Problem 9 - Precalculus with Calculus Previews

Question

Find all solutions of the given trigonometric equation if \(x\) represents a real number. $$ \tan x=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = n\pi\) for any integer \(n\).

Step 1: Identify the Basic Solution

The tangent function, \(\tan x = 0\), is equal to zero wherever the sine function is zero, since \(\tan x = \frac{\sin x}{\cos x}\). This occurs at the angles \(x = 0, \pi, 2\pi, 3\pi, \ldots\). Therefore, the basic solution can be expressed as \(x = n\pi\), where \(n\) is an integer.

Step 2: General Solution Formation

Since tangent has a period of \(\pi\), the equation \(\tan x = 0\) will repeat every \(\pi\) units. Therefore, the general solution is \(x = n\pi\), which incorporates all possible solutions for any integer \(n\). This describes all points where the tangent graph crosses the x-axis.

Step 3: Verify the Solution Set

Check that inserting \(x = n\pi\) into the original equation satisfies \(\tan x = 0\). For any integer \(n\), \(\tan(n\pi) = 0\), confirming that the solutions are correct and complete.

Key Concepts

Tangent Function

The tangent function, commonly denoted as \( \tan x \), arises frequently in trigonometry. It's a ratio of two fundamental trigonometric functions: the sine function and the cosine function. Specifically, the tangent of an angle \( x \) is defined as \( \tan x = \frac{\sin x}{\cos x} \). This means that the tangent of an angle is determined by dividing the sine of that angle by its cosine.

One interesting property of the tangent function is that it can take any real number as its value. However, it becomes undefined whenever the cosine of the angle is zero, which occurs at odd multiples of \( \frac{\pi}{2} \). That's because you'll end up dividing by zero, which is undefined in mathematics.

What's particularly noteworthy about the tangent function in the context of solving the equation \( \tan x = 0 \) is that the sine component must also be zero for the tangent to be zero. This only happens at the integer multiples of \( \pi \) (i.e., \( 0, \pi, 2\pi \), etc.), as identified in the basic solutions of the problem.

Tangent Period

The period of a function describes the interval at which the function repeats its values. Because trigonometric functions are periodic, understanding their periods is key to solving related equations.

For the tangent function, its periodicity is intriguing. It repeats every \( \pi \) units, rather than the \( 2\pi \) period seen with sine and cosine. This means every \( \pi \) interval, the tangent graph starts a new cycle. You can visualize the tangent graph, rising above and falling below the x-axis as it intersects the x-axis at integer multiples of \( \pi \).

By applying this knowledge of the tangent's period, we recognize that solving \( \tan x = 0 \) over all real numbers involves these periodically repeating intercepts at \( x = n\pi \). It's the constant return to zero at each \( \pi \) increment that gives us the confidence in the pattern of solutions.

General Solution

The general solution of a trigonometric equation encompasses all possible solutions for an angle \( x \). To achieve this, we utilize understanding of the function's periodic nature.

In this exercise, finding the general solution for the equation \( \tan x = 0 \) involves considering both its basic solution and period. The basic solution informs us that \( \tan x = 0 \) at \( x = n\pi \) for any integer \( n \). Using the known period, we conclude that these solutions occur every \( \pi \) units as presented: \( x = n\pi \).

This expression of the general solution captures all angles where tangent equals zero. It's not just a single occurrence, but rather an infinite number of alignments along the x-axis at regular intervals. Here, \( n \) can be any integer -- positive, negative, or zero. This principle can be extended to solve a plethora of similarly styled equations, leveraging the periodicity and intercept points of the tangent function.