Problem 11 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Simplify the expressions given that \(x \in\left[0, \frac{\pi}{2}\right]\). (a) \(\tan ^{-1}(\tan (x))\) (b) \(\tan ^{-1}(\tan (-x))\)

Step-by-Step Solution

Answer

\(\tan ^{-1}(\tan (x)) = x\) and \(\tan ^{-1}(\tan (-x)) = -x\)

Step 1: Simplify \(\tan ^{-1}(\tan (x))\)

Since \(x\) is in the range of \(\tan ^{-1}(x)\), the expression \(\tan ^{-1}(\tan (x))\) simplifies directly to \(x\). This is because the inverse tangent function \(\tan ^{-1}(x)\) 'undoes' the tangent function \(\tan(x)\), and vice-versa, within their respective ranges.

Step 2: Simplify \(\tan ^{-1}(\tan (-x))\)

In this case, as \(x\) is in \([0, \frac{\pi}{2}]\), \(-x\) is not in the range of \(\tan ^{-1}(x)\). Thus, the expression \(\tan ^{-1}(\tan (-x))\) does not simplify directly to \(-x\). Instead, since the tangent function is periodic with period \(\pi\), for \(-x\) not in the range of \(\tan ^{-1}(x)\), we have \(\tan y = \tan(-x)\) implies \(y = -x + \pi k\), where \(k\) is an integer. However, because \(y\) must be in the range \(-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}\), the integer \(k\) must be equal to 0, so \(\tan ^{-1}(\tan (-x))\) simplifies to \(-x\).

Key Concepts

Tangent Function

When we talk about functions in trigonometry, the tangent function, represented as \(\tan(x)\), is one of the fundamental aspects. It establishes the ratio of the opposite side to the adjacent side in a right-angled triangle.
The tangent function is closely linked with the sine and cosine functions. Mathematically, it can be expressed in terms of these two functions as:

• \(\tan(x) = \frac{\sin(x)}{\cos(x)}\)

This relationship is crucial, as it can help us relate and convert between different trigonometric functions.
When considering angles, especially on the unit circle, the tangent function can provide the slope of the angle, offering insights into relationships within the circle and various trigonometric identities that may apply.
It should be noted that \(\tan(x)\) has asymptotes, or points where it is not defined, at \(x = \frac{\pi}{2} + n\pi\), where \(n\) is an integer.

Trigonometric Identities

Trigonometric identities are equations that are true for every value of the variable involved. They are useful in simplifying expressions, solving equations, and proving other mathematical assertions involving trigonometric functions.
For example, the identity \(\tan^{-1}(\tan(x)) = x\) holds true for any \(x\) within the range of \(\arctan\), which is \((-\frac{\pi}{2}, \frac{\pi}{2})\).
Key trigonometric identities you should know include:

• Pythagorean Identity: \(\sin^2(x) + \cos^2(x) = 1\)
• Angle Sum and Difference: \(\tan(a \pm b) = \frac{\tan(a) \pm \tan(b)}{1 \mp \tan(a)\tan(b)}\)
• Inversion Identities: \(\tan^{-1}(\tan(x)) = x\), where \(x\) fits within the inverse function's range

By applying these identities, you can solve complex expressions more efficiently and find relationships between angles and sides in geometry problems. The effective use of identities also allows simplifications of equations, often reducing them to a more convenient form for computation.

Periodicity in Trigonometry

Trigonometric functions are known for their periodicity, meaning they repeat values over regular intervals.
The tangent function is periodic with a period of \(\pi\). This implies that \(\tan(x + \pi) = \tan(x)\) for any angle \(x\). This property of repeating every \(\pi\) units is central to solving equations involving the tangent function, such as when finding multiple solutions in a given range.
When working with inverse trigonometric functions, this periodicity is key for interpreting results correctly. The expression \(\tan^{-1}(\tan(-x))\) deals directly with this concept, as -\(x\) might fall outside the primary range of \(\arctan\), but thanks to periodicity, adjustments can be made to find equivalent expressions within an appropriate range. Periodicity ensures functions like \(\tan(x)\) return to their initial values after being displaced by one complete cycle.