Problem 27 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find all solutions of the equation in the interval \([0,2 \pi)\). $$\tan x=-1$$

Step-by-Step Solution

Answer

The solutions of the equation in the interval \([0, 2\pi)\) are \(x = \frac{5\pi}{4}\) and \(x = \frac{7\pi}{4}\).

Step 1: Write down the general solution

In general, \(\tan x = -1\) has solutions where \(x = (2k+1)\frac{\pi}{4}\) for \(k\) an integer. This comes from the fact that tangent is negative in the second and fourth quadrants (specifically at \(\frac{3\pi}{4}\) and \(\frac{7\pi}{4}\) respectively), and the tangent function has a period of \(\pi\).

Step 2: Limit the solution to the given interval

We're looking for solutions in the interval \([0, 2\pi)\). In this interval, the solutions of the equation can be obtained by varying the value of \(k\). Choosing \(k=1\) and \(k=3\), we obtain that \(x = \frac{5\pi}{4}\) and \(x = \frac{7\pi}{4}\) are the solutions within the given interval.

Step 3: Write down the final answer

The solutions of \(\tan x = -1\) in the interval \([0,2 \pi)\) are \(x=\frac{5\pi}{4}\) and \(x = \frac{7\pi}{4}\).

Key Concepts

Tangent Function

The tangent function is a fundamental part of trigonometry and is denoted as \( \tan(x) \). It relates the angle in a right triangle to the ratios of the opposite and adjacent sides. For angles measured in radians, the tangent function is periodic with a period of \( \pi \). This means the function repeats its values every \( \pi \) radians.

• Tangent is unique because it can have values of any real number, including negative numbers.
• The tangent of an angle \( x \) can be calculated using \( \tan(x) = \frac{\sin(x)}{\cos(x)} \).
• The function is undefined for angles where \( \cos(x) = 0 \) because division by zero is undefined.

For the equation \( \tan x = -1 \), the negative value suggests that the solutions will be angles where the sine and cosine have equal magnitudes but opposite signs. This condition typically happens in the second and fourth quadrants of the unit circle.

Unit Circle

The unit circle is a circle with a radius of one, centered at the origin of the coordinate plane. It's a crucial tool for understanding the relationships between the angles and trigonometric functions. Each point on the unit circle signifies an angle \( x \) and its corresponding sine and cosine values, with \( \cos(x) \) as the x-coordinate and \( \sin(x) \) as the y-coordinate.

• Key angles such as \( \frac{\pi}{4} \), \( \frac{3\pi}{4} \), \( \frac{5\pi}{4} \), and \( \frac{7\pi}{4} \) help us determine where functions reach certain values.
• The tangent equals -1 at two specific angles on the unit circle: \( \frac{5\pi}{4} \) and \( \frac{7\pi}{4} \). These angles are located in the third and fourth quadrants.

Understanding where these angles lie on the unit circle is important for solving trigonometric equations. The tangent takes a negative value when sine and cosine have opposite signs, such as at these angles.

Interval Notation

Interval notation is a way of expressing a set of numbers (typically solutions to equations) that lie between two endpoints. It is commonly used in mathematics to specify the increasingly popular solution sets without the need for extensive lists of individual points.

• In the exercise, the use of \([0, 2\pi)\) means we are considering angles starting from \(0\) and ending just before \(2\pi\).
• The brackets \([\) and \()\) indicate inclusiveness or exclusiveness of the endpoints. \([0\) means 0 is included, while \(2\pi)\) denotes that \(2\pi\) is excluded.
• Interval notation is efficient for representing continuous ranges of solutions, rather than listing discrete solutions.

When solving trigonometric equations like \( \tan x = -1 \), interval notation helps us to communicate precisely which intervals contain valid solutions, clarifying where angles such as \( \frac{5\pi}{4} \) and \( \frac{7\pi}{4} \) reside in our interval of interest.