Finding The X-intercept Of Y = Tan(x - 5π/6) A Comprehensive Guide

In the realm of mathematics, understanding the behavior of functions is crucial. Among the various aspects of a function's behavior, $x$-intercepts hold a special significance. An $x$-intercept is a point where the graph of a function intersects the $x$-axis. At this point, the $y$-coordinate is always zero. In simpler terms, it's the value of $x$ for which the function's output, $y$, becomes zero. Finding $x$-intercepts is not just a mathematical exercise; it has practical applications in various fields, including physics, engineering, and economics. For instance, in physics, the $x$-intercept of a projectile's trajectory can represent the point where it hits the ground. In economics, it might represent the break-even point where revenue equals cost.

To find the $x$-intercepts of a function, we set $y = 0$ and solve for $x$. This might involve algebraic manipulation, trigonometric identities, or other techniques depending on the complexity of the function. The solutions we obtain are the $x$-coordinates of the $x$-intercepts. The $x$-intercepts are usually expressed as ordered pairs $(x, 0)$. The process of finding $x$-intercepts can sometimes be challenging, especially for complex functions. However, understanding the fundamental principles and utilizing appropriate techniques can make the task manageable. This article will delve into the specific case of finding the $x$-intercepts of a tangent function, illustrating the steps and concepts involved in solving such problems. By mastering this process, you'll gain a deeper understanding of trigonometric functions and their graphical representations.

The tangent function, denoted as $\tan(x)$, is a fundamental trigonometric function that plays a vital role in various mathematical and scientific applications. It's defined as the ratio of the sine function to the cosine function: $\tan(x) = \frac{\sin(x)}{\cos(x)}$. The tangent function has several key characteristics that distinguish it from other trigonometric functions like sine and cosine. One notable feature is its periodicity. The tangent function repeats its values every $\pi$ radians, unlike sine and cosine, which repeat every $2\pi$ radians. This means that $\tan(x) = \tan(x + n\pi)$ for any integer $n$. Another important characteristic is the presence of vertical asymptotes. The tangent function is undefined at points where $\cos(x) = 0$, which occurs at $x = \frac{(2n + 1)\pi}{2}$, where $n$ is an integer. At these points, the function approaches infinity, resulting in vertical asymptotes on the graph.

The given function, $y = \tan(x - \frac{5\pi}{6})$, is a transformation of the basic tangent function. The term $(x - \frac{5\pi}{6})$ inside the tangent function represents a horizontal phase shift. Specifically, the graph of $y = \tan(x)$ is shifted to the right by $\frac{5\pi}{6}$ units. This shift affects the position of the vertical asymptotes and the $x$-intercepts of the function. Understanding the effect of this phase shift is crucial for accurately determining the $x$-intercepts of the given function. To find the $x$-intercepts, we need to determine the values of $x$ for which $y = 0$. This involves solving the equation $\tan(x - \frac{5\pi}{6}) = 0$. By carefully considering the properties of the tangent function and the phase shift, we can systematically find the solutions and identify the $x$-intercepts of the function. This process demonstrates the interplay between algebraic manipulation and trigonometric concepts in solving mathematical problems.

The primary goal is to find the $x$-intercepts of the function $y = \tan(x - \frac{5\pi}{6})$. As we discussed earlier, $x$-intercepts occur where the function's value, $y$, is equal to zero. Therefore, we need to solve the equation $\tan(x - \frac{5\pi}{6}) = 0$. The tangent function equals zero at integer multiples of $\pi$. That is, $\tan(\theta) = 0$ when $\theta = n\pi$, where $n$ is an integer. Applying this knowledge to our equation, we can set the argument of the tangent function equal to $n\pi$: $x - \frac5\pi}{6} = n\pi$ Next, we need to isolate $x$ to find the values that satisfy this equation. We can do this by adding $\frac{5\pi}{6}$ to both sides of the equation $x = n\pi + \frac{5\pi6}$ This equation gives us a general solution for $x$ in terms of $n$. To find specific $x$-intercepts, we can substitute different integer values for $n$. For example, when $n = 0$, we have $x = 0\pi + \frac{5\pi6} = \frac{5\pi}{6}$ This gives us one $x$-intercept at $x = \frac{5\pi}{6}$. We can also try other values of $n$ to find additional $x$-intercepts. When $n = -1$, we have $x = -1\pi + \frac{5\pi{6} = -\frac{\pi}{6}$ This gives us another $x$-intercept at $x = -\frac{\pi}{6}$. By substituting different integer values for $n$, we can generate a series of $x$-intercepts. Each of these values represents a point where the graph of the function intersects the $x$-axis. The step-by-step approach ensures that we systematically find all possible $x$-intercepts within a given domain or interval. This method highlights the importance of understanding the properties of trigonometric functions and applying algebraic techniques to solve equations.

Now that we have a general solution for the $x$-intercepts, $x = n\pi + \frac{5\pi}{6}$, we can use this to identify the correct answer from the given options. The options are presented as ordered pairs $(x, 0)$, where $x$ is the $x$-coordinate of the $x$-intercept. We need to find the option that matches one of the values generated by our general solution. Let's examine the options:

• A. $\left(-\frac{2\pi}{3}, 0\right)$: We need to check if there is an integer $n$ such that $-\frac{2\pi}{3} = n\pi + \frac{5\pi}{6}$. Subtracting $\frac{5\pi}{6}$ from both sides, we get $-\frac{2\pi}{3} - \frac{5\pi}{6} = n\pi$, which simplifies to $-\frac{9\pi}{6} = n\pi$ or $-\frac{3}{2} = n$. Since $-\frac{3}{2}$ is not an integer, this option is not an $x$-intercept.
• B. $\left(-\frac{\pi}{3}, 0\right)$: Similarly, we check if $-\frac{\pi}{3} = n\pi + \frac{5\pi}{6}$. Subtracting $\frac{5\pi}{6}$ from both sides, we get $-\frac{\pi}{3} - \frac{5\pi}{6} = n\pi$, which simplifies to $-\frac{7\pi}{6} = n\pi$ or $-\frac{7}{6} = n$. Since $-\frac{7}{6}$ is not an integer, this option is also not an $x$-intercept.
• C. $\left(\frac{\pi}{6}, 0\right)$: We check if $\frac{\pi}{6} = n\pi + \frac{5\pi}{6}$. Subtracting $\frac{5\pi}{6}$ from both sides, we get $\frac{\pi}{6} - \frac{5\pi}{6} = n\pi$, which simplifies to $-\frac{4\pi}{6} = n\pi$ or $-\frac{2}{3} = n$. Since $-\frac{2}{3}$ is not an integer, this option is not an $x$-intercept.
• D. $\left(\frac{5\pi}{6}, 0\right)$: We check if $\frac{5\pi}{6} = n\pi + \frac{5\pi}{6}$. Subtracting $\frac{5\pi}{6}$ from both sides, we get $0 = n\pi$, which implies $n = 0$. Since 0 is an integer, this option is an $x$-intercept.

Therefore, the correct answer is D. $\left(\frac{5\pi}{6}, 0\right)$. This systematic approach demonstrates how to use the general solution to verify the given options and identify the correct $x$-intercept. It reinforces the importance of understanding the properties of trigonometric functions and applying algebraic techniques to solve problems.

In this comprehensive guide, we've explored the process of finding the $x$-intercepts of a tangent function, specifically $y = \tan(x - \frac{5\pi}{6})$. We began by understanding the significance of $x$-intercepts as points where the graph of a function intersects the $x$-axis, representing the values of $x$ for which the function's output is zero. This concept has broad applications across various fields, highlighting the practical importance of mastering this skill.

We then delved into the characteristics of the tangent function, including its periodicity and the presence of vertical asymptotes. Understanding the transformation applied to the basic tangent function, a horizontal phase shift of $\frac{5\pi}{6}$, was crucial in accurately determining the $x$-intercepts. By setting the function equal to zero and solving for $x$, we derived a general solution, $x = n\pi + \frac{5\pi}{6}$, where $n$ is an integer. This general solution allowed us to generate a series of $x$-intercepts by substituting different integer values for $n$.

Finally, we applied our general solution to identify the correct $x$-intercept from the given options. By systematically checking each option, we verified that $\left(\frac{5\pi}{6}, 0\right)$ is indeed an $x$-intercept of the function. This step-by-step approach reinforces the importance of combining algebraic manipulation with a solid understanding of trigonometric properties.

Mastering the process of finding $x$-intercepts, especially for trigonometric functions, is a valuable skill in mathematics and its applications. It requires a combination of conceptual understanding, algebraic proficiency, and careful attention to detail. By following the steps outlined in this guide, you can confidently tackle similar problems and deepen your understanding of trigonometric functions and their graphical representations. The ability to find $x$-intercepts not only enhances your problem-solving skills but also provides a foundation for more advanced mathematical concepts and applications.