Given $\cot^2 A + (2 + \sqrt{3})\csc A - (2 + \sqrt{3}) = 0$ And $0 < A < \pi$, Find The Value Of $3 \cot A + 2 \tan \frac{A}{2}$.

In this article, we will delve into the process of solving trigonometric equations and subsequently finding the values of trigonometric expressions. Specifically, we will tackle the equation $\cot^2 A + (2 + \sqrt{3})\csc A - (2 + \sqrt{3}) = 0$, where $0 < A < \pi$. Our goal is to determine the value of the expression $3 \cot A + 2 \tan \frac{A}{2}$. This problem involves manipulating trigonometric identities, solving quadratic equations, and understanding the range of trigonometric functions. Let's embark on this mathematical journey and unravel the solution step by step.

Problem Statement

Given the trigonometric equation $\cot^2 A + (2 + \sqrt{3})\csc A - (2 + \sqrt{3}) = 0$, and the condition $0 < A < \pi$, we aim to find the value of $3 \cot A + 2 \tan \frac{A}{2}$. This problem elegantly combines trigonometric identities, algebraic manipulation, and a touch of problem-solving acumen. The range restriction on $A$ is crucial as it helps us narrow down the possible solutions by considering the quadrants in which the angle lies. The expression we need to evaluate involves both cotangent and tangent functions, adding another layer of complexity to the problem. Let's proceed with a detailed, step-by-step solution to demystify this intriguing problem.

Solution

Step 1: Convert $\cot^2 A$ to $\csc^2 A$

To begin, we utilize the fundamental trigonometric identity that connects cotangent and cosecant: $\cot^2 A = \csc^2 A - 1$. Substituting this identity into the given equation, we transform the equation into a form that involves only the cosecant function. This strategic move simplifies the equation and paves the way for further algebraic manipulation. By expressing the equation in terms of a single trigonometric function, we can leverage algebraic techniques more effectively.

$\cot^2 A + (2 + \sqrt{3})\csc A - (2 + \sqrt{3}) = 0$

$(\csc^2 A - 1) + (2 + \sqrt{3})\csc A - (2 + \sqrt{3}) = 0$

Step 2: Simplify the equation

Now, let's simplify the equation by combining like terms and rearranging the terms. This step is crucial to reveal the underlying structure of the equation. By carefully organizing the terms, we can identify a potential quadratic form, which can be solved using standard algebraic techniques. The goal here is to consolidate the equation into a manageable form that we can solve for $\csc A$.

$\csc^2 A + (2 + \sqrt{3})\csc A - 1 - (2 + \sqrt{3}) = 0$

$\csc^2 A + (2 + \sqrt{3})\csc A - (3 + \sqrt{3}) = 0$

Step 3: Solve the quadratic equation for $\csc A$

We now have a quadratic equation in terms of $\csc A$. To solve it, we can use factoring or the quadratic formula. Factoring is often the preferred method when it's feasible, as it's generally quicker and more elegant. In this case, we can factor the quadratic expression into two binomials. This step is the heart of solving the equation, as it directly leads to the possible values of $\csc A$.

Let $x = \csc A$. Then the equation becomes:

$x^2 + (2 + \sqrt{3})x - (3 + \sqrt{3}) = 0$

We look for two numbers that multiply to $-(3 + \sqrt{3})$ and add up to $2 + \sqrt{3}$. These numbers are $-(1)$ and $3 + \sqrt{3}$.

Therefore, we can factor the quadratic as:

$(x - 1)(x + (3 + \sqrt{3})) = 0$

So, the solutions for $x$ are:

$x = 1$ or $x = -(3 + \sqrt{3})$

Thus, we have:

$\csc A = 1$ or $\csc A = -(3 + \sqrt{3})$

Step 4: Find possible values of $A$

Since $0 < A < \pi$, we analyze the possible values of $A$ based on the values of $\csc A$. Recall that $\csc A = \frac{1}{\sin A}$. The sine function is positive in the first and second quadrants (i.e., $0 < A < \pi$).

If $\csc A = 1$, then $\sin A = 1$, which implies $A = \frac{\pi}{2}$.

If $\csc A = -(3 + \sqrt{3})$, then $\sin A = \frac{-1}{3 + \sqrt{3}}$. However, since $0 < A < \pi$, $\sin A$ must be positive. Therefore, this case yields no solution.

Hence, the only valid solution is $A = \frac{\pi}{2}$.

Step 5: Calculate $\cot A$ and $\tan \frac{A}{2}$

Now that we have found the value of $A$, we can calculate $\cot A$ and $\tan \frac{A}{2}$. These values are essential for evaluating the final expression. The cotangent function is the reciprocal of the tangent function, and the tangent half-angle formula is a crucial tool in this step. By calculating these values, we set the stage for the final computation.

For $A = \frac{\pi}{2}$:

$\cot A = \cot \frac{\pi}{2} = 0$

$\tan \frac{A}{2} = \tan \frac{\pi}{4} = 1$

Step 6: Calculate $3 \cot A + 2 \tan \frac{A}{2}$

Finally, we substitute the values of $\cot A$ and $\tan \frac{A}{2}$ into the expression $3 \cot A + 2 \tan \frac{A}{2}$ to obtain the final answer. This step is a straightforward calculation that brings together all the previous steps. By performing this calculation, we complete the solution to the problem.

$3 \cot A + 2 \tan \frac{A}{2} = 3(0) + 2(1) = 0 + 2 = 2$

Final Answer

Therefore, the value of $3 \cot A + 2 \tan \frac{A}{2}$ is 2.

Conclusion

In this article, we successfully solved the trigonometric equation $\cot^2 A + (2 + \sqrt{3})\csc A - (2 + \sqrt{3}) = 0$ and found the value of $3 \cot A + 2 \tan \frac{A}{2}$ to be 2. The solution involved transforming the equation using trigonometric identities, solving a quadratic equation, and carefully considering the range of the angle $A$. This problem highlights the importance of mastering trigonometric identities and algebraic techniques in solving trigonometric equations. By breaking down the problem into manageable steps, we were able to navigate through the complexities and arrive at the correct solution. Trigonometric problems like these showcase the beauty and power of mathematical reasoning and problem-solving skills. Understanding trigonometric identities is essential for tackling a wide range of problems in mathematics, physics, and engineering. This exercise not only reinforces these concepts but also enhances our ability to approach and solve complex mathematical problems with confidence.