

sin(x/2) [√3 - 2sin(x/2)] = 0

This factored form gives us two possibilities. Either sin(x/2) = 0 or √3 - 2sin(x/2) = 0. Let's consider each of these cases separately. Remember, each case will lead us to potential solutions for x, so we need to investigate both thoroughly. Think of it like having two doors to open, each leading to different possibilities.

Case 1: sin(x/2) = 0

When sin(x/2) = 0, we need to find the angles for which the sine function is zero. We know that sine is zero at integer multiples of π radians (or 180°). So, x/2 must be equal to 0°, 180°, 360°, and so on. Mathematically, we can write this as:

x/2 = n * 180°, where n is an integer.

Multiplying both sides by 2, we get:

x = 2 * n * 180°

Now, let’s find the values of x that fall within our given range of 0° ≤ x < 360°. For n = 0, x = 0°. For n = 1, x = 360°, but this is outside our range, so we only have one solution from this case: x = 0°. It's important to stay within the given boundaries, just like staying on the correct path during a hike.

Case 2: √3 - 2sin(x/2) = 0

Now let's tackle the second case. If √3 - 2sin(x/2) = 0, we can rearrange the equation to isolate sin(x/2):

2sin(x/2) = √3

sin(x/2) = √3 / 2

We need to find the angles x/2 for which the sine function equals √3 / 2. We know that sin(60°) = √3 / 2 and sin(120°) = √3 / 2. So, x/2 could be 60° or 120°. Therefore:

x/2 = 60° or x/2 = 120°

Multiplying both sides by 2, we find the potential solutions for x:

x = 120° or x = 240°

So, from this case, we have two more solutions: x = 120° and x = 240°. Now that we have solutions from both cases, let's consolidate them and make sure we haven't missed anything. It's like gathering all the pieces of a puzzle to see the complete picture.

Combining and Verifying the Solutions

From our analysis, we found three potential solutions for x within the interval 0° ≤ x < 360°:

• x = 0° (from Case 1)
• x = 120° (from Case 2)
• x = 240° (from Case 2)

Now, it's crucial to verify these solutions by plugging them back into the original equation: √3 sin(x/2) + cos(x) = 1. This is like double-checking our work to ensure we didn't make any mistakes along the way. Verification is a critical step in solving any equation, especially trigonometric ones.

Verifying x = 0°:

√3 sin(0°/2) + cos(0°) = √3 * 0 + 1 = 1. This solution checks out!

Verifying x = 120°:

√3 sin(120°/2) + cos(120°) = √3 sin(60°) + cos(120°) = √3 * (√3 / 2) + (-1/2) = 3/2 - 1/2 = 1. This solution also checks out!

Verifying x = 240°:

√3 sin(240°/2) + cos(240°) = √3 sin(120°) + cos(240°) = √3 * (√3 / 2) + (-1/2) = 3/2 - 1/2 = 1. This solution checks out as well!

Since all three potential solutions satisfy the original equation, we can confidently say that these are indeed the correct solutions. It’s always a great feeling when everything lines up perfectly!

Final Solutions

After solving and verifying, we've found the solutions to the equation √3 sin(x/2) + cos(x) = 1 within the interval 0° ≤ x < 360°:

• x = 0°
• x = 120°
• x = 240°

These are the angles that make the equation true within the specified range. Remember, solving trigonometric equations often involves using identities, factoring, and verifying solutions. Each step is important, and practicing these techniques will help you become more confident in tackling these problems. Keep up the great work, guys! You've got this!