Problem 13 - Calculus with Concepts in Calculus

Question

Solve the inequality for \(x\) in \([0,2 \pi)\). $$ \tan x \geq 1 $$

Step-by-Step Solution

Answer

The solution set is \([\frac{\pi}{4}, \frac{5\pi}{4}]\).

Step 1: Solve the Equation

To solve the inequality \( \tan x \geq 1 \), we first solve for the equation \( \tan x = 1 \). The general solution for \( \tan x = 1 \) is \( x = \frac{\pi}{4} + k\pi \), where \( k \) is an integer.

Step 2: Find the Solutions in Problem's Domain

The problem states that \( x \) should be in the interval \([0, 2\pi)\). Calculate the specific values of \( x = \frac{\pi}{4} + k\pi \) that fall within this interval: \- For \( k = 0 \), \( x = \frac{\pi}{4} \).- For \( k = 1 \), \( x = \frac{5\pi}{4} \).(No further values of \( k \) in increasing order will result in \( x < 2\pi \).)

Step 3: Analyze the Inequality in Each Interval

Evaluate the inequality \( \tan x \geq 1 \). The relevant intervals based on the tangent function's period of \( \pi \) and zeros given as solutions are \((\frac{\pi}{4}, \frac{5\pi}{4}) \) and \((\frac{5\pi}{4}, 2\pi) \). In \((\frac{\pi}{4}, \frac{5\pi}{4}) \), \( \tan x \) is greater than or equal to 1, including endpoints. The interval \((\frac{5\pi}{4}, 2\pi) \) is outside the requirement except for where we have \( \tan x \) increasing until another zero at \( 2\pi\). Hence the set of solutions is \( [\frac{\pi}{4}, \frac{5\pi}{4}] \).

Key Concepts

Tangent Function

The tangent function, denoted as \( \tan x \), is one of the primary trigonometric functions. Understanding its behavior is critical when solving trigonometric equations and inequalities. The function is defined by the ratio of the sine and cosine functions: \( \tan x = \frac{\sin x}{\cos x} \). This function is periodic, meaning it repeats its values in a regular cycle. Specifically, \( \tan x \) has a period of \( \pi \), which means after every \( \pi \) units, the function's values start repeating.
Within each period, \( \tan x \) can take all real values, which makes it unique among trigonometric functions. It has vertical asymptotes whenever \( \cos x = 0 \), occurring at odd multiples of \( \frac{\pi}{2} \). At these points, the tangent function is undefined.

• The zeros of \( \tan x \) occur at multiples of \( \pi \), such as \( x = 0, \pi, 2\pi, \ldots \).
• The tangent function is positive in the first quadrant (0 to \( \frac{\pi}{2} \)) and in the third quadrant (\( \pi \) to \( \frac{3\pi}{2} \)).

For solving inequalities like \( \tan x \geq 1 \), knowing where the tangent is positive and increasing helps to determine which intervals to consider.

Trigonometric Equations

Trigonometric equations involve functions like sine, cosine, and tangent. Solving \( \tan x = 1 \) is a typical example where understanding the periodic nature of these functions is crucial. Solving the equation involves:

• Recognizing that \( \tan x = 1 \) occurs at specific angles. For tangent, this happens at angles such as \( x = \frac{\pi}{4} \), since \( \tan \frac{\pi}{4} = \frac{\sin \frac{\pi}{4}}{\cos \frac{\pi}{4}} = 1 \).
• Using the periodicity, the general solution can be expressed as \( x = \frac{\pi}{4} + k\pi \), where \( k \) is an integer.

To solve within a specified domain, such as \([0, 2\pi)\), we find all solutions from the general form that fit within this interval. These specific solutions from the general solution equation must be checked for their positions in the needed range.

Interval Notation

Interval notation is a mathematical language used to describe a range of numbers. It is especially useful in expressing solutions to inequalities. There are several forms, each indicating whether endpoints are included or excluded:

• "[ ]" indicates that the endpoints are included in the interval.
• "( )" implies that the endpoints are not included.

For the inequality \( \tan x \geq 1 \) in \([0, 2\pi)\), two intervals are especially important due to the periodicity of the tangent function: \([\frac{\pi}{4}, \frac{5\pi}{4}]\). Here:

• \([\frac{\pi}{4}, \frac{5\pi}{4}]\) includes both endpoints, meaning \( \tan x \geq 1 \) at \( x = \frac{\pi}{4} \) and \( x = \frac{5\pi}{4} \), as well as all points between.

Understanding this notation helps clarify which values are part of the solution set for an inequality, ensuring accurate representation of the range of solutions.