Problem 116 - Algebra and Trigonometry Real Mathematics, Real People

Question

Prove the identity \(\arctan (-x)=-\arctan x\)

Step-by-Step Solution

Answer

The statement \(\arctan (-x) = -\arctan x\) is proven to be true. This proof relies heavily on the fact that the tangent function is an odd function, i.e. tan(-x) = -tan(x). This property allows us to equate \(\tan(\arctan (-x))\) with \(-\tan(\arctan x)\), which further implies that \(\arctan (-x)\) must equal to \(-\arctan x\).

Step 1: Write down the problem

We are trying to prove that \(\arctan (-x)=-\arctan x\). This means we need to show that the inverse tangent of -x equals the negative of the inverse tangent of x.

Step 2: Use the properties of the inverse trigonometric function

We know that, by definition, \(\arctan (-x)\) is the angle whose tangent is -x. Therefore it can be said that \(\tan(\arctan (-x))= -x\). Now, tangent is an odd function, that means \(\tan(-y) = -\tan(y)\), which implies that we can write -x as \(-\tan(\arctan x)\). So, we have \(-x = -\tan(\arctan x)\)

Step 3: Apply inverse functions

In the previous step, we have seen that the left-hand side and the right-hand side both represent the same number -x. Therefore, the arguments of tangent must also be equal, because the tangent function is one-to-one (which means it maps different inputs to different outputs). Therefore, the angle that we originally called \(\arctan(-x)\) is truly the same as the angle \(-\arctan x\). And this proves the identity \(\arctan (-x) = -\arctan x\)

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are functions that reverse the action of the regular trigonometric functions, like sine, cosine, and tangent. They allow us to find angles from known trigonometric ratios. For example, if we know that the tangent of an angle is a certain value, the inverse tangent function, or \( an^{-1}\), helps us find what that angle is.
This function is crucial in many areas of mathematics, including calculus and engineering, because it bridges the gap between angles and real numbers.
In the problem we are discussing, \( ext{arctan}(-x)\) is used, which represents the angle whose tangent gives the value \-x\. Understanding the inverse tangent function, and its notation as \( ext{arctan} \) or \- \( an^{-1}\), is essential for solving trigonometric equations and verifying identities.

Properties of Odd Functions

Odd functions have a special property: they are symmetric about the origin. This means that if you reflect the graph of the function across both axes, it overlaps itself. Mathematically, a function \( ext{f}(x)\) is odd if \( ext{f}(-x) = - ext{f}(x)\) for all x in its domain.
This characteristic is exactly what is leveraged when proving trigonometric identities involving odd functions. The tangent function, denoted as \( an(x)\), is one such odd function. This means that for any angle y, \( an(-y) = - an(y)\).
This property simplifies the proving of many trigonometric identities, such as \( ext{arctan}(-x) = - ext{arctan} x\), because it naturally relates expressions involving positive and negative inputs.

Tangent Function

The tangent function, \( an(x)\), is a fundamental trigonometric function that deals with the ratio of the opposite to the adjacent side in a right triangle.
Its most distinct feature is its periodic behavior and vertical asymptotes, repeating every \(\pi\) radians. The tangent function can take on any real value, and these characteristics make it useful in describing many periodic phenomena.
Since \( an(x)\) is an odd function, it has particular symmetry that aids in simplifying trigonometric proofs, making it key to understanding the \( ext{arctan}\) identity from the exercise. Recognizing how the properties of \( an(x)\) directly influence the inverse tangent function is essential, as it relates angles with their corresponding tangent values, aiding in the proof process.

Proofs in Mathematics

Mathematical proofs are logical arguments that derive conclusions from given assumptions or axioms. They are crucial tools for validating mathematical statements and ensuring they hold true under specified conditions. Proofs can come in various forms, such as direct proofs, indirect proofs, or proofs by contradiction.
In the exercise, the goal was to prove the trigonometric identity \( ext{arctan}(-x) = - ext{arctan} x\). This was approached through a combination of direct proof and properties of trigonometric functions.
The process began with understanding what needs to be shown, then using known properties about odd functions and the behavior of the tangent and inverse tangent functions, and finally concluding through logical reasoning.
Knowing how to construct and understand proofs is vital in mathematics, as it develops critical thinking and the ability to verify complex relationships or identities.