Solve Tg(3x-2π/3)=1

by Admin 20 views
Solving the Trigonometric Equation tg(3x - 2π/3) = 1

Hey guys, let's dive into solving this gnarly trigonometric equation: tg(3x - 2π/3) = 1. If you're scratching your head wondering how to tackle this, you've come to the right place! We're going to break it down step-by-step, making sure you understand every bit of it. So, buckle up, grab your math notebooks, and let's get this solved!

Understanding the Tangent Function

Before we jump into the equation, let's quickly refresh our memory about the tangent function, or tg. Remember, the tangent of an angle is essentially the ratio of the sine to the cosine of that angle (tg(θ) = sin(θ) / cos(θ)). It has a periodic nature, meaning it repeats its values over intervals. The principal value for which tg(θ) = 1 is π/4 (or 45 degrees). This is a super crucial piece of information we'll be using.

So, when we see tg(something) = 1, we immediately know that 'something' must be equal to π/4 plus any integer multiple of π, because the tangent function has a period of π. This is how we account for all possible solutions. The general solution for tg(θ) = 1 is therefore θ = π/4 + nπ, where 'n' is any integer (..., -2, -1, 0, 1, 2, ...).

Step-by-Step Solution

Now, let's apply this knowledge to our specific equation: tg(3x - 2π/3) = 1.

Here, our 'θ' is the entire expression inside the tangent function, which is (3x - 2π/3). So, following our general rule, we can set up the equation:

3x - 2π/3 = π/4 + nπ

Our main goal now is to isolate 'x'. Let's start by moving the constant term (-2π/3) to the right side of the equation. Remember to flip the sign when you move it across the equals sign!

3x = π/4 + 2π/3 + nπ

Next, we need to add the fractions π/4 and 2π/3. To do this, we need a common denominator, which is 12. So, we'll convert both fractions:

  • π/4 = (π * 3) / (4 * 3) = 3π/12
  • 2π/3 = (2π * 4) / (3 * 4) = 8π/12

Now, add them up:

3x = 3π/12 + 8π/12 + nπ

3x = 11π/12 + nπ

We're almost there, guys! The final step to get 'x' by itself is to divide both sides of the equation by 3. Don't forget to divide everything on the right side by 3, including the 'nπ' term.

x = (11π/12) / 3 + (nπ) / 3

x = 11π/36 + nπ/3

And there you have it! This is the general solution for the equation tg(3x - 2π/3) = 1. Remember that 'n' can be any integer, which means there are infinitely many solutions.

Finding Specific Solutions

Let's say you need to find a few specific solutions, maybe for a particular interval. You can do this by plugging in different integer values for 'n'.

  • If n = 0: x = 11π/36 + (0)π/3 x = 11π/36

  • If n = 1: x = 11π/36 + (1)π/3 To add these, we need a common denominator (36): π/3 = (π * 12) / (3 * 12) = 12π/36 x = 11π/36 + 12π/36 x = 23π/36

  • If n = -1: x = 11π/36 + (-1)π/3 x = 11π/36 - 12π/36 x = -π/36

See? It's pretty straightforward once you get the hang of it. The key is understanding the properties of the tangent function and working carefully with fractions and algebra.

Why This Matters in Algebra

Solving equations like tg(3x - 2π/3) = 1 isn't just about memorizing formulas; it's about developing critical thinking and problem-solving skills. In algebra, and especially in trigonometry, you're constantly dealing with relationships between different quantities. Understanding how functions behave, their periods, their principal values, and how to manipulate equations to isolate variables are fundamental skills that apply way beyond just solving math problems.

Think about it: in the real world, you'll encounter situations that can be modeled by trigonometric functions – from engineering and physics to economics and music. Being able to solve these equations allows you to predict, analyze, and understand these phenomena better. For instance, understanding the periodic nature of trigonometric functions can help in analyzing wave patterns, signal processing, or even the cyclical nature of certain economic indicators. The process of isolating 'x' involves basic algebraic manipulation, which is a cornerstone of mathematics. It teaches you to be systematic, logical, and patient. Every step you take must be justified and correct, building a strong foundation for more complex mathematical reasoning. So, next time you're wrestling with a trig equation, remember you're not just doing math; you're honing skills that are valuable in countless aspects of life. It's all about breaking down complex problems into smaller, manageable steps, just like we did here. We took the core identity of the tangent function, applied it to the given expression, and then used standard algebraic techniques to find the value of 'x'. This methodical approach is what makes math so powerful and applicable to so many different fields. So, keep practicing, keep questioning, and you'll get the hang of it in no time! It’s like building a complex structure; each solved problem adds another brick to your understanding, making your mathematical foundation stronger and more robust. And hey, those 45 points are definitely worth the effort, right? 😉 Keep up the great work, math whizzes!