Question

THOUGHT PROVOKING Use a graphing calculator to graph the function $$ y=\frac{1}{2}\left(\tan \frac{x}{2}+\cot \frac{x}{2}\right) . $$ Use your graph to write a trigonometric identity involving this function. Explain your reasoning.

Short answer

Basing on the observed properties of the graph such as periodicity, symmetry, zero crossings, and vertical asymptotes, it's possible to infer that the function behaves identical to the sine function but stretched by a factor of 2. The inferred trigonometric identity is \(\frac{1}{2}(\tan \frac{x}{2} + \cot \frac{x}{2}) = 2 \sin x\)

Step-by-step solution

Graphing the function

Start by graphing the given function \(y=\frac{1}{2}\left(\tan\frac{x}{2}+\cot\frac{x}{2}\right)\) on a graphing calculator. The function consists of the average of the tangent and cotangent operations applied to half of the input \(x\). Take note of the shape, symmetries, domain, and range displayed by the graph.

Observing the properties of the graph

Notice that the graph of the function shows properties of periodicity with a clear display of vertical asymptotes at an interval of \(\pi\) and zero crossings at x intervals of \(\pi\). It also appears symmetrical about the origin, sharing qualities with the basic sine function graph but with a slightly modulated shape.

Deriving the Identity

Based on the symmetries and characteristics observed from the graph, the function behaves identically to the sine function but stretched by a factor of 2. We can infer an identity from this observation. The function \(\frac{1}{2}(\tan \frac{x}{2} + \cot \frac{x}{2})\) behaves identically to \(2\sin x\) for all values of \(x\) in the domain.

Key concepts

Graphing Functions

Graphing trigonometric functions allows us to visually interpret the behavior and characteristics of functions. When we graph the function \(y=\frac{1}{2}\left(\tan \frac{x}{2}+\cot \frac{x}{2}\right)\), we are essentially plotting the average of two related trigonometric functions: tangent and cotangent. This combination can exhibit interesting properties due to their differences.
The graph helps us see where the function increases, decreases, and where it has vertical asymptotes. Asymptotes are lines that the graph approaches but never crosses, often found in functions involving tangent and cotangent.

• The domain includes all real numbers except where the tangent and cotangent are undefined (asymptotes).
• The range of this function can be comprehensive because the outputs of tangent and cotangent can be very extensive.

Graphing is not just about plotting points; it's also about visualizing the function's behavior including intercepts, asymptotes, and general shape.

Periodicity in Trigonometry

Periodicity is a core concept in trigonometry. It means that the function repeats its values at regular intervals. In the function \(y=\frac{1}{2}\left(\tan \frac{x}{2}+\cot \frac{x}{2}\right)\), the periodicity is evident from the graph. Every \(\pi\) units, the pattern starts again, which is a typical feature of trigonometric functions.
The periodicity of tangent and cotangent, which are originally \(\pi\)-periodic, combines interestingly here because each affects the half-angle \(\frac{x}{2}\), effectively doubling the period to \(2\pi\). However, due to the symmetrical and averaging property of tangent and cotangent in the given function, it maintains the interval pattern of \(\pi\).

• The periodic repetition can help in understanding real-world phenomena that repeat in nature, such as waves or cyclical processes.
• Recognizing periodicity in functions simplifies the process of prediction and analysis.

This concept shows the beauty of trigonometry in simplifying complex repeating systems.

Symmetry in Trigonometric Functions

Symmetry in trigonometric functions helps in simplifying their analysis and understanding their behavior. The function \(y=\frac{1}{2}\left(\tan \frac{x}{2}+\cot \frac{x}{2}\right)\) shows symmetry about the origin, which means that if you rotate the graph 180 degrees around the origin, it looks the same. This is known as origin symmetry or odd symmetry.
Such symmetry is a common feature among certain trigonometric functions, helping identify identities and simplifying calculations.

• Origin symmetry implies that the function satisfies \(f(-x) = -f(x)\).
• Symmetrical functions can often hint at specific identities or transformations that simplify complex function behaviors.

Understanding symmetry takes full advantage of the properties of trigonometric functions, allowing for more intuitive graph interpretations and analysis.