Solving √3(tan X + Cot X) = 4 A Step-by-Step Guide
Hey guys! Let's dive into solving this interesting trigonometric equation: √3(tan x + cot x) = 4. Trigonometric equations can sometimes look intimidating, but with a step-by-step approach and a bit of trigonometric identities, we can crack this nut. We'll break down the problem, explore the key concepts, and walk through the solution together. So, grab your pencils, and let’s get started!
Understanding the Basics
Before we jump into the equation, let’s quickly revisit some fundamental trigonometric concepts that will come in handy.
Tangent and Cotangent
You probably already know that the tangent function, denoted as tan x, is defined as the ratio of the sine function to the cosine function:
tan x = sin x / cos x
Cotangent, denoted as cot x, is the reciprocal of the tangent function. So, it's the cosine function divided by the sine function:
cot x = cos x / sin x
The Reciprocal Relationship
The key relationship here is that cot x is the reciprocal of tan x. Mathematically, this means:
cot x = 1 / tan x
This relationship is super important because it allows us to rewrite our equation in terms of a single trigonometric function, which will make it much easier to solve. Recognizing and using these basic definitions and relationships is crucial in simplifying trigonometric equations. When we see tangent and cotangent together, this reciprocal relationship is often our go-to strategy for simplification. It's like having a secret weapon in our mathematical toolkit!
Why This Matters
Knowing these definitions is essential because they allow us to manipulate and simplify equations. In our given equation, having both tan x and cot x might seem complicated, but recognizing that cot x is just the reciprocal of tan x lets us rewrite the equation in a more manageable form. This is a common strategy in trigonometry: transforming expressions to use fewer different trigonometric functions.
Setting Up the Equation
Now that we've refreshed our memory on the basics, let’s get back to our equation:
√3(tan x + cot x) = 4
Our goal here is to isolate x, but first, we need to simplify the equation. Remember that cot x = 1 / tan x? Let’s use that to rewrite the equation:
√3(tan x + 1/tan x) = 4
This is a crucial step because it transforms the equation into a form that's easier to work with. By expressing everything in terms of tan x, we reduce the number of different trigonometric functions, making the equation simpler and more approachable. It's like translating a sentence into a language you understand better!
Clearing the Fraction
Fractions can be a bit messy, so let's get rid of the fraction inside the parentheses. We can do this by multiplying both terms inside the parentheses by tan x:
√3((tan^2 x + 1) / tan x) = 4
Now, the equation looks a bit cleaner. We have a single fraction, which is easier to handle. Clearing fractions is a common algebraic technique that simplifies equations by removing denominators. This often makes the subsequent steps, such as further simplification or solving for a variable, much more straightforward. In our case, it sets us up nicely for the next step.
Preparing for Substitution
Next, we want to isolate the trigonometric terms. To do this, let’s multiply both sides of the equation by tan x and also divide by √3:
(tan^2 x + 1) = (4 / √3) * tan x
This step prepares us for a substitution that will make the equation look much more familiar. Isolating the trigonometric terms and rearranging the equation in this way is a strategic move. It allows us to see the structure of the equation more clearly and identify opportunities for simplification. We're essentially setting the stage for the next act in our mathematical drama!
Substitution for Simplicity
To make our equation look even simpler, let’s use a substitution. Let's say:
y = tan x
This substitution will transform our trigonometric equation into a more familiar algebraic equation. Substitution is a powerful technique in mathematics that involves replacing a complex expression with a single variable. This simplifies the equation, making it easier to solve. It's like giving a nickname to a long and complicated name!
Transforming the Equation
Now, replace tan x with y in our equation:
y^2 + 1 = (4 / √3) * y
Rearrange this into a standard quadratic form by multiplying both sides by √3 and bringing all terms to one side:
√3 * y^2 - 4y + √3 = 0
Now we have a quadratic equation in terms of y. This transformation is a significant step because we've converted a trigonometric equation into a quadratic equation, which we know how to solve using various methods like factoring, completing the square, or the quadratic formula. It's like switching from a complex recipe to a simpler one!
Solving the Quadratic Equation
Our quadratic equation is:
√3 * y^2 - 4y + √3 = 0
We can solve this using the quadratic formula. Remember the quadratic formula? It’s:
y = (-b ± √(b^2 - 4ac)) / (2a)
In our equation:
a = √3 b = -4 c = √3
Applying the Quadratic Formula
Let's plug these values into the formula:
y = (4 ± √((-4)^2 - 4 * √3 * √3)) / (2 * √3) y = (4 ± √(16 - 12)) / (2√3) y = (4 ± √4) / (2√3) y = (4 ± 2) / (2√3)
So, we have two possible values for y:
y1 = (4 + 2) / (2√3) = 6 / (2√3) = 3 / √3 = √3 y2 = (4 - 2) / (2√3) = 2 / (2√3) = 1 / √3
These are our solutions for y. The quadratic formula is a powerful tool for solving quadratic equations, and it's a fundamental concept in algebra. By applying the formula, we've found the values of y that satisfy our equation. It's like finding the missing pieces of a puzzle!
Back to Trigonometry
Remember that we made a substitution, y = tan x. Now we need to go back and find the values of x that correspond to our values of y.
Finding x for y = √3
First, let's consider y = √3. This means:
tan x = √3
We need to find the angles x whose tangent is √3. From the unit circle or our knowledge of trigonometric values, we know that:
x = π/3 + nπ
where n is an integer. This gives us the general solution for x when tan x = √3. We know that the tangent function has a period of π, so we add nπ to the principal value (π/3) to find all possible solutions. It's like finding all the spots on a clock where the hands make the same angle!
Finding x for y = 1/√3
Next, let's consider y = 1/√3. This means:
tan x = 1/√3
Again, we need to find the angles x whose tangent is 1/√3. From our trigonometric knowledge:
x = π/6 + nπ
where n is an integer. This gives us the general solution for x when tan x = 1/√3. Similar to the previous case, we add nπ to the principal value (π/6) to account for all possible solutions, thanks to the periodic nature of the tangent function. It's like finding all the times a specific event occurs on a repeating schedule!
Final Solutions
So, the solutions to our original trigonometric equation √3(tan x + cot x) = 4 are:
x = π/3 + nπ x = π/6 + nπ
where n is any integer. These are the general solutions that cover all possible values of x that satisfy the equation. We've successfully navigated through the trigonometric terrain, transformed the equation, solved for the variable, and found the general solutions. It's like completing a challenging quest and claiming your treasure!
Conclusion
Alright guys, we’ve successfully solved the trigonometric equation √3(tan x + cot x) = 4. We started by understanding the basics of tangent and cotangent, transformed the equation using trigonometric identities, made a substitution to simplify it, solved the resulting quadratic equation, and finally, found the general solutions for x. Trigonometric equations might seem tricky at first, but with a systematic approach and a good grasp of the fundamentals, you can tackle them like a pro. Keep practicing, and you'll become a trigonometry wizard in no time! Remember, math is like a puzzle – each piece fits together to reveal the solution. Happy problem-solving!
