Problem 21 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find all solutions of the equation in the interval \([0,2 \pi)\). $$\tan x=-\frac{\sqrt{3}}{3}$$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{5\pi}{6}, \frac{11\pi}{6}\)

Step 1: Identify the reference angle

First, note that \( \tan x = -\frac{\sqrt{3}}{3} \) is negative. So, the angle is in either the second or fourth quadrant. Next, look for the reference angle for \( \frac{\sqrt{3}}{3} \). An angle, \(\alpha\) in the first quadrant, with \( \tan \alpha = \frac{\sqrt{3}}{3} \) is \(\frac{\pi}{6}\).

Step 2: Find the angles in the second and fourth quadrant

Using the reference angle, we find the angle in the second quadrant by subtracting the reference angle from \(\pi\), and in the fourth quadrant by subtracting it from \(2\pi\). This gives us \( x = \pi - \frac{\pi}{6} = \frac{5\pi}{6} \) (second quadrant) and \( x = 2\pi - \frac{\pi}{6} = \frac{11\pi}{6} \) (fourth quadrant).

Step 3: Verification

To ensure these are the correct solutions, substitute these values back into the original equation and verify that the tangent of these angles equals \( -\frac{\sqrt{3}}{3} \).

Key Concepts

Tangent Function

The tangent function, commonly denoted as \( \tan \), is one of the fundamental trigonometric functions. It's defined as the ratio of the opposite side to the adjacent side of a right triangle. In terms of the unit circle, it represents the ratio of the y-coordinate to the x-coordinate of a point on the unit circle.

For any angle \( x \), \( \tan(x) \) can be found by drawing a right-angled triangle in the coordinate system where the angle \( x \) opens from the positive x-axis or by using the unit circle. One of the key properties of the tangent function is that it is periodic with a period of \( \pi \), meaning that \( \tan(x + \pi) = \tan(x) \).

In your exercise, identifying that the tangent function is negative helps in determining the correct quadrants where the solutions lie. Since the sine and cosine functions, which make up the \( \tan \) function (\( \tan(x) = \frac{\sin(x)}{\cos(x)} \)), have different signs in the second and fourth quadrants, the tangent function is negative in these quadrants.

Reference Angle

A reference angle is the acute angle formed by the terminal side of an angle \( x \) and the horizontal axis. For any given angle in any quadrant, you can find a positive acute angle that is equivalent to it in the first quadrant, and this is the reference angle.

Using the reference angle is very helpful in solving trigonometric equations because trigonometric functions of the reference angles have the same absolute value as the original angles. Their signs, however, depend on the quadrant in which the terminal side of the original angle lies.

For instance, the equation in your exercise required finding a reference angle for which the tangent function has the value \( \frac{\sqrt{3}}{3} \). This approach significantly simplifies the process of solving for the angle \( x \). The reference angle you identified, \( \frac{\pi}{6} \), is the positive acute angle that corresponds to \( \tan^{-1}\left(\frac{\sqrt{3}}{3}\right) \).

Trigonometry in Quadrants

In trigonometry, understanding the behavior of trigonometric functions in different quadrants is crucial for solving equations. The four quadrants of the Cartesian plane each have specific signs for sine, cosine, and tangent functions:

• First Quadrant: All functions (\( \sin \), \( \cos \), \( \tan \)) are positive.
• Second Quadrant: Sine is positive, but cosine and tangent are negative.
• Third Quadrant: Tangent is positive, but sine and cosine are negative.
• Fourth Quadrant: Cosine is positive, but sine and tangent are negative.

In the context of your exercise, knowing that \( \tan \) is negative immediately narrows down possible solutions to the second and fourth quadrants. By applying the reference angle method to these quadrants, you can accurately find all solutions within the given interval.

Angle Verification

After solving trigonometric equations, it's important to verify that the solutions are correct. This step ensures that any potential mistakes made during the solving process are identified and corrected. Verification typically involves substituting the found angles back into the original trigonometric equation to check if the equation holds true.

For the equation \( \tan x = -\frac{\sqrt{3}}{3} \), once you have determined the potential solutions, you should test these angles to confirm that their tangent values match the equation. Substituting the angles \( \frac{5\pi}{6} \) and \( \frac{11\pi}{6} \) into the tangent function should indeed yield \( -\frac{\sqrt{3}}{3} \), which proves their validity as solutions. This step is essential for complete confidence in the results obtained.