Question

\(y=|\tan x|\)

Short answer

The absolute value of \(\tan(x)\) can be broken down into two intervals. For \(-\pi/2 < x < \pi/2\), \(y = \tan(x)\), and for \(\pi/2 < x < 3\pi/2\), \(y = -\tan(x)\)

Step-by-step solution

Identify the intervals

First, identify the two intervals. The function \(tan(x)\) is positive in the interval \(-\pi/2 < x < \pi/2\) and negative in the interval \(\pi/2 < x < 3\pi/2\). The two intervals will therefore be \(-\pi/2 < x < \pi/2\) and \(\pi/2 < x < 3\pi/2\).

Apply the absolute value function

In the interval \(-\pi/2 < x < \pi/2\), since \(tan(x)\) is positive, applying the absolute value function does not change anything so for this range \(y = \tan x\). In the interval \(\pi/2 < x < 3\pi/2\), since \(tan(x)\) is negative, applying the absolute value function will produce a positive value. So for this range \(y = -\tan x\).

Conclude the solution

So, the absolute value of \(tan(x)\) can be broken down into two intervals. For \(-\pi/2 < x < \pi/2\), \(y = \tan(x)\), and for \(\pi/2 < x < 3\pi/2\), \(y = -\tan(x)\).

Key concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, especially when dealing with triangles and modeling periodic phenomena. The three primary functions are sine, cosine, and tangent, each of which is a ratio of the sides of a right-angled triangle. Specifically, the tangent function, represented by \(\tan{x}\), is the ratio of the opposite side to the adjacent side of an angle in a right triangle. It's known for its periodic behavior with a period of \(\pi\) radians (that is every 180 degrees), and for having asymptotes, points where the function grows without bounds, at intervals of \(\frac{\pi}{2} + k\pi\), where \(k\) is an integer. This behavior is essential when graphing the function or applying transformations, such as taking the absolute value.

Absolute Value

The absolute value function captures the 'distance' a number is from zero on the number line, irrespective of its direction; thus, it's always nonnegative. In notation, the absolute value of \(x\) is shown as \(|x|\). When you apply the absolute value to a trigonometric function, such as the tangent, you're essentially reflecting all the negative portions of the function above the \(x\)-axis, making the resulting function always positive or zero. Understanding the absolute value is crucial for solving various algebraic and trigonometric problems where the direction (sign) of a value doesn't affect the outcome.

Intervals in Trigonometry

In trigonometry, intervals refer to specific ranges of angle measures that dictate the behavior of trigonometric functions. The importance of intervals is especially evident when considering the signs of functions like tangent. For example, the tangent function fluctuates between positive and negative values within its period, and it’s essential to pinpoint these intervals to determine the function's behavior. Intervals help in analyzing functions and in practical applications like solving trigonometric equations where an understanding of the function's sign within specific intervals leads to correct solutions.

Graphing Trigonometric Functions

Graphing is the visual representation of functions, and it's a powerful tool to understand the behavior of trigonometric functions. When you graph the absolute value of the tangent function, you're combining the periodic, undulating shape of a regular tangent curve with the nonnegative restriction imposed by the absolute value. The graph of \(|\tan{x}|\) will show a repeating pattern of peaks but no valleys below the \(x\)-axis. This reflection creates a graphical representation in which the function demonstrates absolute values, and this visual can help students better understand the combined effects of trigonometric behavior and absolute value transformations.