Problem 32 - Algebra and Trigonometry

Question

\(23-44=\) Find the exact value of the expression, if it is defined. \(\tan ^{-1}\left(\tan \left(-\frac{\pi}{4}\right)\right)\)

Step-by-Step Solution

Answer

The exact value is \(-\frac{\pi}{4}\).

Step 1: Understanding the Inverse Tangent Function

The function \( \tan^{-1}(x) \) or \( \operatorname{arctan}(x) \) is the inverse of the tangent function. It returns an angle \( \theta \) such that \( \tan(\theta) = x \). The range for \( \tan^{-1} \)(x) is typically from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), which is important to determine the appropriate angle.

Step 2: Analyze the Tangent of the Given Angle

Calculate \( \tan\left(-\frac{\pi}{4}\right) \). Since the tangent of an angle \( \theta \) is defined by \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \), and using reference angles, \( \tan\left(-\frac{\pi}{4}\right) = -1 \).

Step 3: Apply the Inverse Tangent Function

Now that we have \( \tan \left(-\frac{\pi}{4}\right) = -1 \), apply the inverse tangent function \( \tan^{-1} \) to \(-1\). We need to find \( \theta \) such that \( \tan(\theta) = -1 \) within the range from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).

Step 4: Select the Correct Value within the Range

Within the range \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), the angle \( \theta \) such that \( \tan(\theta) = -1 \) is \(-\frac{\pi}{4}\). This results directly from the property of inverse functions, where the inverse tangent of the tangent of an angle returns the original angle if it is within the principal range.

Key Concepts

Tangent Function

The tangent function, denoted as \( \tan(\theta) \), is a fundamental trigonometric function. It relates to the other trigonometric functions sine and cosine. Specifically, the tangent of an angle \( \theta \) is calculated by the ratio \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). This ratio is particularly significant in right triangles and is also extensively used in calculus.In trigonometry, the tangent function revolves around angles in a circle on the coordinate plane. It's useful to remember:

• The periodic nature of \( \tan \); it repeats every \( \pi \) (or 180 degrees).
• Its behavior around asymptotes; specifically, it is undefined at odd multiples of \( \frac{\pi}{2} \).

The tangent function's graph has a distinct wave-like structure, with vertical asymptotes and periodic peaks. It's essential in understanding the inverse function, which leads us directly to the inverse tangent.

Principal Values

Principal values are important in inverse trigonometric functions to ensure that each function only returns one single output for any input, maintaining the property of being a function.For inverse trigonometric functions like \(\tan^{-1}(x)\), which is also known as \(\operatorname{arctan}(x)\), the principal values range from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). This range is selected because within it, the tangent function is one-to-one and continuous, meaning it does not repeat any values and varies smoothly.Understanding principal values is key because they ensure that when you solve \(\tan^{-1}(\tan(\theta))\), you can correctly find \(\theta\) if it remains in the range. If \(\theta\) is outside this range, bringing it within by accounting for its periodicity ensures accurate results. That way, calculations reflect the simplest and most conventional angle.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the involved variables. They play a crucial role in solving various trigonometric problems, especially when working with inverse functions.One of the vital identities related to the tangent function is the identity \( \tan( heta) = \frac{\sin(\theta)}{\cos(\theta)} \), crucial for transforming and manipulating expressions. Using identities like these allows you to express one trigonometric function in terms of another, simplifying calculations.Another important concept is using reference angles to compute tangent values easily, especially for angles not within the first quadrant. These identities bridge the gap between complex angles and their basic trigonometric principles, ultimately helping understand how different angles relate to one another in their simplest forms.When you're solving an equation like \(\tan^{-1}(\tan(-\frac{\pi}{4}))\), recognizing that \(\tan(-\frac{\pi}{4}) = -1\) and how it relates to \(\tan(\frac{\pi}{4})\) being \(1\) helps identify the angle's correct principal value. It's through identities that these simplifications become manageable and logically sound.