Solve Tan(x) = √3/3: Exact Solutions [0, 2π]

Hey guys! Let's dive into solving trigonometric equations, specifically focusing on finding the exact solutions for tan(x) = √3/3 within the interval [0, 2π]. This might sound intimidating at first, but trust me, we'll break it down step by step. By the end of this guide, you'll not only be able to solve this particular problem but also have a solid understanding of the underlying concepts involved in solving similar trigonometric equations. So, buckle up and let's get started!

Understanding the Tangent Function

Before we jump into solving the equation, it's crucial to have a firm grasp on the tangent function itself. Tangent, often abbreviated as tan, is one of the fundamental trigonometric functions. In a right-angled triangle, the tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side. Mathematically, we express it as tan(θ) = opposite / adjacent. But there's more to it than just right triangles! The tangent function can also be defined using the unit circle, which is super helpful for understanding its behavior over all angles.

Think of a circle with a radius of 1 centered at the origin of a coordinate plane. An angle θ, measured counterclockwise from the positive x-axis, will intersect the unit circle at a point. The coordinates of this point are (cos(θ), sin(θ)). Now, the tangent function is defined as tan(θ) = sin(θ) / cos(θ). This definition is incredibly powerful because it extends the concept of tangent to angles beyond the acute angles found in right triangles. It allows us to think about tangent for any angle, even those greater than 90 degrees or negative angles. Understanding this connection to the unit circle is key to visualizing and solving trigonometric equations.

The tangent function has some unique characteristics that set it apart from sine and cosine. One of the most important is its periodicity. The tangent function repeats its values every π radians (or 180 degrees), meaning that tan(x) = tan(x + π) = tan(x + 2π), and so on. This is different from sine and cosine, which have a period of 2π. This periodicity will be crucial when we're finding all the solutions within a given interval. Another key characteristic is that the tangent function has vertical asymptotes at values where cos(x) = 0, such as x = π/2 and x = 3π/2. At these points, the tangent function is undefined because we would be dividing by zero. This behavior is visualized in the graph of the tangent function, which shows repeating cycles and these vertical asymptotes.

Knowing the common values of the tangent function for special angles is also super helpful. For example, tan(0) = 0, tan(π/6) = √3/3, tan(π/4) = 1, tan(π/3) = √3, and tan(π/2) is undefined. These values are derived from the special right triangles (30-60-90 and 45-45-90) and their relationships on the unit circle. Being familiar with these values allows us to quickly identify solutions to trigonometric equations without relying solely on calculators. So, before we move on, make sure you're comfortable with the definition of the tangent function, its periodicity, its behavior on the unit circle, and the common tangent values for special angles. This foundation will make solving tan(x) = √3/3 a breeze!

Solving tan(x) = √3/3: A Step-by-Step Approach

Alright, let's get down to the nitty-gritty of solving tan(x) = √3/3. We're looking for all the angles x within the interval [0, 2π] (that's from 0 to 360 degrees) that make this equation true. Remember, the key is to combine our knowledge of the tangent function with a systematic approach to finding solutions. We'll break this down into a few manageable steps, so you can follow along easily.

Step 1: Identify the Reference Angle The first thing we need to do is find the reference angle. The reference angle is the acute angle (an angle between 0 and π/2 radians) whose tangent is √3/3. Think back to those special angles we talked about earlier. Do you remember which angle has a tangent of √3/3? If you said π/6 (or 30 degrees), you're absolutely right! This is our reference angle. You can also use the inverse tangent function on your calculator (usually labeled as arctan or tan⁻¹) to find this angle. Make sure your calculator is in radian mode if you want the answer in radians. So, the reference angle, which we'll call α, is α = π/6.

Step 2: Determine the Quadrants Now that we have the reference angle, we need to figure out in which quadrants the tangent function will have a positive value. Remember that tan(x) = sin(x) / cos(x). Tangent is positive when both sine and cosine have the same sign (both positive or both negative). Sine is positive in the first and second quadrants, and cosine is positive in the first and fourth quadrants. Therefore, tangent is positive in the first quadrant (where both sine and cosine are positive) and in the third quadrant (where both sine and cosine are negative). So, we're looking for solutions in the first and third quadrants within our interval [0, 2π].

Step 3: Find the Solutions in the Interval [0, 2π] We know our reference angle is π/6, and we need to find angles in the first and third quadrants that have this reference angle. In the first quadrant, the angle is simply the reference angle itself. So, our first solution is x₁ = π/6. Easy peasy!

Now, let's find the solution in the third quadrant. To get an angle in the third quadrant with a reference angle of π/6, we add π to the reference angle. This is because the third quadrant starts at π radians (180 degrees). So, our second solution is x₂ = π + π/6. To add these, we need a common denominator, so we rewrite π as 6π/6. This gives us x₂ = 6π/6 + π/6 = 7π/6. Voila! We have our second solution.

Step 4: Verify the Solutions It's always a good idea to double-check our solutions to make sure they're correct. We can do this by plugging them back into the original equation and seeing if they satisfy it. Let's start with x₁ = π/6. We need to check if tan(π/6) = √3/3. We already know this is true from our knowledge of special angles, but it's good to confirm. Now, let's check x₂ = 7π/6. We need to see if tan(7π/6) = √3/3. Since 7π/6 is in the third quadrant, both its sine and cosine are negative. The reference angle is π/6, so tan(7π/6) will have the same magnitude as tan(π/6), which is √3/3. Since both sine and cosine are negative, the tangent will be positive, so tan(7π/6) = √3/3. Awesome! Both solutions check out.

Step 5: State the Solutions Finally, we can state our solutions. The exact solutions to the equation tan(x) = √3/3 in the interval [0, 2π] are x = π/6 and x = 7π/6. And that's it! We've successfully solved the equation. See, it wasn't so bad after all, right?

By following these steps – identifying the reference angle, determining the quadrants, finding the solutions, verifying them, and stating the answers – you can confidently tackle similar trigonometric equations. Remember, practice makes perfect, so the more you solve these types of problems, the more comfortable you'll become with the process.

Visualizing Solutions on the Unit Circle

To truly master solving trigonometric equations, it's super beneficial to visualize the solutions on the unit circle. We've already touched on how the unit circle relates to the tangent function, but let's dive deeper into how it helps us see the solutions to our equation, tan(x) = √3/3. Visualizing these solutions not only reinforces our understanding but also makes it easier to solve similar problems in the future. So, grab your mental protractor and let's explore the unit circle!

Remember, the unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. An angle x, measured counterclockwise from the positive x-axis, intersects the circle at a point with coordinates (cos(x), sin(x)). The tangent of x is then given by tan(x) = sin(x) / cos(x). This means that the tangent value is related to the ratio of the y-coordinate (sine) to the x-coordinate (cosine) of the point on the unit circle. When we're solving tan(x) = √3/3, we're essentially looking for points on the unit circle where this ratio equals √3/3.

We already found that the reference angle is π/6. On the unit circle, this corresponds to a point in the first quadrant. If you draw a line from the origin through this point, it will form an angle of π/6 with the positive x-axis. The coordinates of this point are (√3/2, 1/2). If you calculate sin(π/6) / cos(π/6), you'll indeed find it equals √3/3. So, this point visually represents our first solution, x = π/6.

But remember, the tangent function is also positive in the third quadrant. This means there's another point on the unit circle where the ratio of the y-coordinate to the x-coordinate is √3/3. To find this point, we go to the third quadrant and find the angle that has a reference angle of π/6. As we determined earlier, this angle is 7π/6. If you draw a line from the origin through the point corresponding to 7π/6, you'll see that it's directly opposite the point for π/6 across the origin. The coordinates of this point are (-√3/2, -1/2). Notice that both the x and y coordinates are negative, but their ratio is still positive √3/3, confirming that tan(7π/6) = √3/3. This point visually represents our second solution, x = 7π/6.

By visualizing these solutions on the unit circle, you can see why there are two solutions within the interval [0, 2π]. The tangent function repeats its values every π radians, so these two points are π radians apart on the circle. This visual representation also helps you understand the symmetry of the solutions and how the reference angle plays a crucial role in finding all the solutions. Furthermore, this visualization technique can be applied to solving other trigonometric equations, not just those involving tangent. It's a powerful tool for understanding the behavior of trigonometric functions and their solutions.

So, the next time you're solving a trigonometric equation, take a moment to sketch the unit circle and visualize the solutions. It'll deepen your understanding and make the process much more intuitive. Trust me, it's like having a cheat sheet right in your mind!

Common Mistakes and How to Avoid Them

Okay, guys, let's talk about some common pitfalls people stumble into when solving trigonometric equations, especially ones like tan(x) = √3/3. Knowing these common mistakes beforehand will help you avoid them and ensure you get the correct solutions every time. Think of this as our