Problem 44 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the graph of the function to determine whether the function is even, odd, or neither. \(f(x)=\cot 2 x\)

Step-by-Step Solution

Answer

The function \(f(x) = \cot(2x)\) is an odd function.

Step 1: Understanding Even and Odd Functions

An even function is a type of function where \(f(x) = f(-x)\). Graphically, an even function exhibits symmetry about the y-axis. An odd function, on the other hand, is a type of function where \(f(x) = -f(-x)\). Visually, odd functions exhibit symmetry about the origin. These two properties will be our path to determine if the function is even, odd, or neither.

Step 2: Analyzing the Function

The function in question is \(f(x) = \cot(2x)\). Let's attempt to apply the even and odd characteristics on this function. For it to be even, \(f(x) = f(-x)\), but \(\cot(2x) \neq \cot(-2x)\) since cotangent is an odd function. The other property to check is for oddness, \(f(x) = -f(-x)\) that means \(\cot(2x) = -\cot(-2x)\), this equation holds true because cotangent of any negative angle is equal to the negative cotangent of the angle. Hence, function is odd.

Step 3: Understanding the Graph of the function

If we analyze the graph of \(\cot(2x)\) or any cotangent function, the symmetric property about the origin confirms that \(\cot(2x)\) is an odd function.

Key Concepts

Cotangent Function

The cotangent function, denoted as \textbf{cot(x)}, is one of the six fundamental trigonometric functions. It is the reciprocal of the tangent function and is defined whenever the tangent function is not zero. Mathematically, it's expressed as \( \cot(x) = \frac{1}{\tan(x)} = \frac{\cos(x)}{\sin(x)} \).

Like tangent, the cotangent function exhibits certain periodic properties since it arises from the unit circle. It has a period of \( \pi \) because its values repeat every \( \pi \) radians. This function has undefined values at multiples of \( \pi \) where the sine function is zero, and it diverges at these points, resulting in asymptotes in the graph.

In the context of even and odd functions, the cotangent function behaves like an odd function. This means that \( \cot(-x) = -\cot(x) \), reflecting the property that cotangent is symmetric with respect to the origin. This symmetry plays a crucial role when analyzing the behavior of functions derived from cotangent under transformations.

Symmetry in Graphs

The concept of symmetry in graphs is a significant aspect of function analysis. It allows us to understand how functions behave without necessarily computing every point's value. There are two primary types of symmetry which are commonly discussed with respect to the coordinate axes: symmetry about the \textbf{y-axis} and symmetry about the \textbf{origin}.

\textbf{Symmetry about the y-axis} occurs when the left and right sides of the graph are mirror images of each other. This typically signifies an \textbf{even function}, which mathematically satisfies the condition \( f(x) = f(-x) \).

\textbf{Symmetry about the origin} involves rotating the graph 180 degrees about the origin point (0,0) to obtain the same graph. This indicates an \textbf{odd function}, holding to the condition \( f(x) = -f(-x) \).

Understanding these symmetries can drastically simplify the process of graph analysis and is particularly helpful in identifying the properties of trigonometric functions, such as the cotangent function.

Trigonometric Properties

Trigonometric functions are the foundation of trigonometry and involve ratios of sides in right-angled triangles as well as properties of circles. They have unique properties that allow us to solve various problems in geometry, physics, and engineering. The properties often exploited in trigonometry include \textbf{periodicity, symmetry, and reciprocal identities}.

\textbf{Periodicity} refers to the fact that trigonometric functions repeat their values in regular intervals, called periods. For example, the sine and cosine functions have periods of \( 2\pi \) radians, while the tangent and cotangent functions have shorter periods of \( \pi \) radians.

\textbf{Symmetry} is another key property. We talk about symmetry in terms of functions being even or odd, which relates to how their graphs appear under reflection about the y-axis or rotation about the origin, respectively.

Lastly, \textbf{reciprocal identities} are expressions that show the reciprocal relationship between some trigonometric functions, such as \( \sin(x) \) and \( \csc(x) \) or \( \tan(x) \) and \( \cot(x) \). These properties are useful not only for understanding the behavior of these functions but also when solving trigonometric equations or verifying identities.