Problem 54 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the properties of inverse functions to find the exact value of the expression, if possible. tan(arctan 25)

Step-by-Step Solution

Answer

The exact value of the expression \(tan(arctan 25)\) is 25.

Step 1: Understand the nature of inverse functions

First, recall the key property of inverse functions. If function f and function g are inverses, then \(f(g(x)) = x\). This is true for any value of x where both \(f(x)\) and \(g(x)\) exist.

Step 2: Apply the property of inverse functions

In this problem, the tangent function is the function \(f\) and the arctan function is its inverse \(g\). The value 25 is the \(x\) in the relationship \(f(g(x)) = x\). So we know that \(tan(arctan 25) = 25\) because the tangent and arctan are inverse functions.

Key Concepts

Tangent Function

The tangent function, often noted as \( \tan \), is one of the basic trigonometric functions that relates to a right-angled triangle. It is defined as the ratio of the opposite side to the adjacent side. This means that if you have a right triangle, and you are interested in the angle \( \theta \), the tangent of \( \theta \) is calculated as:

• opposite side/adjacent side

However, the tangent function is also significant beyond right triangles. It can map real numbers to all real numbers, covering all angles from \(-\infty\) to \(\infty\) over its range.
This function is periodic with a period of \(\pi\) and has vertical asymptotes where the function goes to infinity. This occurs every \(\pi\) unit, starting from \((2n+1)\pi/2\) for all integers \(n\).
The tangent function plays a vital role in many fields including engineering, physics, and navigation, as it can model waves, oscillations, and periodic phenomena.

Arctan Function

Arctan, or the inverse tangent function, is represented as \( \text{arctan} \) or sometimes \( \tan^{-1} \). It is the inverse of the tangent function, which essentially means it reverts or "undoes" the tangent operation.
If you have an output from the \( \tan \) function, the \( \text{arctan} \) lets you compute the original input angle. An example is finding an angle whose tangent is 1; using arctan, you could compute it as \( \text{arctan}(1) = \frac{\pi}{4} \), since the tangent of \( \frac{\pi}{4} \) is 1.
The result of arctan is known as the angle or phase. This angle is usually limited to the range \((-\frac{\pi}{2}, \frac{\pi}{2})\), which means it provides all possible outputs within this interval, sometimes referred as its principal value.

Inverse Trigonometric Functions

Inverse trigonometric functions, including \(\text{arctan}\), are crucial in mathematics. They allow functions such as sine, cosine, and tangent to be reversible. This is important when solving equations that involve trigonometric functions. By using their inverses, you can find angles corresponding to given trigonometric values.
The primary inverse trigonometric functions include:

• \( \text{arcsin} \) or \( \sin^{-1} \)
• \( \text{arccos} \) or \( \cos^{-1} \)
• \( \text{arctan} \) or \( \tan^{-1} \)

Each has a conventional range to ensure they are functions and not multivalued. This ensures every input from their respective range has a unique output.
For the inverse trigonometric functions to work effectively, it's crucial to understand their limited range, which corresponds to the principal values. Knowing these ranges helps prevent potential confusion, especially when working with complex mathematical problems or when values exceed the standard intervals.