Solving Tan(x)(cot(x) - √3/3) = 0: Find X In [0, 2π)

Hey guys! Ever get tangled up in trigonometric equations? Don't worry, we've all been there. Today, we're going to break down how to solve the equation ${\tan x\left(\cot x-\frac{\sqrt{3}}{3}\right)=0}$ within the interval ${[0,2 \pi)}$. This might seem daunting at first, but trust me, with a step-by-step approach, it’s totally manageable. Let's dive in and get those x values sorted!

Understanding the Equation

Before we jump into solving, let's make sure we understand what the equation is telling us. We have ${\tan x\left(\cot x-\frac{\sqrt{3}}{3}\right)=0}$. This equation is essentially saying that the product of two expressions is zero. Remember from basic algebra, if ${a * b = 0}$, then either ${a = 0}$ or ${b = 0}$ (or both!). So, we can break our equation into two separate cases:

1. ${\tan x = 0}$
2. ${\cot x - \frac{\sqrt{3}}{3} = 0}$

Now, let's tackle each of these cases individually. We'll start by exploring the first case, focusing on where the tangent function equals zero within our specified interval. This foundational step is crucial in unraveling the complete solution set for our original equation. By understanding the behavior of the tangent and cotangent functions, we can systematically identify the values of x that satisfy the given conditions. This methodical approach not only simplifies the problem but also enhances our comprehension of trigonometric principles. So, stick with me as we dissect each part, ensuring a clear and confident path to the solutions. This journey through trigonometric equations is not just about finding answers; it's about deepening our understanding of the mathematical concepts that govern these functions.

Case 1: Solving tan(x) = 0

Okay, let's tackle the first case: ${\tan x = 0}$. To solve this, we need to think about where the tangent function equals zero. Remember, ${\tan x}$ is defined as ${\frac{\sin x}{\cos x}}$. So, ${\tan x = 0}$ when ${\sin x = 0}$ (because a fraction is zero only when the numerator is zero).

Within the interval ${[0, 2\pi)}$, the sine function is zero at two points:

• ${x = 0}$
• ${x = \pi}$

So, we have two potential solutions from this case. It’s super important to visualize the unit circle here. Think about where the y-coordinate (which represents ${\sin x}$) is zero. It’s at the points where the circle intersects the x-axis, which are at 0 and ${\pi}$. This visual aid can really solidify your understanding and help you recall these values quickly. Now, let's keep these solutions in mind as we move on to the second case. We're building our solution set piece by piece, and it's crucial to consider each case thoroughly to ensure we don't miss any potential answers. The beauty of solving trigonometric equations lies in this methodical approach, where each step brings us closer to the complete picture. So, let's carry this momentum forward and explore the next part of our puzzle!

Case 2: Solving cot(x) - √3/3 = 0

Now, let's dive into the second case: ${\cot x - \frac{\sqrt{3}}{3} = 0}$. First, we need to isolate ${\cot x}$. We can do this by adding ${\frac{\sqrt{3}}{3}}$ to both sides of the equation:

${\cot x = \frac{\sqrt{3}}{3}}$

Now, remember that ${\cot x}$ is the reciprocal of ${\tan x}$, so ${\cot x = \frac{1}{\tan x}}$. We can rewrite our equation as:

${\frac{1}{\tan x} = \frac{\sqrt{3}}{3}}$

To make things easier, let's take the reciprocal of both sides. This gives us:

${\tan x = \frac{3}{\sqrt{3}}}$

We can simplify the right side by multiplying the numerator and denominator by ${\sqrt{3}}$:

${\tan x = \frac{3\sqrt{3}}{3}}$

${\tan x = \sqrt{3}}$

Alright, we've simplified the equation to ${\tan x = \sqrt{3}}$. Now, we need to figure out where the tangent function equals ${\sqrt{3}}$ within our interval ${[0, 2\pi)}$. Think back to your unit circle and special triangles! The tangent function is ${\sqrt{3}}$ at:

• ${x = \frac{\pi}{3}}$
• ${x = \frac{4\pi}{3}}$

These are the two solutions for this case. It's super helpful to visualize the unit circle and the 30-60-90 triangle to remember these values. The tangent is the ratio of the opposite side to the adjacent side, so we're looking for angles where this ratio is ${\sqrt{3}}$. Now that we've found the solutions for both cases, we're in the home stretch! Let's bring it all together and state our final answer. We've navigated through the complexities of trigonometric functions, simplified equations, and identified key values. This process not only gives us the solutions but also strengthens our understanding of the underlying principles. So, let's move forward with confidence and consolidate our findings.

Combining the Solutions

Okay, we've done the hard work! We found the solutions for each case separately. Now, we need to combine them to get the complete solution set for the original equation.

From Case 1, we have:

• ${x = 0}$
• ${x = \pi}$

From Case 2, we have:

• ${x = \frac{\pi}{3}}$
• ${x = \frac{4\pi}{3}}$

So, the complete solution set for ${\tan x\left(\cot x-\frac{\sqrt{3}}{3}\right)=0}$ on the interval ${[0,2 \pi)}$ is:

${x = \left\{0, \frac{\pi}{3}, \pi, \frac{4\pi}{3}\right\}}$

And that's it! We've successfully solved the equation. Remember, the key is to break down the problem into manageable parts, understand the definitions of trigonometric functions, and use the unit circle as a visual aid. You guys nailed it! Give yourselves a pat on the back for sticking with it. We've not only found the solutions but also reinforced our understanding of trigonometric equations. This knowledge will serve you well as you tackle more complex problems in the future. So, keep practicing, keep exploring, and remember that every equation is just a puzzle waiting to be solved. Onwards and upwards in our mathematical journey!

Final Answer

The solutions to the equation ${\tan x\left(\cot x-\frac{\sqrt{3}}{3}\right)=0}$ on the interval ${[0,2 \pi)}$ are:

${x = 0, \frac{\pi}{3}, \pi, \frac{4\pi}{3}}$