Problem 9 - Advanced Engineering Mathematics

Question

In Problems 1-14, find all values of the given quantity. $$ \tan ^{-1} 1 $$

Step-by-Step Solution

Answer

The angles are \( \theta = \frac{\pi}{4} + k\pi \) for any integer \( k \).

Step 1: Understand the Problem

The problem asks us to find all angles \( \theta \) such that \( \tan(\theta) = 1 \). This usually means finding the inverse tangent or arctangent of 1.

Step 2: Recall the Definition of Inverse Tangent

The function \( \tan^{-1}(x) \) or \( \arctan(x) \) returns an angle \( \theta \) such that the tangent of \( \theta \) is \( x \). For \( \tan^{-1}(1) \), we need to find an angle where \( \tan(\theta) = 1 \).

Step 3: Identify the Principal Value of \( \tan^{-1}(1) \)

Recall that the principal value range for \( \tan^{-1}(\theta) \) is \(-\frac{\pi}{2} < \theta < \frac{\pi}{2} \). Within this range, the angle whose tangent is 1 is \( \theta = \frac{\pi}{4} \).

Step 4: Determine All Solutions Based on the Periodicity of Tangent

The tangent function has a period of \( \pi \), meaning \( \tan(\theta) = \tan(\theta + k\pi) \) for any integer \( k \). Therefore, the complete set of solutions for \( \tan(\theta) = 1 \) is \( \theta = \frac{\pi}{4} + k\pi \) where \( k \) is any integer.

Step 5: Express the General Solution

Combining the results, we can express the solutions as \( \theta = \frac{\pi}{4} + k\pi \) for any integer \( k \).

Key Concepts

tangent function

The tangent function is a fundamental trigonometric function used widely in mathematics. It arises from the opposite and adjacent sides of a right-angled triangle. The tangent of an angle \( \theta \) is defined as the ratio of the opposite side to the adjacent side: \[\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\]For example, in a right triangle, if the opposite side to the angle \( \theta \) measures 1 unit and the adjacent side measures 1 unit, then the tangent of \( \theta \) is 1. This means \( \tan(\theta) = 1 \), which corresponds to an angle \( \theta \) of \( \frac{\pi}{4} \) in radians or 45 degrees.

• Tangent is undefined at 90 degrees and 270 degrees, where the adjacent side is 0.
• \( \tan(\theta) \) is periodic with a period of \( \pi \), repeating every \( 180 \degree \) or \( \pi \) radians.

principal value

When solving equations involving trigonometric functions, it's vital to understand the principal value. The principal value is the range of values for an inverse function that is continuously increasing or decreasing, offering a single solution per input value. For the inverse tangent function, \( \tan^{-1}(x) \), the principal value is the range of angles between \(-\frac{\pi}{2} < \theta < \frac{\pi}{2}\).Within this interval, the angle that provides a tangent of 1 is \( \frac{\pi}{4} \), often referred to as the principal value of \( \tan^{-1}(1) \). Understanding the principal value is crucial for correctly interpreting the solutions of inverse trigonometric equations. It ensures we are using the unambiguous, commonly accepted 'main' solution within a specific interval.

periodicity

The concept of periodicity is essential in trigonometry as it explains how functions repeat over intervals. For the tangent function, this concept is particularly interesting.\( \tan(\theta) \) repeats its values every \( \pi \) radians (180 degrees). This means that if \( \tan(\theta) \) is equal to some value at a certain angle, it will have the same value again at \( \theta + k\pi \), where \( k \) is an integer.By understanding periodicity:

• One can determine all the solutions to \( \tan(\theta) = 1 \) by considering the principal value \( \frac{\pi}{4} \) and adding integer multiples of \( \pi \), leading to \( \theta = \frac{\pi}{4} + k\pi \).
• This infinite set of solutions reflects the repeating nature of the tangent function.

To fully grasp tangent's behavior, it's key to think about its period and how it leads to multiple cyclic solutions in trigonometric equations.