Problem 25 - Precalculus Mathematics for Calculus

Question

Find the exact value of the expression, if it is defined. $$\tan \left(\tan ^{-1} 5\right)$$

Step-by-Step Solution

Answer

The exact value is 5.

Step 1: Understand the Inverse Function

The expression includes \(\tan^{-1} 5\), which represents the angle \(\theta\) whose tangent is 5. This means that \(\theta = \tan^{-1} 5\).

Step 2: Apply the Inverse Function Property

Inverse trigonometric functions have a property where if \( y = \tan^{-1}(x) \), then \( \tan(y) = x \). In this case, \( y = \tan^{-1}(5) \) implies \( \tan(\tan^{-1}(5)) = 5 \).

Step 3: Verify the Defined Domain

Check if the expression \( \tan^{-1}(5) \) is within the valid range of \(-\frac{\pi}{2} < \theta < \frac{\pi}{2} \), which it is. So the expression is defined.

Step 4: Determine the Exact Value

Since \( \tan(\tan^{-1}(5)) = 5 \) and the inverse function is within its valid range, the exact value of \( \tan(\tan^{-1}(5)) \) is indeed 5.

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental relationships between trigonometric functions that are true for every value of the occurring variables. They form the basis for solving many mathematical problems involving angles and lengths in geometry. One of the key identities is the inverse trigonometric function identity. This identity is grounded in the idea that an inverse function undoes the action of the original function:

\[y = an^{-1}(x) \\Rightarrow an(y) = x\]

In the case of the original exercise, we make use of this identity. By understanding and applying these identities, we can simplify complex expressions and solve equations involving trigonometric functions efficiently. This interplay between function and its inverse is essential in navigating many trigonometric problems seamlessly.

Tangent Function

The tangent function, commonly written as \( an(x)\), is a fundamental trigonometric function. It is defined as the ratio of the sine and cosine functions:

\[\tan(x) = \frac{\sin(x)}{\cos(x)}\]

This relation not only helps in computations but also reveals that wherever cosine attains the value of zero, the tangent function becomes undefined due to division by zero.

Thus, the tangent function has vertical asymptotes at \(x = \frac{\pi}{2} + k\pi\), where \(k\) is an integer. The graph of the tangent function repeats every \(\pi\), illustrating its periodic nature.

Moreover, understanding the tangent function's behavior helps in solving equations, especially when dealing with its inverse—\(\tan^{-1}(x)\). In the exercise context, calculating the tangent of its inverse simply provides the original input value, harnessing the property of the inverse function.

Domain and Range of Functions

When discussing functions, it's crucial to understand their domain and range. The domain represents all the possible input values (\(x\)-values), while the range represents all possible output values (\(y\)-values).

For the tangent function \(\tan(x)\), the domain excludes \(\frac{\pi}{2} + k\pi\), where \(k\) is any integer, due to vertical asymptotes caused by division by zero.

The range of the tangent function is all real numbers, as it can take on any real value spanning from \(-\infty\) to \(\infty\). Conversely, for its inverse \(\tan^{-1}(x)\), the domain includes all real numbers, while the range is restricted to \(-\frac{\pi}{2}, \frac{\pi}{2}\).

This reverse relationship allows us to confidently say that in the exercise, \(\tan(\tan^{-1}(5))\) not only remains within valid boundaries but also directly returns the input value, exemplifying the straightforwardness of this concept when approaching problems involving inverse trigonometric functions.